ws change

[PyX/mjg.git] / pyx / path.py

#!/usr/bin/env python
# -*- coding: ISO-8859-1 -*-
#
#
# Copyright (C) 2002-2005 Jörg Lehmann <joergl@users.sourceforge.net>
# Copyright (C) 2003-2004 Michael Schindler <m-schindler@users.sourceforge.net>
# Copyright (C) 2002-2005 André Wobst <wobsta@users.sourceforge.net>
#
# This file is part of PyX (http://pyx.sourceforge.net/).
#
# PyX is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# PyX is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with PyX; if not, write to the Free Software
# Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

#       - correct bbox for curveto and normcurve
#         (maybe we still need the current bbox implementation (then maybe called
#          cbox = control box) for normcurve for the use during the
#          intersection of bpaths)

from __future__ import nested_scopes

import math
from math import cos, sin, tan, acos, pi
try:
    from math import radians, degrees
except ImportError:
    # fallback implementation for Python 2.1
    def radians(x): return x*pi/180
    def degrees(x): return x*180/pi
import bbox, canvas, trafo, unit

try:
    sum([])
except NameError:
    # fallback implementation for Python 2.2 and below
    def sum(list):
        return reduce(lambda x, y: x+y, list, 0)

try:
    enumerate([])
except NameError:
    # fallback implementation for Python 2.2 and below
    def enumerate(list):
        return zip(xrange(len(list)), list)

# use new style classes when possible
__metaclass__ = type

################################################################################

# global epsilon (default precision of normsubpaths)
_epsilon = 1e-5

def set(epsilon=None):
    global _epsilon
    if epsilon is not None:
        _epsilon = epsilon

################################################################################
# Bezier helper functions
################################################################################

def _arctobcurve(x_pt, y_pt, r_pt, phi1, phi2):
    """generate the best bezier curve corresponding to an arc segment"""

    dphi = phi2-phi1

    if dphi==0: return None

    # the two endpoints should be clear
    x0_pt, y0_pt = x_pt+r_pt*cos(phi1), y_pt+r_pt*sin(phi1)
    x3_pt, y3_pt = x_pt+r_pt*cos(phi2), y_pt+r_pt*sin(phi2)

    # optimal relative distance along tangent for second and third
    # control point
    l = r_pt*4*(1-cos(dphi/2))/(3*sin(dphi/2))

    x1_pt, y1_pt = x0_pt-l*sin(phi1), y0_pt+l*cos(phi1)
    x2_pt, y2_pt = x3_pt+l*sin(phi2), y3_pt-l*cos(phi2)

    return normcurve_pt(x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt)


def _arctobezierpath(x_pt, y_pt, r_pt, phi1, phi2, dphimax=45):
    apath = []

    phi1 = radians(phi1)
    phi2 = radians(phi2)
    dphimax = radians(dphimax)

    if phi2<phi1:
        # guarantee that phi2>phi1 ...
        phi2 = phi2 + (math.floor((phi1-phi2)/(2*pi))+1)*2*pi
    elif phi2>phi1+2*pi:
        # ... or remove unnecessary multiples of 2*pi
        phi2 = phi2 - (math.floor((phi2-phi1)/(2*pi))-1)*2*pi

    if r_pt == 0 or phi1-phi2 == 0: return []

    subdivisions = abs(int((1.0*(phi1-phi2))/dphimax))+1

    dphi = (1.0*(phi2-phi1))/subdivisions

    for i in range(subdivisions):
        apath.append(_arctobcurve(x_pt, y_pt, r_pt, phi1+i*dphi, phi1+(i+1)*dphi))

    return apath

#
# we define one exception
#

class PathException(Exception): pass

################################################################################
# _currentpoint: current point during walk along path
################################################################################

class _invalidcurrentpointclass:

    def invalid1(self):
        raise PathException("current point not defined (path must start with moveto or the like)")
    __str__ = __repr__ = __neg__ = invalid1

    def invalid2(self, other):
        self.invalid1()
    __cmp__ = __add__ = __iadd__ = __sub__ = __isub__ = __mul__ = __imul__ = __div__ = __idiv__ = invalid2

_invalidcurrentpoint = _invalidcurrentpointclass()


class _currentpoint:

    """current point during walk along path"""

    __slots__ = "x_pt", "y_pt"

    def __init__(self, x_pt=_invalidcurrentpoint, y_pt=_invalidcurrentpoint):
        """initialize current point

        By default the current point is marked invalid.
        """
        self.x_pt = x_pt
        self.y_pt = y_pt

    def invalidate(self):
        """mark current point invalid"""
        self.x_pt = _invalidcurrentpoint

    def valid(self):
        """checks whether the current point is invalid"""
        return self.x_pt is not _invalidcurrentpoint


################################################################################
# pathitem: element of a PS style path
################################################################################

class pathitem:

    """element of a PS style path"""

    def _updatecurrentpoint(self, currentpoint):
        """update current point of during walk along pathitem

        changes currentpoint in place
        """
        raise NotImplementedError()


    def _bbox(self, currentpoint):
        """return bounding box of pathitem

        currentpoint: current point along path
        """
        raise NotImplementedError()

    def _normalized(self, currentpoint):
        """return list of normalized version of pathitem

        currentpoint: current point along path

        Returns the path converted into a list of normline or normcurve
        instances. Additionally instances of moveto_pt and closepath are
        contained, which act as markers.
        """
        raise NotImplementedError()

    def outputPS(self, file, writer, context):
        """write PS code corresponding to pathitem to file, using writer and context"""
        raise NotImplementedError()

    def outputPDF(self, file, writer, context):
        """write PDF code corresponding to pathitem to file

        Since PDF is limited to lines and curves, _normalized is used to
        generate PDF outout. Thus only moveto_pt and closepath need to
        implement the outputPDF method."""
        raise NotImplementedError()


# various pathitems
#
# Each one comes in two variants:
#  - one with suffix _pt. This one requires the coordinates
#    to be already in pts (mainly used for internal purposes)
#  - another which accepts arbitrary units


class closepath(pathitem):

    """Connect subpath back to its starting point"""

    __slots__ = ()

    def __str__(self):
        return "closepath()"

    def _updatecurrentpoint(self, currentpoint):
        if not currentpoint.valid():
            raise PathException("closepath on an empty path")
        currentpoint.invalidate()

    def _bbox(self, currentpoint):
        return None

    def _normalized(self, currentpoint):
        return [self]

    def outputPS(self, file, writer, context):
        file.write("closepath\n")

    def outputPDF(self, file, writer, context):
        file.write("h\n")


class moveto_pt(pathitem):

    """Set current point to (x_pt, y_pt) (coordinates in pts)"""

    __slots__ = "x_pt", "y_pt"

    def __init__(self, x_pt, y_pt):
        self.x_pt = x_pt
        self.y_pt = y_pt

    def __str__(self):
        return "moveto_pt(%g, %g)" % (self.x_pt, self.y_pt)

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt = self.x_pt
        currentpoint.y_pt = self.y_pt

    def _bbox(self, currentpoint):
        return None

    def _normalized(self, currentpoint):
        return [moveto_pt(self.x_pt, self.y_pt)]

    def outputPS(self, file, writer, context):
        file.write("%g %g moveto\n" % (self.x_pt, self.y_pt) )

    def outputPDF(self, file, writer, context):
        file.write("%f %f m\n" % (self.x_pt, self.y_pt) )


class lineto_pt(pathitem):

    """Append straight line to (x_pt, y_pt) (coordinates in pts)"""

    __slots__ = "x_pt", "y_pt"

    def __init__(self, x_pt, y_pt):
        self.x_pt = x_pt
        self.y_pt = y_pt

    def __str__(self):
        return "lineto_pt(%g, %g)" % (self.x_pt, self.y_pt)

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt = self.x_pt
        currentpoint.y_pt = self.y_pt

    def _bbox(self, currentpoint):
        return bbox.bbox_pt(min(currentpoint.x_pt, self.x_pt),
                            min(currentpoint.y_pt, self.y_pt),
                            max(currentpoint.x_pt, self.x_pt),
                            max(currentpoint.y_pt, self.y_pt))

    def _normalized(self, currentpoint):
        return [normline_pt(currentpoint.x_pt, currentpoint.y_pt, self.x_pt, self.y_pt)]

    def outputPS(self, file, writer, context):
        file.write("%g %g lineto\n" % (self.x_pt, self.y_pt) )


class curveto_pt(pathitem):

    """Append curveto (coordinates in pts)"""

    __slots__ = "x1_pt", "y1_pt", "x2_pt", "y2_pt", "x3_pt", "y3_pt"

    def __init__(self, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt):
        self.x1_pt = x1_pt
        self.y1_pt = y1_pt
        self.x2_pt = x2_pt
        self.y2_pt = y2_pt
        self.x3_pt = x3_pt
        self.y3_pt = y3_pt

    def __str__(self):
        return "curveto_pt(%g,%g, %g, %g, %g, %g)" % (self.x1_pt, self.y1_pt,
                                                      self.x2_pt, self.y2_pt,
                                                      self.x3_pt, self.y3_pt)

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt = self.x3_pt
        currentpoint.y_pt = self.y3_pt

    def _bbox(self, currentpoint):
        return bbox.bbox_pt(min(currentpoint.x_pt, self.x1_pt, self.x2_pt, self.x3_pt),
                            min(currentpoint.y_pt, self.y1_pt, self.y2_pt, self.y3_pt),
                            max(currentpoint.x_pt, self.x1_pt, self.x2_pt, self.x3_pt),
                            max(currentpoint.y_pt, self.y1_pt, self.y2_pt, self.y3_pt))

    def _normalized(self, currentpoint):
        return [normcurve_pt(currentpoint.x_pt, currentpoint.y_pt,
                             self.x1_pt, self.y1_pt,
                             self.x2_pt, self.y2_pt,
                             self.x3_pt, self.y3_pt)]

    def outputPS(self, file, writer, context):
        file.write("%g %g %g %g %g %g curveto\n" % ( self.x1_pt, self.y1_pt,
                                                     self.x2_pt, self.y2_pt,
                                                     self.x3_pt, self.y3_pt ) )


class rmoveto_pt(pathitem):

    """Perform relative moveto (coordinates in pts)"""

    __slots__ = "dx_pt", "dy_pt"

    def __init__(self, dx_pt, dy_pt):
         self.dx_pt = dx_pt
         self.dy_pt = dy_pt

    def __str__(self):
        return "rmoveto_pt(%g, %g)" % (self.dx_pt, self.dy_pt)

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt += self.dx_pt
        currentpoint.y_pt += self.dy_pt

    def _bbox(self, currentpoint):
        return None

    def _normalized(self, currentpoint):
        return [moveto_pt(currentpoint.x_pt + self.dx_pt, currentpoint.y_pt + self.dy_pt)]

    def outputPS(self, file, writer, context):
        file.write("%g %g rmoveto\n" % (self.dx_pt, self.dy_pt) )


class rlineto_pt(pathitem):

    """Perform relative lineto (coordinates in pts)"""

    __slots__ = "dx_pt", "dy_pt"

    def __init__(self, dx_pt, dy_pt):
        self.dx_pt = dx_pt
        self.dy_pt = dy_pt

    def __str__(self):
        return "rlineto_pt(%g %g)" % (self.dx_pt, self.dy_pt)

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt += self.dx_pt
        currentpoint.y_pt += self.dy_pt

    def _bbox(self, currentpoint):
        x_pt = currentpoint.x_pt + self.dx_pt
        y_pt = currentpoint.y_pt + self.dy_pt
        return bbox.bbox_pt(min(currentpoint.x_pt, x_pt),
                            min(currentpoint.y_pt, y_pt),
                            max(currentpoint.x_pt, x_pt),
                            max(currentpoint.y_pt, y_pt))

    def _normalized(self, currentpoint):
        return [normline_pt(currentpoint.x_pt, currentpoint.y_pt,
                            currentpoint.x_pt + self.dx_pt, currentpoint.y_pt + self.dy_pt)]

    def outputPS(self, file, writer, context):
        file.write("%g %g rlineto\n" % (self.dx_pt, self.dy_pt) )


class rcurveto_pt(pathitem):

    """Append rcurveto (coordinates in pts)"""

    __slots__ = "dx1_pt", "dy1_pt", "dx2_pt", "dy2_pt", "dx3_pt", "dy3_pt"

    def __init__(self, dx1_pt, dy1_pt, dx2_pt, dy2_pt, dx3_pt, dy3_pt):
        self.dx1_pt = dx1_pt
        self.dy1_pt = dy1_pt
        self.dx2_pt = dx2_pt
        self.dy2_pt = dy2_pt
        self.dx3_pt = dx3_pt
        self.dy3_pt = dy3_pt

    def __str__(self):
        return "rcurveto_pt(%g, %g, %g, %g, %g, %g)" % (self.dx1_pt, self.dy1_pt,
                                                        self.dx2_pt, self.dy2_pt,
                                                        self.dx3_pt, self.dy3_pt)

