2 * SGI FREE SOFTWARE LICENSE B (Version 2.0, Sept. 18, 2008)
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10 * Software is furnished to do so, subject to the following conditions:
12 * The above copyright notice including the dates of first publication and
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14 * http://oss.sgi.com/projects/FreeB/
15 * shall be included in all copies or substantial portions of the Software.
17 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
18 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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21 * WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF
22 * OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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28 * Silicon Graphics, Inc.
31 ** Author: Eric Veach, July 1994.
43 int __gl_vertLeq( GLUvertex
*u
, GLUvertex
*v
)
45 /* Returns TRUE if u is lexicographically <= v. */
47 return VertLeq( u
, v
);
50 GLdouble
__gl_edgeEval( GLUvertex
*u
, GLUvertex
*v
, GLUvertex
*w
)
52 /* Given three vertices u,v,w such that VertLeq(u,v) && VertLeq(v,w),
53 * evaluates the t-coord of the edge uw at the s-coord of the vertex v.
54 * Returns v->t - (uw)(v->s), i.e. the signed distance from uw to v.
55 * If uw is vertical (and thus passes through v), the result is zero.
57 * The calculation is extremely accurate and stable, even when v
58 * is very close to u or w. In particular if we set v->t = 0 and
59 * let r be the negated result (this evaluates (uw)(v->s)), then
60 * r is guaranteed to satisfy MIN(u->t,w->t) <= r <= MAX(u->t,w->t).
64 assert( VertLeq( u
, v
) && VertLeq( v
, w
));
69 if( gapL
+ gapR
> 0 ) {
71 return (v
->t
- u
->t
) + (u
->t
- w
->t
) * (gapL
/ (gapL
+ gapR
));
73 return (v
->t
- w
->t
) + (w
->t
- u
->t
) * (gapR
/ (gapL
+ gapR
));
80 GLdouble
__gl_edgeSign( GLUvertex
*u
, GLUvertex
*v
, GLUvertex
*w
)
82 /* Returns a number whose sign matches EdgeEval(u,v,w) but which
83 * is cheaper to evaluate. Returns > 0, == 0 , or < 0
84 * as v is above, on, or below the edge uw.
88 assert( VertLeq( u
, v
) && VertLeq( v
, w
));
93 if( gapL
+ gapR
> 0 ) {
94 return (v
->t
- w
->t
) * gapL
+ (v
->t
- u
->t
) * gapR
;
101 /***********************************************************************
102 * Define versions of EdgeSign, EdgeEval with s and t transposed.
105 GLdouble
__gl_transEval( GLUvertex
*u
, GLUvertex
*v
, GLUvertex
*w
)
107 /* Given three vertices u,v,w such that TransLeq(u,v) && TransLeq(v,w),
108 * evaluates the t-coord of the edge uw at the s-coord of the vertex v.
109 * Returns v->s - (uw)(v->t), i.e. the signed distance from uw to v.
110 * If uw is vertical (and thus passes through v), the result is zero.
112 * The calculation is extremely accurate and stable, even when v
113 * is very close to u or w. In particular if we set v->s = 0 and
114 * let r be the negated result (this evaluates (uw)(v->t)), then
115 * r is guaranteed to satisfy MIN(u->s,w->s) <= r <= MAX(u->s,w->s).
119 assert( TransLeq( u
, v
) && TransLeq( v
, w
));
124 if( gapL
+ gapR
> 0 ) {
126 return (v
->s
- u
->s
) + (u
->s
- w
->s
) * (gapL
/ (gapL
+ gapR
));
128 return (v
->s
- w
->s
) + (w
->s
- u
->s
) * (gapR
/ (gapL
+ gapR
));
135 GLdouble
__gl_transSign( GLUvertex
*u
, GLUvertex
*v
, GLUvertex
*w
)
137 /* Returns a number whose sign matches TransEval(u,v,w) but which
138 * is cheaper to evaluate. Returns > 0, == 0 , or < 0
139 * as v is above, on, or below the edge uw.
143 assert( TransLeq( u
, v
) && TransLeq( v
, w
));
148 if( gapL
+ gapR
> 0 ) {
149 return (v
->s
- w
->s
) * gapL
+ (v
->s
- u
->s
) * gapR
;
156 int __gl_vertCCW( GLUvertex
*u
, GLUvertex
*v
, GLUvertex
*w
)
158 /* For almost-degenerate situations, the results are not reliable.