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt += self.dx3_pt
        currentpoint.y_pt += self.dy3_pt

    def _bbox(self, currentpoint):
        x1_pt = currentpoint.x_pt + self.dx1_pt
        y1_pt = currentpoint.y_pt + self.dy1_pt
        x2_pt = currentpoint.x_pt + self.dx2_pt
        y2_pt = currentpoint.y_pt + self.dy2_pt
        x3_pt = currentpoint.x_pt + self.dx3_pt
        y3_pt = currentpoint.y_pt + self.dy3_pt
        return bbox.bbox_pt(min(currentpoint.x_pt, x1_pt, x2_pt, x3_pt),
                            min(currentpoint.y_pt, y1_pt, y2_pt, y3_pt),
                            max(currentpoint.x_pt, x1_pt, x2_pt, x3_pt),
                            max(currentpoint.y_pt, y1_pt, y2_pt, y3_pt))

    def _normalized(self, currentpoint):
        return [normcurve_pt(currentpoint.x_pt, currentpoint.y_pt,
                             currentpoint.x_pt + self.dx1_pt, currentpoint.y_pt + self.dy1_pt,
                             currentpoint.x_pt + self.dx2_pt, currentpoint.y_pt + self.dy2_pt,
                             currentpoint.x_pt + self.dx3_pt, currentpoint.y_pt + self.dy3_pt)]

    def outputPS(self, file, writer, context):
        file.write("%g %g %g %g %g %g rcurveto\n" % (self.dx1_pt, self.dy1_pt,
                                                     self.dx2_pt, self.dy2_pt,
                                                     self.dx3_pt, self.dy3_pt))


class arc_pt(pathitem):

    """Append counterclockwise arc (coordinates in pts)"""

    __slots__ = "x_pt", "y_pt", "r_pt", "angle1", "angle2"

    def __init__(self, x_pt, y_pt, r_pt, angle1, angle2):
        self.x_pt = x_pt
        self.y_pt = y_pt
        self.r_pt = r_pt
        self.angle1 = angle1
        self.angle2 = angle2

    def __str__(self):
        return "arc_pt(%g, %g, %g, %g, %g)" % (self.x_pt, self.y_pt, self.r_pt,
                                               self.angle1, self.angle2)

    def _sarc(self):
        """return starting point of arc segment"""
        return (self.x_pt+self.r_pt*cos(radians(self.angle1)),
                self.y_pt+self.r_pt*sin(radians(self.angle1)))

    def _earc(self):
        """return end point of arc segment"""
        return (self.x_pt+self.r_pt*cos(radians(self.angle2)),
                self.y_pt+self.r_pt*sin(radians(self.angle2)))

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt, currentpoint.y_pt = self._earc()

    def _bbox(self, currentpoint):
        phi1 = radians(self.angle1)
        phi2 = radians(self.angle2)

        # starting end end point of arc segment
        sarcx_pt, sarcy_pt = self._sarc()
        earcx_pt, earcy_pt = self._earc()

        # Now, we have to determine the corners of the bbox for the
        # arc segment, i.e. global maxima/mimima of cos(phi) and sin(phi)
        # in the interval [phi1, phi2]. These can either be located
        # on the borders of this interval or in the interior.

        if phi2 < phi1:
            # guarantee that phi2>phi1
            phi2 = phi2 + (math.floor((phi1-phi2)/(2*pi))+1)*2*pi

        # next minimum of cos(phi) looking from phi1 in counterclockwise
        # direction: 2*pi*floor((phi1-pi)/(2*pi)) + 3*pi

        if phi2 < (2*math.floor((phi1-pi)/(2*pi))+3)*pi:
            minarcx_pt = min(sarcx_pt, earcx_pt)
        else:
            minarcx_pt = self.x_pt-self.r_pt

        # next minimum of sin(phi) looking from phi1 in counterclockwise
        # direction: 2*pi*floor((phi1-3*pi/2)/(2*pi)) + 7/2*pi

        if phi2 < (2*math.floor((phi1-3.0*pi/2)/(2*pi))+7.0/2)*pi:
            minarcy_pt = min(sarcy_pt, earcy_pt)
        else:
            minarcy_pt = self.y_pt-self.r_pt

        # next maximum of cos(phi) looking from phi1 in counterclockwise
        # direction: 2*pi*floor((phi1)/(2*pi))+2*pi

        if phi2 < (2*math.floor((phi1)/(2*pi))+2)*pi:
            maxarcx_pt = max(sarcx_pt, earcx_pt)
        else:
            maxarcx_pt = self.x_pt+self.r_pt

        # next maximum of sin(phi) looking from phi1 in counterclockwise
        # direction: 2*pi*floor((phi1-pi/2)/(2*pi)) + 1/2*pi

        if phi2 < (2*math.floor((phi1-pi/2)/(2*pi))+5.0/2)*pi:
            maxarcy_pt = max(sarcy_pt, earcy_pt)
        else:
            maxarcy_pt = self.y_pt+self.r_pt

        # Finally, we are able to construct the bbox for the arc segment.
        # Note that if a current point is defined, we also
        # have to include the straight line from this point
        # to the first point of the arc segment.

        if currentpoint.valid():
            return (bbox.bbox_pt(min(currentpoint.x_pt, sarcx_pt),
                                 min(currentpoint.y_pt, sarcy_pt),
                                 max(currentpoint.x_pt, sarcx_pt),
                                 max(currentpoint.y_pt, sarcy_pt)) +
                    bbox.bbox_pt(minarcx_pt, minarcy_pt, maxarcx_pt, maxarcy_pt) )
        else:
            return  bbox.bbox_pt(minarcx_pt, minarcy_pt, maxarcx_pt, maxarcy_pt)

    def _normalized(self, currentpoint):
        # get starting and end point of arc segment and bpath corresponding to arc
        sarcx_pt, sarcy_pt = self._sarc()
        earcx_pt, earcy_pt = self._earc()
        barc = _arctobezierpath(self.x_pt, self.y_pt, self.r_pt, self.angle1, self.angle2)

        # convert to list of curvetos omitting movetos
        nbarc = []

        for bpathitem in barc:
            nbarc.append(normcurve_pt(bpathitem.x0_pt, bpathitem.y0_pt,
                                      bpathitem.x1_pt, bpathitem.y1_pt,
                                      bpathitem.x2_pt, bpathitem.y2_pt,
                                      bpathitem.x3_pt, bpathitem.y3_pt))

        # Note that if a current point is defined, we also
        # have to include the straight line from this point
        # to the first point of the arc segment.
        # Otherwise, we have to add a moveto at the beginning.

        if currentpoint.valid():
            return [normline_pt(currentpoint.x_pt, currentpoint.y_pt, sarcx_pt, sarcy_pt)] + nbarc
        else:
            return [moveto_pt(sarcx_pt, sarcy_pt)] + nbarc

    def outputPS(self, file, writer, context):
        file.write("%g %g %g %g %g arc\n" % (self.x_pt, self.y_pt,
                                             self.r_pt,
                                             self.angle1,
                                             self.angle2))


class arcn_pt(pathitem):

    """Append clockwise arc (coordinates in pts)"""

    __slots__ = "x_pt", "y_pt", "r_pt", "angle1", "angle2"

    def __init__(self, x_pt, y_pt, r_pt, angle1, angle2):
        self.x_pt = x_pt
        self.y_pt = y_pt
        self.r_pt = r_pt
        self.angle1 = angle1
        self.angle2 = angle2

    def __str__(self):
        return "arcn_pt(%g, %g, %g, %g, %g)" % (self.x_pt, self.y_pt, self.r_pt,
                                                self.angle1, self.angle2)

    def _sarc(self):
        """return starting point of arc segment"""
        return (self.x_pt+self.r_pt*cos(radians(self.angle1)),
                self.y_pt+self.r_pt*sin(radians(self.angle1)))

    def _earc(self):
        """return end point of arc segment"""
        return (self.x_pt+self.r_pt*cos(radians(self.angle2)),
                self.y_pt+self.r_pt*sin(radians(self.angle2)))

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt, currentpoint.y_pt = self._earc()

    def _bbox(self, currentpoint):
        # in principle, we obtain bbox of an arcn element from
        # the bounding box of the corrsponding arc element with
        # angle1 and angle2 interchanged. Though, we have to be carefull
        # with the straight line segment, which is added if a current point
        # is defined.

        # Hence, we first compute the bbox of the arc without this line:

        a = arc_pt(self.x_pt, self.y_pt, self.r_pt,
                   self.angle2,
                   self.angle1)

        sarcx_pt, sarcy_pt = self._sarc()
        arcbb = a._bbox(_currentpoint())

        # Then, we repeat the logic from arc.bbox, but with interchanged
        # start and end points of the arc
        # XXX: I found the code to be equal! (AW, 31.1.2005)

        if currentpoint.valid():
            return  bbox.bbox_pt(min(currentpoint.x_pt, sarcx_pt),
                                 min(currentpoint.y_pt, sarcy_pt),
                                 max(currentpoint.x_pt, sarcx_pt),
                                 max(currentpoint.y_pt, sarcy_pt)) + arcbb
        else:
            return arcbb

    def _normalized(self, currentpoint):
        # get starting and end point of arc segment and bpath corresponding to arc
        sarcx_pt, sarcy_pt = self._sarc()
        earcx_pt, earcy_pt = self._earc()
        barc = _arctobezierpath(self.x_pt, self.y_pt, self.r_pt, self.angle2, self.angle1)
        barc.reverse()

        # convert to list of curvetos omitting movetos
        nbarc = []

        for bpathitem in barc:
            nbarc.append(normcurve_pt(bpathitem.x3_pt, bpathitem.y3_pt,
                                      bpathitem.x2_pt, bpathitem.y2_pt,
                                      bpathitem.x1_pt, bpathitem.y1_pt,
                                      bpathitem.x0_pt, bpathitem.y0_pt))

        # Note that if a current point is defined, we also
        # have to include the straight line from this point
        # to the first point of the arc segment.
        # Otherwise, we have to add a moveto at the beginning.

        if currentpoint.valid():
            return [normline_pt(currentpoint.x_pt, currentpoint.y_pt, sarcx_pt, sarcy_pt)] + nbarc
        else:
            return [moveto_pt(sarcx_pt, sarcy_pt)] + nbarc


    def outputPS(self, file, writer, context):
        file.write("%g %g %g %g %g arcn\n" % (self.x_pt, self.y_pt,
                                              self.r_pt,
                                              self.angle1,
                                              self.angle2))


class arct_pt(pathitem):

    """Append tangent arc (coordinates in pts)"""

    __slots__ = "x1_pt", "y1_pt", "x2_pt", "y2_pt", "r_pt"

    def __init__(self, x1_pt, y1_pt, x2_pt, y2_pt, r_pt):
        self.x1_pt = x1_pt
        self.y1_pt = y1_pt
        self.x2_pt = x2_pt
        self.y2_pt = y2_pt
        self.r_pt = r_pt

    def __str__(self):
        return "arct_pt(%g, %g, %g, %g, %g)" % (self.x1_pt, self.y1_pt,
                                                self.x2_pt, self.y2_pt,
                                                self.r_pt)

    def _pathitem(self, currentpoint):
        """return pathitem which corresponds to arct with the given currentpoint.

        The return value is either a arc_pt, a arcn_pt or a line_pt instance.

        This is a helper routine for _updatecurrentpoint, _bbox and _normalized,
        which will all delegate the work to the constructed pathitem.
        """

        # direction of tangent 1
        dx1_pt, dy1_pt = self.x1_pt-currentpoint.x_pt, self.y1_pt-currentpoint.y_pt
        l1_pt = math.hypot(dx1_pt, dy1_pt)
        dx1, dy1 = dx1_pt/l1_pt, dy1_pt/l1_pt

        # direction of tangent 2
        dx2_pt, dy2_pt = self.x2_pt-self.x1_pt, self.y2_pt-self.y1_pt
        l2_pt = math.hypot(dx2_pt, dy2_pt)
        dx2, dy2 = dx2_pt/l2_pt, dy2_pt/l2_pt

        # intersection angle between two tangents in the range (-pi, pi).
        # We take the orientation from the sign of the vector product.
        # Negative (positive) angles alpha corresponds to a turn to the right (left)
        # as seen from currentpoint.
        if dx1*dy2-dy1*dx2 > 0:
            alpha = acos(dx1*dx2+dy1*dy2) 
        else:
            alpha = -acos(dx1*dx2+dy1*dy2) 

        try:
            # two tangent points
            xt1_pt = self.x1_pt - dx1*self.r_pt*tan(abs(alpha)/2)
            yt1_pt = self.y1_pt - dy1*self.r_pt*tan(abs(alpha)/2)
            xt2_pt = self.x1_pt + dx2*self.r_pt*tan(abs(alpha)/2)
            yt2_pt = self.y1_pt + dy2*self.r_pt*tan(abs(alpha)/2)

            # direction point 1 -> center of arc
            dmx_pt = 0.5*(xt1_pt+xt2_pt) - self.x1_pt
            dmy_pt = 0.5*(yt1_pt+yt2_pt) - self.y1_pt
            lm_pt = math.hypot(dmx_pt, dmy_pt)
            dmx, dmy = dmx_pt/lm_pt, dmy_pt/lm_pt

            # center of arc
            mx_pt = self.x1_pt + dmx*self.r_pt/cos(alpha/2)
            my_pt = self.y1_pt + dmy*self.r_pt/cos(alpha/2)

            # angle around which arc is centered
            phi = degrees(math.atan2(-dmy, -dmx))

            # half angular width of arc
            deltaphi = degrees(alpha)/2

            if alpha > 0:
                return arc_pt(mx_pt, my_pt, self.r_pt, phi-deltaphi, phi+deltaphi)
            else:
                return arcn_pt(mx_pt, my_pt, self.r_pt, phi-deltaphi, phi+deltaphi)

        except ZeroDivisionError:
            # in the degenerate case, we just return a line as specified by the PS 
            # language reference
            return lineto_pt(self.x1_pt, self.y1_pt)

    def _updatecurrentpoint(self, currentpoint):
        self._pathitem(currentpoint)._updatecurrentpoint(currentpoint)

    def _bbox(self, currentpoint):
        return self._pathitem(currentpoint)._bbox(currentpoint)

    def _normalized(self, currentpoint):
        return self._pathitem(currentpoint)._normalized(currentpoint)

    def outputPS(self, file, writer, context):
        file.write("%g %g %g %g %g arct\n" % (self.x1_pt, self.y1_pt,
                                              self.x2_pt, self.y2_pt,
                                              self.r_pt))

#
# now the pathitems that convert from user coordinates to pts
#

class moveto(moveto_pt):

    """Set current point to (x, y)"""

    __slots__ = "x_pt", "y_pt"

    def __init__(self, x, y):
        moveto_pt.__init__(self, unit.topt(x), unit.topt(y))


class lineto(lineto_pt):

    """Append straight line to (x, y)"""

    __slots__ = "x_pt", "y_pt"

    def __init__(self, x, y):
        lineto_pt.__init__(self, unit.topt(x), unit.topt(y))


class curveto(curveto_pt):

    """Append curveto"""

    __slots__ = "x1_pt", "y1_pt", "x2_pt", "y2_pt", "x3_pt", "y3_pt"

    def __init__(self, x1, y1, x2, y2, x3, y3):
        curveto_pt.__init__(self,
                            unit.topt(x1), unit.topt(y1),
                            unit.topt(x2), unit.topt(y2),
                            unit.topt(x3), unit.topt(y3))

class rmoveto(rmoveto_pt):

    """Perform relative moveto"""

    __slots__ = "dx_pt", "dy_pt"

    def __init__(self, dx, dy):
        rmoveto_pt.__init__(self, unit.topt(dx), unit.topt(dy))


class rlineto(rlineto_pt):

    """Perform relative lineto"""

    __slots__ = "dx_pt", "dy_pt"

    def __init__(self, dx, dy):
        rlineto_pt.__init__(self, unit.topt(dx), unit.topt(dy))


class rcurveto(rcurveto_pt):

    """Append rcurveto"""

    __slots__ = "dx1_pt", "dy1_pt", "dx2_pt", "dy2_pt", "dx3_pt", "dy3_pt"

    def __init__(self, dx1, dy1, dx2, dy2, dx3, dy3):
        rcurveto_pt.__init__(self,
                             unit.topt(dx1), unit.topt(dy1),
                             unit.topt(dx2), unit.topt(dy2),
                             unit.topt(dx3), unit.topt(dy3))


class arcn(arcn_pt):

    """Append clockwise arc"""

    __slots__ = "x_pt", "y_pt", "r_pt", "angle1", "angle2"

    def __init__(self, x, y, r, angle1, angle2):
        arcn_pt.__init__(self, unit.topt(x), unit.topt(y), unit.topt(r), angle1, angle2)


class arc(arc_pt):

    """Append counterclockwise arc"""

    __slots__ = "x_pt", "y_pt", "r_pt", "angle1", "angle2"

    def __init__(self, x, y, r, angle1, angle2):
        arc_pt.__init__(self, unit.topt(x), unit.topt(y), unit.topt(r), angle1, angle2)


class arct(arct_pt):

    """Append tangent arc"""

    __slots__ = "x1_pt", "y1_pt", "x2_pt", "y2_pt", "r_pt"

    def __init__(self, x1, y1, x2, y2, r):
        arct_pt.__init__(self, unit.topt(x1), unit.topt(y1),
                         unit.topt(x2), unit.topt(y2), unit.topt(r))

#
# "combined" pathitems provided for performance reasons
#

class multilineto_pt(pathitem):

    """Perform multiple linetos (coordinates in pts)"""

    __slots__ = "points_pt"

    def __init__(self, points_pt):
        self.points_pt = points_pt

    def __str__(self):
        result = []
        for point_pt in self.points_pt:
            result.append("(%g, %g)" % point_pt )
        return "multilineto_pt([%s])" % (", ".join(result))

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt, currentpoint.y_pt = self.points_pt[-1]

    def _bbox(self, currentpoint):
        xs_pt = [point[0] for point in self.points_pt]
        ys_pt = [point[1] for point in self.points_pt]
        return bbox.bbox_pt(min(currentpoint.x_pt, *xs_pt),
                            min(currentpoint.y_pt, *ys_pt),
                            max(currentpoint.x_pt, *xs_pt),
                            max(currentpoint.y_pt, *ys_pt))

    def _normalized(self, currentpoint):
        result = []
        x0_pt = currentpoint.x_pt
        y0_pt = currentpoint.y_pt
        for x1_pt, y1_pt in self.points_pt:
            result.append(normline_pt(x0_pt, y0_pt, x1_pt, y1_pt))
            x0_pt, y0_pt = x1_pt, y1_pt
        return result

    def outputPS(self, file, writer, context):
        for point_pt in self.points_pt:
            file.write("%g %g lineto\n" % point_pt )


class multicurveto_pt(pathitem):

    """Perform multiple curvetos (coordinates in pts)"""

    __slots__ = "points_pt"

    def __init__(self, points_pt):
        self.points_pt = points_pt

    def __str__(self):
        result = []
        for point_pt in self.points_pt:
            result.append("(%g, %g, %g, %g, %g, %g)" % point_pt )
        return "multicurveto_pt([%s])" % (", ".join(result))

    def _updatecurrentpoint(self, currentpoint):
        currentpoint.x_pt, currentpoint.y_pt = self.points_pt[-1]

    def _bbox(self, currentpoint):
        xs_pt = ( [point[0] for point in self.points_pt] +
                  [point[2] for point in self.points_pt] +
                  [point[4] for point in self.points_pt] )
        ys_pt = ( [point[1] for point in self.points_pt] +
                  [point[3] for point in self.points_pt] +
                  [point[5] for point in self.points_pt] )
        return bbox.bbox_pt(min(currentpoint.x_pt, *xs_pt),
                            min(currentpoint.y_pt, *ys_pt),
                            max(currentpoint.x_pt, *xs_pt),
                            max(currentpoint.y_pt, *ys_pt))

    def _normalized(self, currentpoint):
        result = []
        x0_pt = currentpoint.x_pt
        y0_pt = currentpoint.y_pt
        for point_pt in self.points_pt:
            result.append(normcurve_pt(x_pt, y_pt, *point_pt))
            x_pt, y_pt = point_pt[4:]
        return result

    def outputPS(self, file, writer, context):
        for point_pt in self.points_pt:
            file.write("%g %g %g %g %g %g curveto\n" % point_pt)


################################################################################
# path: PS style path
################################################################################

class path(canvas.canvasitem):

    """PS style path"""

    __slots__ = "path", "_normpath"

    def __init__(self, *pathitems):
        """construct a path from pathitems *args"""

        for apathitem in pathitems:
            assert isinstance(apathitem, pathitem), "only pathitem instances allowed"

        self.pathitems = list(pathitems)
        # normpath cache
        self._normpath = None

    def __add__(self, other):
        """create new path out of self and other"""
        return path(*(self.pathitems + other.path().pathitems))

    def __iadd__(self, other):
        """add other inplace

        If other is a normpath instance, it is converted to a path before
        being added.
        """
        self.pathitems += other.path().pathitems
        self._normpath = None
        return self

    def __getitem__(self, i):
        """return path item i"""
        return self.pathitems[i]

    def __len__(self):
        """return the number of path items"""
        return len(self.pathitems)

    def __str__(self):
        l = ", ".join(map(str, self.pathitems))
        return "path(%s)" % l

    def append(self, apathitem):
        """append a path item"""
        assert isinstance(apathitem, pathitem), "only pathitem instance allowed"
        self.pathitems.append(apathitem)
        self._normpath = None

    def arclen_pt(self):
        """return arc length in pts"""
        return self.normpath().arclen_pt()

    def arclen(self):
        """return arc length"""
        return self.normpath().arclen()

    def arclentoparam_pt(self, lengths_pt):
        """return the param(s) matching the given length(s)_pt in pts"""
        return self.normpath().arclentoparam_pt(lengths_pt)

    def arclentoparam(self, lengths):
        """return the param(s) matching the given length(s)"""
        return self.normpath().arclentoparam(lengths)

    def at_pt(self, params):
        """return coordinates of path in pts at param(s) or arc length(s) in pts"""
        return self.normpath().at_pt(params)

    def at(self, params):
        """return coordinates of path at param(s) or arc length(s)"""
        return self.normpath().at(params)

    def atbegin_pt(self):
        """return coordinates of the beginning of first subpath in path in pts"""
        return self.normpath().atbegin_pt()

    def atbegin(self):
        """return coordinates of the beginning of first subpath in path"""
        return self.normpath().atbegin()

    def atend_pt(self):
        """return coordinates of the end of last subpath in path in pts"""
        return self.normpath().atend_pt()

    def atend(self):
        """return coordinates of the end of last subpath in path"""
        return self.normpath().atend()

    def bbox(self):
        """return bbox of path"""
        currentpoint = _currentpoint()
        abbox = None

        for pitem in self.pathitems:
            nbbox =  pitem._bbox(currentpoint)
            pitem._updatecurrentpoint(currentpoint)
            if abbox is None:
                abbox = nbbox
            elif nbbox:
                abbox += nbbox

        return abbox

    def begin(self):
        """return param corresponding of the beginning of the path"""
        return self.normpath().begin()

    def curveradius_pt(self, params):
        """return the curvature radius in pts at param(s) or arc length(s) in pts

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""
        return self.normpath().curveradius_pt(params)

    def curveradius(self, params):
        """return the curvature radius at param(s) or arc length(s)

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""
        return self.normpath().curveradius(params)

    def end(self):
        """return param corresponding of the end of the path"""
        return self.normpath().end()

    def extend(self, pathitems):
        """extend path by pathitems"""
        for apathitem in pathitems:
            assert isinstance(apathitem, pathitem), "only pathitem instance allowed"
        self.pathitems.extend(pathitems)
        self._normpath = None

    def intersect(self, other):
        """intersect self with other path

        Returns a tuple of lists consisting of the parameter values
        of the intersection points of the corresponding normpath.
        """
        return self.normpath().intersect(other)

    def join(self, other):
        """join other path/normpath inplace

        If other is a normpath instance, it is converted to a path before
        being joined.
        """
        self.pathitems = self.joined(other).path().pathitems
        self._normpath = None
        return self

    def joined(self, other):
        """return path consisting of self and other joined together"""
        return self.normpath().joined(other).path()