159 * Unless the floating-point arithmetic can be performed without
160 * rounding errors, *any* implementation will give incorrect results
161 * on some degenerate inputs, so the client must have some way to
162 * handle this situation.
164 return (u
->s
*(v
->t
- w
->t
) + v
->s
*(w
->t
- u
->t
) + w
->s
*(u
->t
- v
->t
)) >= 0;
167 /* Given parameters a,x,b,y returns the value (b*x+a*y)/(a+b),
168 * or (x+y)/2 if a==b==0. It requires that a,b >= 0, and enforces
169 * this in the rare case that one argument is slightly negative.
170 * The implementation is extremely stable numerically.
171 * In particular it guarantees that the result r satisfies
172 * MIN(x,y) <= r <= MAX(x,y), and the results are very accurate
173 * even when a and b differ greatly in magnitude.
175 #define Interpolate(a,x,b,y) \
176 (a = (a < 0) ? 0 : a, b = (b < 0) ? 0 : b, \
177 ((a <= b) ? ((b == 0) ? ((x+y) / 2) \
178 : (x + (y-x) * (a/(a+b)))) \
179 : (y + (x-y) * (b/(a+b)))))
181 #define Swap(a,b) do { GLUvertex *t = a; a = b; b = t; } while (0)
183 void __gl_edgeIntersect( GLUvertex
*o1
, GLUvertex
*d1
,
184 GLUvertex
*o2
, GLUvertex
*d2
,
186 /* Given edges (o1,d1) and (o2,d2), compute their point of intersection.
187 * The computed point is guaranteed to lie in the intersection of the
188 * bounding rectangles defined by each edge.
193 /* This is certainly not the most efficient way to find the intersection
194 * of two line segments, but it is very numerically stable.
196 * Strategy: find the two middle vertices in the VertLeq ordering,
197 * and interpolate the intersection s-value from these. Then repeat
198 * using the TransLeq ordering to find the intersection t-value.
201 if( ! VertLeq( o1
, d1
)) { Swap( o1
, d1
); }
202 if( ! VertLeq( o2
, d2
)) { Swap( o2
, d2
); }
203 if( ! VertLeq( o1
, o2
)) { Swap( o1
, o2
); Swap( d1
, d2
); }
205 if( ! VertLeq( o2
, d1
)) {
206 /* Technically, no intersection -- do our best */
207 v
->s
= (o2
->s
+ d1
->s
) / 2;
208 } else if( VertLeq( d1
, d2
)) {
209 /* Interpolate between o2 and d1 */
210 z1
= EdgeEval( o1
, o2
, d1
);
211 z2
= EdgeEval( o2
, d1
, d2
);
212 if( z1
+z2
< 0 ) { z1
= -z1
; z2
= -z2
; }
213 v
->s
= Interpolate( z1
, o2
->s
, z2
, d1
->s
);
215 /* Interpolate between o2 and d2 */
216 z1
= EdgeSign( o1
, o2
, d1
);
217 z2
= -EdgeSign( o1
, d2
, d1
);
218 if( z1
+z2
< 0 ) { z1
= -z1
; z2
= -z2
; }
219 v
->s
= Interpolate( z1
, o2
->s
, z2
, d2
->s
);
222 /* Now repeat the process for t */
224 if( ! TransLeq( o1
, d1
)) { Swap( o1
, d1
); }
225 if( ! TransLeq( o2
, d2
)) { Swap( o2
, d2
); }
226 if( ! TransLeq( o1
, o2
)) { Swap( o1
, o2
); Swap( d1
, d2
); }
228 if( ! TransLeq( o2
, d1
)) {
229 /* Technically, no intersection -- do our best */
230 v
->t
= (o2
->t
+ d1
->t
) / 2;
231 } else if( TransLeq( d1
, d2
)) {
232 /* Interpolate between o2 and d1 */
233 z1
= TransEval( o1
, o2
, d1
);
234 z2
= TransEval( o2
, d1
, d2
);
235 if( z1
+z2
< 0 ) { z1
= -z1
; z2
= -z2
; }
236 v
->t
= Interpolate( z1
, o2
->t
, z2
, d1
->t
);
238 /* Interpolate between o2 and d2 */
239 z1
= TransSign( o1
, o2
, d1
);
240 z2
= -TransSign( o1
, d2
, d1
);
241 if( z1
+z2
< 0 ) { z1
= -z1
; z2
= -z2
; }
242 v
->t
= Interpolate( z1
, o2
->t
, z2
, d2
->t
);