    # << operator also designates joining
    __lshift__ = joined

    def normpath(self, epsilon=None):
        """convert the path into a normpath"""
        # use cached value if existent
        if self._normpath is not None:
            return self._normpath
        # split path in sub paths
        subpaths = []
        currentsubpathitems = []
        currentpoint = _currentpoint()
        for pitem in self.pathitems:
            for npitem in pitem._normalized(currentpoint):
                if isinstance(npitem, moveto_pt):
                    if currentsubpathitems:
                        # append open sub path
                        subpaths.append(normsubpath(currentsubpathitems, closed=0, epsilon=epsilon))
                    # start new sub path
                    currentsubpathitems = []
                elif isinstance(npitem, closepath):
                    if currentsubpathitems:
                        # append closed sub path
                        currentsubpathitems.append(normline_pt(currentpoint.x_pt, currentpoint.y_pt,
                                                               *currentsubpathitems[0].atbegin_pt()))
                    subpaths.append(normsubpath(currentsubpathitems, closed=1, epsilon=epsilon))
                    currentsubpathitems = []
                else:
                    currentsubpathitems.append(npitem)
            pitem._updatecurrentpoint(currentpoint)

        if currentsubpathitems:
            # append open sub path
            subpaths.append(normsubpath(currentsubpathitems, 0, epsilon))
        self._normpath = normpath(subpaths)
        return self._normpath

    def paramtoarclen_pt(self, params):
        """return arc lenght(s) in pts matching the given param(s)"""
        return self.normpath().paramtoarclen_pt(lengths_pt)

    def paramtoarclen(self, params):
        """return arc lenght(s) matching the given param(s)"""
        return self.normpath().paramtoarclen(lengths_pt)

    def path(self):
        """return corresponding path, i.e., self"""
        return self

    def reversed(self):
        """return reversed normpath"""
        # TODO: couldn't we try to return a path instead of converting it
        #       to a normpath (but this might not be worth the trouble)
        return self.normpath().reversed()

    def split_pt(self, params):
        """split normpath at param(s) or arc length(s) in pts and return list of normpaths"""
        return self.normpath().split(params)

    def split(self, params):
        """split normpath at param(s) or arc length(s) and return list of normpaths"""
        return self.normpath().split(params)

    def tangent_pt(self, params, length=None):
        """return tangent vector of path at param(s) or arc length(s) in pts

        If length in pts is not None, the tangent vector will be scaled to
        the desired length.
        """
        return self.normpath().tangent_pt(params, length)

    def tangent(self, params, length=None):
        """return tangent vector of path at param(s) or arc length(s)

        If length is not None, the tangent vector will be scaled to
        the desired length.
        """
        return self.normpath().tangent(params, length)

    def trafo_pt(self, params):
        """return transformation at param(s) or arc length(s) in pts"""
        return self.normpath().trafo(params)

    def trafo(self, params):
        """return transformation at param(s) or arc length(s)"""
        return self.normpath().trafo(params)

    def transformed(self, trafo):
        """return transformed path"""
        return self.normpath().transformed(trafo)

    def outputPS(self, file, writer, context):
        """write PS code to file"""
        for pitem in self.pathitems:
            pitem.outputPS(file, writer, context)

    def outputPDF(self, file, writer, context):
        """write PDF code to file"""
        # PDF only supports normsubpathitems but instead of
        # converting to a normpath, which will fail for short
        # closed paths, we use outputPDF of the normalized paths
        currentpoint = _currentpoint()
        for pitem in self.pathitems:
            for npitem in pitem._normalized(currentpoint):
                npitem.outputPDF(file, writer, context)
            pitem._updatecurrentpoint(currentpoint)


#
# some special kinds of path, again in two variants
#

class line_pt(path):

   """straight line from (x1_pt, y1_pt) to (x2_pt, y2_pt) in pts"""

   def __init__(self, x1_pt, y1_pt, x2_pt, y2_pt):
       path.__init__(self, moveto_pt(x1_pt, y1_pt), lineto_pt(x2_pt, y2_pt))


class curve_pt(path):

   """bezier curve with control points (x0_pt, y1_pt),..., (x3_pt, y3_pt) in pts"""

   def __init__(self, x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt):
       path.__init__(self,
                     moveto_pt(x0_pt, y0_pt),
                     curveto_pt(x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt))


class rect_pt(path):

   """rectangle at position (x, y) with width and height in pts"""

   def __init__(self, x, y, width, height):
       path.__init__(self, moveto_pt(x, y),
                           lineto_pt(x+width, y),
                           lineto_pt(x+width, y+height),
                           lineto_pt(x, y+height),
                           closepath())


class circle_pt(path):

   """circle with center (x, y) and radius in pts"""

   def __init__(self, x, y, radius):
       path.__init__(self, arc_pt(x, y, radius, 0, 360), closepath())


class line(line_pt):

   """straight line from (x1, y1) to (x2, y2)"""

   def __init__(self, x1, y1, x2, y2):
       line_pt.__init__(self, unit.topt(x1), unit.topt(y1),
                              unit.topt(x2), unit.topt(y2))


class curve(curve_pt):

   """bezier curve with control points (x0, y1),..., (x3, y3)"""

   def __init__(self, x0, y0, x1, y1, x2, y2, x3, y3):
       curve_pt.__init__(self, unit.topt(x0), unit.topt(y0),
                               unit.topt(x1), unit.topt(y1),
                               unit.topt(x2), unit.topt(y2),
                               unit.topt(x3), unit.topt(y3))


class rect(rect_pt):

   """rectangle at position (x,y) with width and height"""

   def __init__(self, x, y, width, height):
       rect_pt.__init__(self, unit.topt(x), unit.topt(y),
                              unit.topt(width), unit.topt(height))


class circle(circle_pt):

   """circle with center (x,y) and radius"""

   def __init__(self, x, y, radius):
       circle_pt.__init__(self, unit.topt(x), unit.topt(y), unit.topt(radius))


################################################################################
# normsubpathitems
################################################################################

class normsubpathitem:

    """element of a normalized sub path

    Various operations on normsubpathitems might be subject of
    approximitions. Those methods get the finite precision epsilon,
    which is the accuracy needed expressed as a length in pts.

    normsubpathitems should never be modified inplace, since references
    might be shared betweeen several normsubpaths.
    """

    def arclen_pt(self, epsilon):
        """return arc length in pts"""
        pass

    def _arclentoparam_pt(self, lengths_pt, epsilon):
        """return a tuple of params and the total length arc length in pts"""
        pass

    def at_pt(self, params):
        """return coordinates at params in pts"""
        pass

    def atbegin_pt(self):
        """return coordinates of first point in pts"""
        pass

    def atend_pt(self):
        """return coordinates of last point in pts"""
        pass

    def bbox(self):
        """return bounding box of normsubpathitem"""
        pass

    def curveradius_pt(self, params):
        """return the curvature radius at params in pts

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""
        pass

    def intersect(self, other, epsilon):
        """intersect self with other normsubpathitem"""
        pass

    def modifiedbegin_pt(self, x_pt, y_pt):
        """return a normsubpathitem with a modified beginning point"""
        pass

    def modifiedend_pt(self, x_pt, y_pt):
        """return a normsubpathitem with a modified end point"""
        pass

    def _paramtoarclen_pt(self, param, epsilon):
        """return a tuple of arc lengths and the total arc length in pts"""
        pass

    def pathitem(self):
        """return pathitem corresponding to normsubpathitem"""

    def reversed(self):
        """return reversed normsubpathitem"""
        pass

    def segments(self, params):
        """return segments of the normsubpathitem

        The returned list of normsubpathitems for the segments between
        the params. params need to contain at least two values.
        """
        pass

    def trafo(self, params):
        """return transformations at params"""

    def transformed(self, trafo):
        """return transformed normsubpathitem according to trafo"""
        pass

    def outputPS(self, file, writer, context):
        """write PS code corresponding to normsubpathitem to file"""
        pass

    def outputPDF(self, file, writer, context):
        """write PDF code corresponding to normsubpathitem to file"""
        pass


class normline_pt(normsubpathitem):

    """Straight line from (x0_pt, y0_pt) to (x1_pt, y1_pt) (coordinates in pts)"""

    __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt"

    def __init__(self, x0_pt, y0_pt, x1_pt, y1_pt):
        self.x0_pt = x0_pt
        self.y0_pt = y0_pt
        self.x1_pt = x1_pt
        self.y1_pt = y1_pt

    def __str__(self):
        return "normline_pt(%g, %g, %g, %g)" % (self.x0_pt, self.y0_pt, self.x1_pt, self.y1_pt)

    def _arclentoparam_pt(self, lengths, epsilon):
        # do self.arclen_pt inplace for performance reasons
        l = math.hypot(self.x0_pt-self.x1_pt, self.y0_pt-self.y1_pt)
        return [length/l for length in lengths], l

    def arclen_pt(self,  epsilon):
        return math.hypot(self.x0_pt-self.x1_pt, self.y0_pt-self.y1_pt)

    def at_pt(self, params):
        return [(self.x0_pt+(self.x1_pt-self.x0_pt)*t, self.y0_pt+(self.y1_pt-self.y0_pt)*t)
                for t in params]

    def atbegin_pt(self):
        return self.x0_pt, self.y0_pt

    def atend_pt(self):
        return self.x1_pt, self.y1_pt

    def bbox(self):
        return bbox.bbox_pt(min(self.x0_pt, self.x1_pt), min(self.y0_pt, self.y1_pt),
                            max(self.x0_pt, self.x1_pt), max(self.y0_pt, self.y1_pt))

    def curveradius_pt(self, params):
        return [None] * len(params)

    def intersect(self, other, epsilon):
        if isinstance(other, normline_pt):
            a_deltax_pt = self.x1_pt - self.x0_pt
            a_deltay_pt = self.y1_pt - self.y0_pt

            b_deltax_pt = other.x1_pt - other.x0_pt
            b_deltay_pt = other.y1_pt - other.y0_pt
            try:
                det = 1.0 / (b_deltax_pt * a_deltay_pt - b_deltay_pt * a_deltax_pt)
            except ArithmeticError:
                return []

            ba_deltax0_pt = other.x0_pt - self.x0_pt
            ba_deltay0_pt = other.y0_pt - self.y0_pt

            a_t = (b_deltax_pt * ba_deltay0_pt - b_deltay_pt * ba_deltax0_pt) * det
            b_t = (a_deltax_pt * ba_deltay0_pt - a_deltay_pt * ba_deltax0_pt) * det

            # check for intersections out of bound
            # TODO: we might allow for a small out of bound errors.
            if not (0<=a_t<=1 and 0<=b_t<=1):
                return []

            # return parameters of intersection
            return [(a_t, b_t)]
        else:
            return [(s_t, o_t) for o_t, s_t in other.intersect(self, epsilon)]

    def modifiedbegin_pt(self, x_pt, y_pt):
        return normline_pt(x_pt, y_pt, self.x1_pt, self.y1_pt)

    def modifiedend_pt(self, x_pt, y_pt):
        return normline_pt(self.x0_pt, self.y0_pt, x_pt, y_pt)

    def _paramtoarclen_pt(self, params, epsilon):
        totalarclen_pt = self.arclen_pt(epsilon)
        arclens_pt = [totalarclen_pt * param for param in params + [1]]
        return arclens_pt[:-1], arclens_pt[-1]

    def pathitem(self):
        return lineto_pt(self.x1_pt, self.y1_pt)

    def reversed(self):
        return normline_pt(self.x1_pt, self.y1_pt, self.x0_pt, self.y0_pt)

    def segments(self, params):
        if len(params) < 2:
            raise ValueError("at least two parameters needed in segments")
        result = []
        xl_pt = yl_pt = None
        for t in params:
            xr_pt = self.x0_pt + (self.x1_pt-self.x0_pt)*t
            yr_pt = self.y0_pt + (self.y1_pt-self.y0_pt)*t
            if xl_pt is not None:
                result.append(normline_pt(xl_pt, yl_pt, xr_pt, yr_pt))
            xl_pt = xr_pt
            yl_pt = yr_pt
        return result

    def trafo(self, params):
        rotate = trafo.rotate(degrees(math.atan2(self.y1_pt-self.y0_pt, self.x1_pt-self.x0_pt)))
        return [trafo.translate_pt(*at_pt) * rotate
                for param, at_pt in zip(params, self.at_pt(params))]

    def transformed(self, trafo):
        return normline_pt(*(trafo._apply(self.x0_pt, self.y0_pt) + trafo._apply(self.x1_pt, self.y1_pt)))

    def outputPS(self, file, writer, context):
        file.write("%g %g lineto\n" % (self.x1_pt, self.y1_pt))

    def outputPDF(self, file, writer, context):
        file.write("%f %f l\n" % (self.x1_pt, self.y1_pt))


class normcurve_pt(normsubpathitem):

    """Bezier curve with control points x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt (coordinates in pts)"""

    __slots__ = "x0_pt", "y0_pt", "x1_pt", "y1_pt", "x2_pt", "y2_pt", "x3_pt", "y3_pt"

    def __init__(self, x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt):
        self.x0_pt = x0_pt
        self.y0_pt = y0_pt
        self.x1_pt = x1_pt
        self.y1_pt = y1_pt
        self.x2_pt = x2_pt
        self.y2_pt = y2_pt
        self.x3_pt = x3_pt
        self.y3_pt = y3_pt

    def __str__(self):
        return "normcurve_pt(%g, %g, %g, %g, %g, %g, %g, %g)" % (self.x0_pt, self.y0_pt, self.x1_pt, self.y1_pt,
                                                                 self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt)

    def _midpointsplit(self, epsilon):
        """split curve into two parts

        Helper method to reduce the complexity of a problem by turning
        a normcurve_pt into several normline_pt segments. This method
        returns normcurve_pt instances only, when they are not yet straight
        enough to be replaceable by normcurve_pt instances. Thus a recursive
        midpointsplitting will turn a curve into line segments with the
        given precision epsilon.
        """

        # first, we have to calculate the  midpoints between adjacent
        # control points
        x01_pt = 0.5*(self.x0_pt + self.x1_pt)
        y01_pt = 0.5*(self.y0_pt + self.y1_pt)
        x12_pt = 0.5*(self.x1_pt + self.x2_pt)
        y12_pt = 0.5*(self.y1_pt + self.y2_pt)
        x23_pt = 0.5*(self.x2_pt + self.x3_pt)
        y23_pt = 0.5*(self.y2_pt + self.y3_pt)

        # In the next iterative step, we need the midpoints between 01 and 12
        # and between 12 and 23
        x01_12_pt = 0.5*(x01_pt + x12_pt)
        y01_12_pt = 0.5*(y01_pt + y12_pt)
        x12_23_pt = 0.5*(x12_pt + x23_pt)
        y12_23_pt = 0.5*(y12_pt + y23_pt)

        # Finally the midpoint is given by
        xmidpoint_pt = 0.5*(x01_12_pt + x12_23_pt)
        ymidpoint_pt = 0.5*(y01_12_pt + y12_23_pt)

        # Before returning the normcurves we check whether we can
        # replace them by normlines within an error of epsilon pts.
        # The maximal error value is given by the modulus of the
        # difference between the length of the control polygon
        # (i.e. |P1-P0|+|P2-P1|+|P3-P2|), which consitutes an upper
        # bound for the length, and the length of the straight line
        # between start and end point of the normcurve (i.e. |P3-P1|),
        # which represents a lower bound.
        upperlen1 = (math.hypot(x01_pt - self.x0_pt, y01_pt - self.y0_pt) +
                     math.hypot(x01_12_pt - x01_pt, y01_12_pt - y01_pt) +
                     math.hypot(xmidpoint_pt - x01_12_pt, ymidpoint_pt - y01_12_pt))
        lowerlen1 = math.hypot(xmidpoint_pt - self.x0_pt, ymidpoint_pt - self.y0_pt)
        if upperlen1-lowerlen1 < epsilon:
            c1 = normline_pt(self.x0_pt, self.y0_pt, xmidpoint_pt, ymidpoint_pt)
        else:
            c1 = normcurve_pt(self.x0_pt, self.y0_pt,
                              x01_pt, y01_pt,
                              x01_12_pt, y01_12_pt,
                              xmidpoint_pt, ymidpoint_pt)

        upperlen2 = (math.hypot(x12_23_pt - xmidpoint_pt, y12_23_pt - ymidpoint_pt) +
                     math.hypot(x23_pt - x12_23_pt, y23_pt - y12_23_pt) +
                     math.hypot(self.x3_pt - x23_pt, self.y3_pt - y23_pt))
        lowerlen2 = math.hypot(self.x3_pt - xmidpoint_pt, self.y3_pt - ymidpoint_pt)
        if upperlen2-lowerlen2 < epsilon:
            c2 = normline_pt(xmidpoint_pt, ymidpoint_pt, self.x3_pt, self.y3_pt)
        else:
            c2 = normcurve_pt(xmidpoint_pt, ymidpoint_pt,
                              x12_23_pt, y12_23_pt,
                              x23_pt, y23_pt,
                              self.x3_pt, self.y3_pt)

        return c1, c2

    def _arclentoparam_pt(self, lengths_pt, epsilon):
        a, b = self._midpointsplit(epsilon)
        params_a, arclen_a = a._arclentoparam_pt(lengths_pt, epsilon)
        params_b, arclen_b = b._arclentoparam_pt([length_pt - arclen_a for length_pt in lengths_pt], epsilon)
        params = []
        for param_a, param_b, length_pt in zip(params_a, params_b, lengths_pt):
            if length_pt > arclen_a:
                params.append(0.5+0.5*param_b)
            else:
                params.append(0.5*param_a)
        return params, arclen_a + arclen_b

    def arclen_pt(self, epsilon):
        a, b = self._midpointsplit(epsilon)
        return a.arclen_pt(epsilon) + b.arclen_pt(epsilon)

    def at_pt(self, params):
        return [( (-self.x0_pt+3*self.x1_pt-3*self.x2_pt+self.x3_pt)*t*t*t +
                  (3*self.x0_pt-6*self.x1_pt+3*self.x2_pt          )*t*t +
                  (-3*self.x0_pt+3*self.x1_pt                      )*t +
                  self.x0_pt,
                  (-self.y0_pt+3*self.y1_pt-3*self.y2_pt+self.y3_pt)*t*t*t +
                  (3*self.y0_pt-6*self.y1_pt+3*self.y2_pt          )*t*t +
                  (-3*self.y0_pt+3*self.y1_pt                      )*t +
                  self.y0_pt )
                for t in params]

    def atbegin_pt(self):
        return self.x0_pt, self.y0_pt

    def atend_pt(self):
        return self.x3_pt, self.y3_pt

    def bbox(self):
        return bbox.bbox_pt(min(self.x0_pt, self.x1_pt, self.x2_pt, self.x3_pt),
                            min(self.y0_pt, self.y1_pt, self.y2_pt, self.y3_pt),
                            max(self.x0_pt, self.x1_pt, self.x2_pt, self.x3_pt),
                            max(self.y0_pt, self.y1_pt, self.y2_pt, self.y3_pt))

    def curveradius_pt(self, params):
        result = []
        for param in params:
            xdot = ( 3 * (1-param)*(1-param) * (-self.x0_pt + self.x1_pt) +
                     6 * (1-param)*param * (-self.x1_pt + self.x2_pt) +
                     3 * param*param * (-self.x2_pt + self.x3_pt) )
            ydot = ( 3 * (1-param)*(1-param) * (-self.y0_pt + self.y1_pt) +
                     6 * (1-param)*param * (-self.y1_pt + self.y2_pt) +
                     3 * param*param * (-self.y2_pt + self.y3_pt) )
            xddot = ( 6 * (1-param) * (self.x0_pt - 2*self.x1_pt + self.x2_pt) +
                      6 * param * (self.x1_pt - 2*self.x2_pt + self.x3_pt) )
            yddot = ( 6 * (1-param) * (self.y0_pt - 2*self.y1_pt + self.y2_pt) +
                      6 * param * (self.y1_pt - 2*self.y2_pt + self.y3_pt) )

            try:
                radius = (xdot**2 + ydot**2)**1.5 / (xdot*yddot - ydot*xddot)
            except:
                radius = None

            result.append(radius)

        return result

    def intersect(self, other, epsilon):
        # we can immediately quit when the bboxes are not overlapping
        if not self.bbox().intersects(other.bbox()):
            return []
        a, b = self._midpointsplit(epsilon)
        # To improve the performance in the general case we alternate the
        # splitting process between the two normsubpathitems
        return ( [(    0.5*a_t, o_t) for o_t, a_t in other.intersect(a, epsilon)] +
                 [(0.5+0.5*b_t, o_t) for o_t, b_t in other.intersect(b, epsilon)] )

    def modifiedbegin_pt(self, x_pt, y_pt):
        return normcurve_pt(x_pt, y_pt,
                            self.x1_pt, self.y1_pt,
                            self.x2_pt, self.y2_pt,
                            self.x3_pt, self.y3_pt)

    def modifiedend_pt(self, x_pt, y_pt):
        return normcurve_pt(self.x0_pt, self.y0_pt,
                            self.x1_pt, self.y1_pt,
                            self.x2_pt, self.y2_pt,
                            x_pt, y_pt)

    def _paramtoarclen_pt(self, params, epsilon):
        arclens_pt = [segment.arclen_pt(epsilon) for segment in self.segments([0] + list(params) + [1])]
        for i in range(1, len(arclens_pt)):
            arclens_pt[i] += arclens_pt[i-1]
        return arclens_pt[:-1], arclens_pt[-1]

    def pathitem(self):
        return curveto_pt(self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt)

    def reversed(self):
        return normcurve_pt(self.x3_pt, self.y3_pt, self.x2_pt, self.y2_pt, self.x1_pt, self.y1_pt, self.x0_pt, self.y0_pt)

    def segments(self, params):
        if len(params) < 2:
            raise ValueError("at least two parameters needed in segments")

        # first, we calculate the coefficients corresponding to our
        # original bezier curve. These represent a useful starting
        # point for the following change of the polynomial parameter
        a0x_pt = self.x0_pt
        a0y_pt = self.y0_pt
        a1x_pt = 3*(-self.x0_pt+self.x1_pt)
        a1y_pt = 3*(-self.y0_pt+self.y1_pt)
        a2x_pt = 3*(self.x0_pt-2*self.x1_pt+self.x2_pt)
        a2y_pt = 3*(self.y0_pt-2*self.y1_pt+self.y2_pt)
        a3x_pt = -self.x0_pt+3*(self.x1_pt-self.x2_pt)+self.x3_pt
        a3y_pt = -self.y0_pt+3*(self.y1_pt-self.y2_pt)+self.y3_pt

        result = []

        for i in range(len(params)-1):
            t1 = params[i]
            dt = params[i+1]-t1

            # [t1,t2] part
            #
            # the new coefficients of the [t1,t1+dt] part of the bezier curve
            # are then given by expanding
            #  a0 + a1*(t1+dt*u) + a2*(t1+dt*u)**2 +
            #  a3*(t1+dt*u)**3 in u, yielding
            #
            #   a0 + a1*t1 + a2*t1**2 + a3*t1**3        +
            #   ( a1 + 2*a2 + 3*a3*t1**2 )*dt    * u    +
            #   ( a2 + 3*a3*t1 )*dt**2           * u**2 +
            #   a3*dt**3                         * u**3
            #
            # from this values we obtain the new control points by inversion
            #
            # TODO: we could do this more efficiently by reusing for
            # (x0_pt, y0_pt) the control point (x3_pt, y3_pt) from the previous
            # Bezier curve

            x0_pt = a0x_pt + a1x_pt*t1 + a2x_pt*t1*t1 + a3x_pt*t1*t1*t1
            y0_pt = a0y_pt + a1y_pt*t1 + a2y_pt*t1*t1 + a3y_pt*t1*t1*t1
            x1_pt = (a1x_pt+2*a2x_pt*t1+3*a3x_pt*t1*t1)*dt/3.0 + x0_pt
            y1_pt = (a1y_pt+2*a2y_pt*t1+3*a3y_pt*t1*t1)*dt/3.0 + y0_pt
            x2_pt = (a2x_pt+3*a3x_pt*t1)*dt*dt/3.0 - x0_pt + 2*x1_pt
            y2_pt = (a2y_pt+3*a3y_pt*t1)*dt*dt/3.0 - y0_pt + 2*y1_pt
            x3_pt = a3x_pt*dt*dt*dt + x0_pt - 3*x1_pt + 3*x2_pt
            y3_pt = a3y_pt*dt*dt*dt + y0_pt - 3*y1_pt + 3*y2_pt

            result.append(normcurve_pt(x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt))

        return result

    def trafo(self, params):
        result = []
        for param, at_pt in zip(params, self.at_pt(params)):
            tdx_pt = (3*(  -self.x0_pt+3*self.x1_pt-3*self.x2_pt+self.x3_pt)*param*param +
                      2*( 3*self.x0_pt-6*self.x1_pt+3*self.x2_pt           )*param +
                        (-3*self.x0_pt+3*self.x1_pt                        ))
            tdy_pt = (3*(  -self.y0_pt+3*self.y1_pt-3*self.y2_pt+self.y3_pt)*param*param +
                      2*( 3*self.y0_pt-6*self.y1_pt+3*self.y2_pt           )*param +
                        (-3*self.y0_pt+3*self.y1_pt                        ))
            result.append(trafo.translate_pt(*at_pt) * trafo.rotate(degrees(math.atan2(tdy_pt, tdx_pt))))
        return result

    def transformed(self, trafo):
        x0_pt, y0_pt = trafo._apply(self.x0_pt, self.y0_pt)
        x1_pt, y1_pt = trafo._apply(self.x1_pt, self.y1_pt)
        x2_pt, y2_pt = trafo._apply(self.x2_pt, self.y2_pt)
        x3_pt, y3_pt = trafo._apply(self.x3_pt, self.y3_pt)
        return normcurve_pt(x0_pt, y0_pt, x1_pt, y1_pt, x2_pt, y2_pt, x3_pt, y3_pt)

    def outputPS(self, file, writer, context):
        file.write("%g %g %g %g %g %g curveto\n" % (self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt))

    def outputPDF(self, file, writer, context):
        file.write("%f %f %f %f %f %f c\n" % (self.x1_pt, self.y1_pt, self.x2_pt, self.y2_pt, self.x3_pt, self.y3_pt))


################################################################################
# normsubpath
################################################################################

class normsubpath:

    """sub path of a normalized path

    A subpath consists of a list of normsubpathitems, i.e., normlines_pt and
    normcurves_pt and can either be closed or not.

    Some invariants, which have to be obeyed:
    - All normsubpathitems have to be longer than epsilon pts.
    - At the end there may be a normline (stored in self.skippedline) whose
      length is shorter than epsilon -- it has to be taken into account
      when adding further normsubpathitems
    - The last point of a normsubpathitem and the first point of the next
      element have to be equal.
    - When the path is closed, the last point of last normsubpathitem has
      to be equal to the first point of the first normsubpathitem.
    """

    __slots__ = "normsubpathitems", "closed", "epsilon", "skippedline"

    def __init__(self, normsubpathitems=[], closed=0, epsilon=None):
        """construct a normsubpath"""
        if epsilon is None:
            epsilon = _epsilon
        self.epsilon = epsilon
        # If one or more items appended to the normsubpath have been
        # skipped (because their total length was shorter than epsilon),
        # we remember this fact by a line because we have to take it
        # properly into account when appending further normsubpathitems
        self.skippedline = None

        self.normsubpathitems = []
        self.closed = 0

        # a test (might be temporary)
        for anormsubpathitem in normsubpathitems:
            assert isinstance(anormsubpathitem, normsubpathitem), "only list of normsubpathitem instances allowed"

        self.extend(normsubpathitems)

        if closed:
            self.close()

    def __getitem__(self, i):
        """return normsubpathitem i"""
        return self.normsubpathitems[i]

    def __len__(self):
        """return number of normsubpathitems"""
        return len(self.normsubpathitems)

    def __str__(self):
        l = ", ".join(map(str, self.normsubpathitems))
        if self.closed:
            return "normsubpath([%s], closed=1)" % l
        else:
            return "normsubpath([%s])" % l

    def _distributeparams(self, params):
        """return a dictionary mapping normsubpathitemindices to a tuple
        of a paramindices and normsubpathitemparams.

        normsubpathitemindex specifies a normsubpathitem containing
        one or several positions.  paramindex specify the index of the
        param in the original list and normsubpathitemparam is the
        parameter value in the normsubpathitem.
        """

        result = {}
        for i, param in enumerate(params):
            if param > 0:
                index = int(param)
                if index > len(self.normsubpathitems) - 1:
                    index = len(self.normsubpathitems) - 1
            else:
                index = 0
            result.setdefault(index, ([], []))
            result[index][0].append(i)
            result[index][1].append(param - index)
        return result

    def append(self, anormsubpathitem):
        """append normsubpathitem

        Fails on closed normsubpath.
        """
        # consitency tests (might be temporary)
        assert isinstance(anormsubpathitem, normsubpathitem), "only normsubpathitem instances allowed"
        if self.skippedline:
            assert math.hypot(*[x-y for x, y in zip(self.skippedline.atend_pt(), anormsubpathitem.atbegin_pt())]) < self.epsilon, "normsubpathitems do not match"
        elif self.normsubpathitems:
            assert math.hypot(*[x-y for x, y in zip(self.normsubpathitems[-1].atend_pt(), anormsubpathitem.atbegin_pt())]) < self.epsilon, "normsubpathitems do not match"

        if self.closed:
            raise PathException("Cannot append to closed normsubpath")

        if self.skippedline:
            xs_pt, ys_pt = self.skippedline.atbegin_pt()
        else:
            xs_pt, ys_pt = anormsubpathitem.atbegin_pt()
        xe_pt, ye_pt = anormsubpathitem.atend_pt()

        if (math.hypot(xe_pt-xs_pt, ye_pt-ys_pt) >= self.epsilon or
            anormsubpathitem.arclen_pt(self.epsilon) >= self.epsilon):
            if self.skippedline:
                anormsubpathitem = anormsubpathitem.modifiedbegin_pt(xs_pt, ys_pt)
            self.normsubpathitems.append(anormsubpathitem)
            self.skippedline = None
        else:
            self.skippedline = normline_pt(xs_pt, ys_pt, xe_pt, ye_pt)

    def arclen_pt(self):
        """return arc length in pts"""
        return sum([npitem.arclen_pt(self.epsilon) for npitem in self.normsubpathitems])

    def _arclentoparam_pt(self, lengths_pt):
        """return a tuple of params and the total length arc length in pts"""
        # work on a copy which is counted down to negative values
        lengths_pt = lengths_pt[:]
        results = [None] * len(lengths_pt)

        totalarclen = 0
        for normsubpathindex, normsubpathitem in enumerate(self.normsubpathitems):
            params, arclen = normsubpathitem._arclentoparam_pt(lengths_pt, self.epsilon)
            for i in range(len(results)):
                if results[i] is None:
                    lengths_pt[i] -= arclen
                    if lengths_pt[i] < 0 or normsubpathindex == len(self.normsubpathitems) - 1:
                        # overwrite the results until the length has become negative
                        results[i] = normsubpathindex + params[i]
            totalarclen += arclen

        return results, totalarclen

    def at_pt(self, params):
        """return coordinates at params in pts"""
        result = [None] * len(params)
        for normsubpathitemindex, (indices, params) in self._distributeparams(params).items():
            for index, point_pt in zip(indices, self.normsubpathitems[normsubpathitemindex].at_pt(params)):
                result[index] = point_pt
        return result

    def atbegin_pt(self):
        """return coordinates of first point in pts"""
        if not self.normsubpathitems and self.skippedline:
            return self.skippedline.atbegin_pt()
        return self.normsubpathitems[0].atbegin_pt()

    def atend_pt(self):
        """return coordinates of last point in pts"""
        if self.skippedline:
            return self.skippedline.atend_pt()
        return self.normsubpathitems[-1].atend_pt()

    def bbox(self):
        """return bounding box of normsubpath"""
        if self.normsubpathitems:
            abbox = self.normsubpathitems[0].bbox()
            for anormpathitem in self.normsubpathitems[1:]:
                abbox += anormpathitem.bbox()
            return abbox
        else:
            return None

    def close(self):
        """close subnormpath

        Fails on closed normsubpath.
        """
        if self.closed:
            raise PathException("Cannot close already closed normsubpath")
        if not self.normsubpathitems:
            if self.skippedline is None:
                raise PathException("Cannot close empty normsubpath")
            else:
                raise PathException("Normsubpath too short, cannot be closed")

        xs_pt, ys_pt = self.normsubpathitems[-1].atend_pt()
        xe_pt, ye_pt = self.normsubpathitems[0].atbegin_pt()
        self.append(normline_pt(xs_pt, ys_pt, xe_pt, ye_pt))

        # the append might have left a skippedline, which we have to remove
        # from the end of the closed path
        if self.skippedline:
            self.normsubpathitems[-1] = self.normsubpathitems[-1].modifiedend_pt(*self.skippedline.atend_pt())
            self.skippedline = None

        self.closed = 1

    def copy(self):
        """return copy of normsubpath"""
        # Since normsubpathitems are never modified inplace, we just
        # need to copy the normsubpathitems list. We do not pass the
        # normsubpathitems to the constructor to not repeat the checks
        # for minimal length of each normsubpathitem.
        result = normsubpath(epsilon=self.epsilon)
        result.normsubpathitems = self.normsubpathitems[:]
        result.closed = self.closed

        # We can share the reference to skippedline, since it is a
        # normsubpathitem as well and thus not modified in place either.
        result.skippedline = self.skippedline

        return result

    def curveradius_pt(self, params):
        """return the curvature radius at params in pts

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""
        result = [None] * len(params)
        for normsubpathitemindex, (indices, params) in self._distributeparams(params).items():
            for index, radius_pt in zip(indices, self.normsubpathitems[normsubpathitemindex].curveradius_pt(params)):
                result[index] = radius_pt
        return result

    def extend(self, normsubpathitems):
        """extend path by normsubpathitems

        Fails on closed normsubpath.
        """
        for normsubpathitem in normsubpathitems:
            self.append(normsubpathitem)

    def intersect(self, other):
        """intersect self with other normsubpath

        Returns a tuple of lists consisting of the parameter values
        of the intersection points of the corresponding normsubpath.
        """
        intersections_a = []
        intersections_b = []
        epsilon = min(self.epsilon, other.epsilon)
        # Intersect all subpaths of self with the subpaths of other, possibly including
        # one intersection point several times
        for t_a, pitem_a  in enumerate(self.normsubpathitems):
            for t_b, pitem_b in enumerate(other.normsubpathitems):
                for intersection_a, intersection_b in pitem_a.intersect(pitem_b, epsilon):
                    intersections_a.append(intersection_a + t_a)
                    intersections_b.append(intersection_b + t_b)

        # although intersectipns_a are sorted for the different normsubpathitems,
        # within a normsubpathitem, the ordering has to be ensured separately:
        intersections = zip(intersections_a, intersections_b)
        intersections.sort()
        intersections_a = [a for a, b in intersections]
        intersections_b = [b for a, b in intersections]

        # for symmetry reasons we enumerate intersections_a as well, although
        # they are already sorted (note we do not need to sort intersections_a)
        intersections_a = zip(intersections_a, range(len(intersections_a)))
        intersections_b = zip(intersections_b, range(len(intersections_b)))
        intersections_b.sort()

        # a helper function to join two normsubpaths
        def joinnormsubpaths(nsp1, nsp2):
            # we do not have closed paths
            assert not nsp1.closed and not nsp2.closed
            result = normsubpath()
            result.normsubpathitems = nsp1.normsubpathitems[:]
            result.epsilon = nsp1.epsilon
            result.skippedline = self.skippedline
            result.extend(nsp2.normsubpathitems)
            if nsp2.skippedline:
                result.append(nsp2.skippedline)
            return result

        # now we search for intersections points which are closer together than epsilon
        # This task is handled by the following function
        def closepoints(normsubpath, intersections):
            split = normsubpath.segments([0] + [intersection for intersection, index in intersections] + [len(normsubpath)])
            result = []
            if normsubpath.closed:
                # note that the number of segments of a closed path is off by one
                # compared to an open path
                i = 0
                while i < len(split):
                    splitnormsubpath = split[i]
                    j = i
                    while not splitnormsubpath.normsubpathitems: # i.e. while "is short"
                        ip1, ip2 = intersections[i-1][1], intersections[j][1]
                        if ip1<ip2:
                            result.append((ip1, ip2))
                        else:
                            result.append((ip2, ip1))
                        j += 1
                        if j == len(split):
                            j = 0
                        if j < len(split):
                            splitnormsubpath = splitnormsubpath.joined(split[j])
                        else:
                            break
                    i += 1
            else:
                i = 1
                while i < len(split)-1:
                    splitnormsubpath = split[i]
                    j = i
                    while not splitnormsubpath.normsubpathitems: # i.e. while "is short"
                        ip1, ip2 = intersections[i-1][1], intersections[j][1]
                        if ip1<ip2:
                            result.append((ip1, ip2))
                        else:
                            result.append((ip2, ip1))
                        j += 1
                        if j < len(split)-1:
                            splitnormsubpath = splitnormsubpath.joined(split[j])
                        else:
                            break
                    i += 1
            return result

        closepoints_a = closepoints(self, intersections_a)
        closepoints_b = closepoints(other, intersections_b)

        # map intersection point to lowest point which is equivalent to the
        # point
        equivalentpoints = list(range(len(intersections_a)))

        for closepoint_a in closepoints_a:
            for closepoint_b in closepoints_b:
                if closepoint_a == closepoint_b:
                    for i in range(closepoint_a[1], len(equivalentpoints)):
                        if equivalentpoints[i] == closepoint_a[1]:
                            equivalentpoints[i] = closepoint_a[0]

        # determine the remaining intersection points
        intersectionpoints = {}
        for point in equivalentpoints:
            intersectionpoints[point] = 1

        # build result
        result = []
        intersectionpointskeys = intersectionpoints.keys()
        intersectionpointskeys.sort()
        for point in intersectionpointskeys:
            for intersection_a, index_a in intersections_a:
                if index_a == point:
                    result_a = intersection_a
            for intersection_b, index_b in intersections_b:
                if index_b == point:
                    result_b = intersection_b
            result.append((result_a, result_b))
        # note that the result is sorted in a, since we sorted
        # intersections_a in the very beginning

        return [x for x, y in result], [y for x, y in result]

    def join(self, other):
        """join other normsubpath inplace

        Fails on closed normsubpath. Fails to join closed normsubpath.
        """
        if other.closed:
            raise PathException("Cannot join closed normsubpath")

        # insert connection line
        x0_pt, y0_pt = self.atend_pt()
        x1_pt, y1_pt = other.atbegin_pt()
        self.append(normline_pt(x0_pt, y0_pt, x1_pt, y1_pt))

        # append other normsubpathitems
        self.extend(other.normsubpathitems)
        if other.skippedline:
            self.append(other.skippedline)

    def joined(self, other):
        """return joined self and other

        Fails on closed normsubpath. Fails to join closed normsubpath.
        """
        result = self.copy()
        result.join(other)
        return result

    def _paramtoarclen_pt(self, params):
        """return a tuple of arc lengths and the total arc length in pts"""
        result = [None] * len(params)
        totalarclen_pt = 0
        distributeparams = self._distributeparams(params)
        for normsubpathitemindex in range(len(self.normsubpathitems)):
            if distributeparams.has_key(normsubpathitemindex):
                indices, params = distributeparams[normsubpathitemindex]
                arclens_pt, normsubpathitemarclen_pt = self.normsubpathitems[normsubpathitemindex]._paramtoarclen_pt(params, self.epsilon)
                for index, arclen_pt in zip(indices, arclens_pt):
                    result[index] = totalarclen_pt + arclen_pt
                totalarclen_pt += normsubpathitemarclen_pt
            else:
                totalarclen_pt += self.normsubpathitems[normsubpathitemindex].arclen_pt(self.epsilon)
        return result, totalarclen_pt

    def pathitems(self):
        """return list of pathitems"""
        if not self.normsubpathitems:
            return []

        # remove trailing normline_pt of closed subpaths
        if self.closed and isinstance(self.normsubpathitems[-1], normline_pt):
            normsubpathitems = self.normsubpathitems[:-1]
        else:
            normsubpathitems = self.normsubpathitems

        result = [moveto_pt(*self.atbegin_pt())]
        for normsubpathitem in normsubpathitems:
            result.append(normsubpathitem.pathitem())
        if self.closed:
            result.append(closepath())
        return result

    def reversed(self):
        """return reversed normsubpath"""
        nnormpathitems = []
        for i in range(len(self.normsubpathitems)):
            nnormpathitems.append(self.normsubpathitems[-(i+1)].reversed())
        return normsubpath(nnormpathitems, self.closed)

    def segments(self, params):
        """return segments of the normsubpath

        The returned list of normsubpaths for the segments between
        the params. params need to contain at least two values.

        For a closed normsubpath the last segment result is joined to
        the first one when params starts with 0 and ends with len(self).
        or params starts with len(self) and ends with 0. Thus a segments
        operation on a closed normsubpath might properly join those the
        first and the last part to take into account the closed nature of
        the normsubpath. However, for intermediate parameters, closepath
        is not taken into account, i.e. when walking backwards you do not
        loop over the closepath forwardly. The special values 0 and
        len(self) for the first and the last parameter should be given as
        integers, i.e. no finite precision is used when checking for
        equality."""

        if len(params) < 2:
            raise ValueError("at least two parameters needed in segments")

        result = [normsubpath(epsilon=self.epsilon)]

        # instead of distribute the parameters, we need to keep their
        # order and collect parameters for the needed segments of
        # normsubpathitem with index collectindex
        collectparams = []
        collectindex = None
        for param in params:
            # calculate index and parameter for corresponding normsubpathitem
            if param > 0:
                index = int(param)
                if index > len(self.normsubpathitems) - 1:
                    index = len(self.normsubpathitems) - 1
                param -= index
            else:
                index = 0
            if index != collectindex:
                if collectindex is not None:
                    # append end point depening on the forthcoming index
                    if index > collectindex:
                        collectparams.append(1)
                    else:
                        collectparams.append(0)
                    # get segments of the normsubpathitem and add them to the result
                    segments = self.normsubpathitems[collectindex].segments(collectparams)
                    result[-1].append(segments[0])
                    result.extend([normsubpath([segment], epsilon=self.epsilon) for segment in segments[1:]])
                    # add normsubpathitems and first segment parameter to close the
                    # gap to the forthcoming index
                    if index > collectindex:
                        for i in range(collectindex+1, index):
                            result[-1].append(self.normsubpathitems[i])
                        collectparams = [0]
                    else:
                        for i in range(collectindex-1, index, -1):
                            result[-1].append(self.normsubpathitems[i].reversed())
                        collectparams = [1]
                collectindex = index
            collectparams.append(param)
        # add remaining collectparams to the result
        segments = self.normsubpathitems[collectindex].segments(collectparams)
        result[-1].append(segments[0])
        result.extend([normsubpath([segment], epsilon=self.epsilon) for segment in segments[1:]])

        if self.closed:
            # join last and first segment together if the normsubpath was
            # originally closed and first and the last parameters are the
            # beginning and end points of the normsubpath
            if ( ( params[0] == 0 and params[-1] == len(self.normsubpathitems) ) or
                 ( params[-1] == 0 and params[0] == len(self.normsubpathitems) ) ):
                result[-1].normsubpathitems.extend(result[0].normsubpathitems)
                result = result[-1:] + result[1:-1]

        return result

    def trafo(self, params):
        """return transformations at params"""
        result = [None] * len(params)
        for normsubpathitemindex, (indices, params) in self._distributeparams(params).items():
            for index, trafo in zip(indices, self.normsubpathitems[normsubpathitemindex].trafo(params)):
                result[index] = trafo
        return result

    def transformed(self, trafo):
        """return transformed path"""
        nnormsubpath = normsubpath(epsilon=self.epsilon)
        for pitem in self.normsubpathitems:
            nnormsubpath.append(pitem.transformed(trafo))
        if self.closed:
            nnormsubpath.close()
        elif self.skippedline is not None:
            nnormsubpath.append(self.skippedline.transformed(trafo))
        return nnormsubpath

    def outputPS(self, file, writer, context):
        # if the normsubpath is closed, we must not output a normline at
        # the end
        if not self.normsubpathitems:
            return
        if self.closed and isinstance(self.normsubpathitems[-1], normline_pt):
            assert len(self.normsubpathitems) > 1, "a closed normsubpath should contain more than a single normline_pt"
            normsubpathitems = self.normsubpathitems[:-1]
        else:
            normsubpathitems = self.normsubpathitems
        file.write("%g %g moveto\n" % self.atbegin_pt())
        for anormsubpathitem in normsubpathitems:
            anormsubpathitem.outputPS(file, writer, context)
        if self.closed:
            file.write("closepath\n")

    def outputPDF(self, file, writer, context):
        # if the normsubpath is closed, we must not output a normline at
        # the end
        if not self.normsubpathitems:
            return
        if self.closed and isinstance(self.normsubpathitems[-1], normline_pt):
            assert len(self.normsubpathitems) > 1, "a closed normsubpath should contain more than a single normline_pt"
            normsubpathitems = self.normsubpathitems[:-1]
        else:
            normsubpathitems = self.normsubpathitems
        file.write("%f %f m\n" % self.atbegin_pt())
        for anormsubpathitem in normsubpathitems:
            anormsubpathitem.outputPDF(file, writer, context)
        if self.closed:
            file.write("h\n")


################################################################################
# normpath
################################################################################

class normpathparam:

    """parameter of a certain point along a normpath"""

    __slots__ = "normpath", "normsubpathindex", "normsubpathparam"

    def __init__(self, normpath, normsubpathindex, normsubpathparam):
        self.normpath = normpath
        self.normsubpathindex = normsubpathindex
        self.normsubpathparam = normsubpathparam
        float(normsubpathparam)

    def __str__(self):
        return "normpathparam(%s, %s, %s)" % (self.normpath, self.normsubpathindex, self.normsubpathparam)

    def __add__(self, other):
        if isinstance(other, normpathparam):
            assert self.normpath is other.normpath, "normpathparams have to belong to the same normpath"
            return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) +
                                                  other.normpath.paramtoarclen_pt(other))
        else:
            return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) + unit.topt(other))

    __radd__ = __add__

    def __sub__(self, other):
        if isinstance(other, normpathparam):
            assert self.normpath is other.normpath, "normpathparams have to belong to the same normpath"
            return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) -
                                                  other.normpath.paramtoarclen_pt(other))
        else:
            return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) - unit.topt(other))

    def __rsub__(self, other):
        # other has to be a length in this case
        return self.normpath.arclentoparam_pt(-self.normpath.paramtoarclen_pt(self) + unit.topt(other))

    def __mul__(self, factor):
        return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) * factor)

    __rmul__ = __mul__

    def __div__(self, divisor):
        return self.normpath.arclentoparam_pt(self.normpath.paramtoarclen_pt(self) / divisor)

    def __neg__(self):
        return self.normpath.arclentoparam_pt(-self.normpath.paramtoarclen_pt(self))

    def __cmp__(self, other):
        if isinstance(other, normpathparam):
            assert self.normpath is other.normpath, "normpathparams have to belong to the same normpath"
            return cmp((self.normsubpathindex, self.normsubpathparam), (other.normsubpathindex, other.normsubpathparam))
        else:
            return cmp(self.normpath.paramtoarclen_pt(self), unit.topt(other))

    def arclen_pt(self):
        """return arc length in pts corresponding to the normpathparam """
        return self.normpath.paramtoarclen_pt(self)

    def arclen(self):
        """return arc length corresponding to the normpathparam """
        return self.normpath.paramtoarclen(self)


def _valueorlistmethod(method):
    """Creates a method which takes a single argument or a list and
    returns a single value or a list out of method, which always
    works on lists."""

    def wrappedmethod(self, valueorlist, *args, **kwargs):
        try:
            for item in valueorlist:
                break
        except:
            return method(self, [valueorlist], *args, **kwargs)[0]
        return method(self, valueorlist, *args, **kwargs)
    return wrappedmethod


class normpath(canvas.canvasitem):

    """normalized path

    A normalized path consists of a list of normsubpaths.
    """

    def __init__(self, normsubpaths=None):
        """construct a normpath from a list of normsubpaths"""

        if normsubpaths is None:
            self.normsubpaths = [] # make a fresh list
        else:
            self.normsubpaths = normsubpaths
            for subpath in normsubpaths:
                assert isinstance(subpath, normsubpath), "only list of normsubpath instances allowed"

    def __add__(self, other):
        """create new normpath out of self and other"""
        result = self.copy()
        result += other
        return result

    def __iadd__(self, other):
        """add other inplace"""
        for normsubpath in other.normpath().normsubpaths:
            self.normsubpaths.append(normsubpath.copy())
        return self

    def __getitem__(self, i):
        """return normsubpath i"""
        return self.normsubpaths[i]

    def __len__(self):
        """return the number of normsubpaths"""
        return len(self.normsubpaths)

    def __str__(self):
        return "normpath([%s])" % ", ".join(map(str, self.normsubpaths))

    def _convertparams(self, params, convertmethod):
        """return params with all non-normpathparam arguments converted by convertmethod

        usecases:
        - self._convertparams(params, self.arclentoparam_pt)
        - self._convertparams(params, self.arclentoparam)
        """

        converttoparams = []
        convertparamindices = []
        for i, param in enumerate(params):
            if not isinstance(param, normpathparam):
                converttoparams.append(param)
                convertparamindices.append(i)
        if converttoparams:
            params = params[:]
            for i, param in zip(convertparamindices, convertmethod(converttoparams)):
                params[i] = param
        return params

    def _distributeparams(self, params):
        """return a dictionary mapping subpathindices to a tuple of a paramindices and subpathparams

        subpathindex specifies a subpath containing one or several positions.
        paramindex specify the index of the normpathparam in the original list and
        subpathparam is the parameter value in the subpath.
        """

        result = {}
        for i, param in enumerate(params):
            assert param.normpath is self, "normpathparam has to belong to this path"
            result.setdefault(param.normsubpathindex, ([], []))
            result[param.normsubpathindex][0].append(i)
            result[param.normsubpathindex][1].append(param.normsubpathparam)
        return result

    def append(self, anormsubpath):
        """append a normsubpath by a normsubpath or a pathitem"""
        if isinstance(anormsubpath, normsubpath):
            # the normsubpaths list can be appended by a normsubpath only
            self.normsubpaths.append(anormsubpath)
        else:
            # ... but we are kind and allow for regular path items as well
            # in order to make a normpath to behave more like a regular path

            for pathitem in anormsubpath._normalized(_currentpoint(*self.normsubpaths[-1].atend_pt())):
                if isinstance(pathitem, closepath):
                    self.normsubpaths[-1].close()
                elif isinstance(pathitem, moveto_pt):
                    self.normsubpaths.append(normsubpath([normline_pt(pathitem.x_pt, pathitem.y_pt,
                                                                   pathitem.x_pt, pathitem.y_pt)]))
                else:
                    self.normsubpaths[-1].append(pathitem)

    def arclen_pt(self):
        """return arc length in pts"""
        return sum([normsubpath.arclen_pt() for normsubpath in self.normsubpaths])

    def arclen(self):
        """return arc length"""
        return self.arclen_pt() * unit.t_pt

    def _arclentoparam_pt(self, lengths_pt):
        """return the params matching the given lengths_pt"""
        # work on a copy which is counted down to negative values
        lengths_pt = lengths_pt[:]
        results = [None] * len(lengths_pt)

        for normsubpathindex, normsubpath in enumerate(self.normsubpaths):
            params, arclen = normsubpath._arclentoparam_pt(lengths_pt)
            done = 1
            for i, result in enumerate(results):
                if results[i] is None:
                    lengths_pt[i] -= arclen
                    if lengths_pt[i] < 0 or normsubpathindex == len(self.normsubpaths) - 1:
                        # overwrite the results until the length has become negative
                        results[i] = normpathparam(self, normsubpathindex, params[i])
                    done = 0
            if done:
                break

        return results

    def arclentoparam_pt(self, lengths_pt):
        """return the param(s) matching the given length(s)_pt in pts"""
        pass
    arclentoparam_pt = _valueorlistmethod(_arclentoparam_pt)

    def arclentoparam(self, lengths):
        """return the param(s) matching the given length(s)"""
        return self._arclentoparam_pt([unit.topt(l) for l in lengths])
    arclentoparam = _valueorlistmethod(arclentoparam)

    def _at_pt(self, params):
        """return coordinates of normpath in pts at params"""
        result = [None] * len(params)
        for normsubpathindex, (indices, params) in self._distributeparams(params).items():
            for index, point_pt in zip(indices, self.normsubpaths[normsubpathindex].at_pt(params)):
                result[index] = point_pt
        return result

    def at_pt(self, params):
        """return coordinates of normpath in pts at param(s) or lengths in pts"""
        return self._at_pt(self._convertparams(params, self.arclentoparam_pt))
    at_pt = _valueorlistmethod(at_pt)

    def at(self, params):
        """return coordinates of normpath at param(s) or arc lengths"""
        return [(x_pt * unit.t_pt, y_pt * unit.t_pt)
                for x_pt, y_pt in self._at_pt(self._convertparams(params, self.arclentoparam))]
    at = _valueorlistmethod(at)

    def atbegin_pt(self):
        """return coordinates of the beginning of first subpath in normpath in pts"""
        if self.normsubpaths:
            return self.normsubpaths[0].atbegin_pt()
        else:
            raise PathException("cannot return first point of empty path")

    def atbegin(self):
        """return coordinates of the beginning of first subpath in normpath"""
        x, y = self.atbegin_pt()
        return x * unit.t_pt, y * unit.t_pt

    def atend_pt(self):
        """return coordinates of the end of last subpath in normpath in pts"""
        if self.normsubpaths:
            return self.normsubpaths[-1].atend_pt()
        else:
            raise PathException("cannot return last point of empty path")

    def atend(self):
        """return coordinates of the end of last subpath in normpath"""
        x, y = self.atend_pt()
        return x * unit.t_pt, y * unit.t_pt

    def bbox(self):
        """return bbox of normpath"""
        abbox = None
        for normsubpath in self.normsubpaths:
            nbbox =  normsubpath.bbox()
            if abbox is None:
                abbox = nbbox
            elif nbbox:
                abbox += nbbox
        return abbox

    def begin(self):
        """return param corresponding of the beginning of the normpath"""
        if self.normsubpaths:
            return normpathparam(self, 0, 0)
        else:
            raise PathException("empty path")

    def copy(self):
        """return copy of normpath"""
        result = normpath()
        for normsubpath in self.normsubpaths:
            result.append(normsubpath.copy())
        return result

    def _curveradius_pt(self, params):
        """return the curvature radius at params in pts

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""

        result = [None] * len(params)
        for normsubpathindex, (indices, params) in self._distributeparams(params).items():
            for index, radius_pt in zip(indices, self.normsubpaths[normsubpathindex].curveradius_pt(params)):
                result[index] = radius_pt
        return result

    def curveradius_pt(self, params):
        """return the curvature radius in pts at param(s) or arc length(s) in pts

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""

        return self._curveradius_pt(self._convertparams(params, self.arclentoparam_pt))
    curveradius_pt = _valueorlistmethod(curveradius_pt)

    def curveradius(self, params):
        """return the curvature radius at param(s) or arc length(s)

        The curvature radius is the inverse of the curvature. When the
        curvature is 0, None is returned. Note that this radius can be negative
        or positive, depending on the sign of the curvature."""

        result = []
        for radius_pt in self._curveradius_pt(self._convertparams(params, self.arclentoparam)):
            if radius_pt is not None:
                result.append(radius_pt * unit.t_pt)
            else:
                result.append(None)
        return result
    curveradius = _valueorlistmethod(curveradius)

    def end(self):
        """return param corresponding of the end of the path"""
        if self.normsubpaths:
            return normpathparam(self, len(self)-1, len(self.normsubpaths[-1]))
        else:
            raise PathException("empty path")

    def extend(self, normsubpaths):
        """extend path by normsubpaths or pathitems"""
        for anormsubpath in normsubpaths:
            # use append to properly handle regular path items as well as normsubpaths
            self.append(anormsubpath)

    def intersect(self, other):
        """intersect self with other path

        Returns a tuple of lists consisting of the parameter values
        of the intersection points of the corresponding normpath.
        """
        other = other.normpath()

        # here we build up the result
        intersections = ([], [])

        # Intersect all normsubpaths of self with the normsubpaths of
        # other.
        for ia, normsubpath_a in enumerate(self.normsubpaths):
            for ib, normsubpath_b in enumerate(other.normsubpaths):
                for intersection in zip(*normsubpath_a.intersect(normsubpath_b)):
                    intersections[0].append(normpathparam(self, ia, intersection[0]))
                    intersections[1].append(normpathparam(other, ib, intersection[1]))
        return intersections

    def join(self, other):
        """join other normsubpath inplace

        Both normpaths must contain at least one normsubpath.
        The last normsubpath of self will be joined to the first
        normsubpath of other.
        """
        if not self.normsubpaths:
            raise PathException("cannot join to empty path")
        if not other.normsubpaths:
            raise PathException("cannot join empty path")
        self.normsubpaths[-1].join(other.normsubpaths[0])
        self.normsubpaths.extend(other.normsubpaths[1:])

    def joined(self, other):
        """return joined self and other

        Both normpaths must contain at least one normsubpath.
        The last normsubpath of self will be joined to the first
        normsubpath of other.
        """
        result = self.copy()
        result.join(other.normpath())
        return result

    # << operator also designates joining
    __lshift__ = joined

    def normpath(self):
        """return a normpath, i.e. self"""
        return self

    def _paramtoarclen_pt(self, params):
        """return arc lengths in pts matching the given params"""
        result = [None] * len(params)
        totalarclen_pt = 0
        distributeparams = self._distributeparams(params)
        for normsubpathindex in range(max(distributeparams.keys()) + 1):
            if distributeparams.has_key(normsubpathindex):
                indices, params = distributeparams[normsubpathindex]
                arclens_pt, normsubpatharclen_pt = self.normsubpaths[normsubpathindex]._paramtoarclen_pt(params)
                for index, arclen_pt in zip(indices, arclens_pt):
                    result[index] = totalarclen_pt + arclen_pt
                totalarclen_pt += normsubpatharclen_pt
            else:
                totalarclen_pt += self.normsubpaths[normsubpathindex].arclen_pt()
        return result

    def paramtoarclen_pt(self, params):
        """return arc length(s) in pts matching the given param(s)"""
    paramtoarclen_pt = _valueorlistmethod(_paramtoarclen_pt)

    def paramtoarclen(self, params):
        """return arc length(s) matching the given param(s)"""
        return [arclen_pt * unit.t_pt for arclen_pt in self._paramtoarclen_pt(params)]
    paramtoarclen = _valueorlistmethod(paramtoarclen)

    def path(self):
        """return path corresponding to normpath"""
        pathitems = []
        for normsubpath in self.normsubpaths:
            pathitems.extend(normsubpath.pathitems())
        return path(*pathitems)

    def reversed(self):
        """return reversed path"""
        nnormpath = normpath()
        for i in range(len(self.normsubpaths)):
            nnormpath.normsubpaths.append(self.normsubpaths[-(i+1)].reversed())
        return nnormpath

    def _split_pt(self, params):
        """split path at params and return list of normpaths"""

        # instead of distributing the parameters, we need to keep their
        # order and collect parameters for splitting of normsubpathitem
        # with index collectindex
        collectindex = None
        for param in params:
            if param.normsubpathindex != collectindex:
                if collectindex is not None:
                    # append end point depening on the forthcoming index
                    if param.normsubpathindex > collectindex:
                        collectparams.append(len(self.normsubpaths[collectindex]))
                    else:
                        collectparams.append(0)
                    # get segments of the normsubpath and add them to the result
                    segments = self.normsubpaths[collectindex].segments(collectparams)
                    result[-1].append(segments[0])
                    result.extend([normpath([segment]) for segment in segments[1:]])
                    # add normsubpathitems and first segment parameter to close the
                    # gap to the forthcoming index
                    if param.normsubpathindex > collectindex:
                        for i in range(collectindex+1, param.normsubpathindex):
                            result[-1].append(self.normsubpaths[i])
                        collectparams = [0]
                    else:
                        for i in range(collectindex-1, param.normsubpathindex, -1):
                            result[-1].append(self.normsubpaths[i].reversed())
                        collectparams = [len(self.normsubpaths[param.normsubpathindex])]
                else:
                    result = [normpath(self.normsubpaths[:param.normsubpathindex])]
                    collectparams = [0]
                collectindex = param.normsubpathindex
            collectparams.append(param.normsubpathparam)
        # add remaining collectparams to the result
        collectparams.append(len(self.normsubpaths[collectindex]))
        segments = self.normsubpaths[collectindex].segments(collectparams)
        result[-1].append(segments[0])
        result.extend([normpath([segment]) for segment in segments[1:]])
        result[-1].extend(self.normsubpaths[collectindex+1:])
        return result

    def split_pt(self, params):
        """split path at param(s) or arc length(s) in pts and return list of normpaths"""
        try:
            for param in params:
                break
        except:
            params = [params]
        return self._split_pt(self._convertparams(params, self.arclentoparam_pt))

    def split(self, params):
        """split path at param(s) or arc length(s) and return list of normpaths"""
        try:
            for param in params:
                break
        except:
            params = [params]
        return self._split_pt(self._convertparams(params, self.arclentoparam))

    def _tangent(self, params, length=None):
        """return tangent vector of path at params

        If length is not None, the tangent vector will be scaled to
        the desired length.
        """

        result = [None] * len(params)
        tangenttemplate = line_pt(0, 0, 1, 0).normpath()
        for normsubpathindex, (indices, params) in self._distributeparams(params).items():
            for index, atrafo in zip(indices, self.normsubpaths[normsubpathindex].trafo(params)):
                tangentpath = tangenttemplate.transformed(atrafo)
                if length is not None:
                    sfactor = unit.topt(length)/tangentpath.arclen_pt()
                    tangentpath = tangentpath.transformed(trafo.scale_pt(sfactor, sfactor, *tangentpath.atbegin_pt()))
                result[index] = tangentpath
        return result

    def tangent_pt(self, params, length=None):
        """return tangent vector of path at param(s) or arc length(s) in pts

        If length in pts is not None, the tangent vector will be scaled to
        the desired length.
        """
        return self._tangent(self._convertparams(params, self.arclentoparam_pt), length)
    tangent_pt = _valueorlistmethod(tangent_pt)

    def tangent(self, params, length=None):
        """return tangent vector of path at param(s) or arc length(s)

        If length is not None, the tangent vector will be scaled to
        the desired length.
        """
        return self._tangent(self._convertparams(params, self.arclentoparam), length)
    tangent = _valueorlistmethod(tangent)

    def _trafo(self, params):
        """return transformation at params"""
        result = [None] * len(params)
        for normsubpathindex, (indices, params) in self._distributeparams(params).items():
            for index, trafo in zip(indices, self.normsubpaths[normsubpathindex].trafo(params)):
                result[index] = trafo
        return result

    def trafo_pt(self, params):
        """return transformation at param(s) or arc length(s) in pts"""
        return self._trafo(self._convertparams(params, self.arclentoparam_pt))
    trafo_pt = _valueorlistmethod(trafo_pt)

    def trafo(self, params):
        """return transformation at param(s) or arc length(s)"""
        return self._trafo(self._convertparams(params, self.arclentoparam))
    trafo = _valueorlistmethod(trafo)

    def transformed(self, trafo):
        """return transformed normpath"""
        return normpath([normsubpath.transformed(trafo) for normsubpath in self.normsubpaths])

    def outputPS(self, file, writer, context):
        for normsubpath in self.normsubpaths:
            normsubpath.outputPS(file, writer, context)

    def outputPDF(self, file, writer, context):
        for normsubpath in self.normsubpaths:
            normsubpath.outputPDF(file, writer, context)