1/*
2*/
3
4General Polygon Tesselation
5---------------------------
6
7 This note describes a tesselator for polygons consisting of one or
8 more closed contours. It is backward-compatible with the current
9 OpenGL Utilities tesselator, and is intended to replace it. Here is
10 a summary of the major differences:
11
12 - input contours can be intersecting, self-intersecting, or degenerate.
13
14 - supports a choice of several winding rules for determining which parts
15 of the polygon are on the "interior". This makes it possible to do
16 CSG operations on polygons.
17
18 - boundary extraction: instead of tesselating the polygon, returns a
19 set of closed contours which separate the interior from the exterior.
20
21 - returns the output as a small number of triangle fans and strips,
22 rather than a list of independent triangles (when possible).
23
24 - output is available as an explicit mesh (a quad-edge structure),
25 in addition to the normal callback interface.
26
27 - the algorithm used is extremely robust.
28
29
30The interface
31-------------
32
33 The tesselator state is maintained in a "tesselator object".
34 These are allocated and destroyed using
35
36 GLUtesselator *gluNewTess( void );
37 void gluDeleteTess( GLUtesselator *tess );
38
39 Several tesselator objects may be used simultaneously.
40
41 Inputs
42 ------
43
44 The input contours are specified with the following routines:
45
46 void gluTessBeginPolygon( GLUtesselator *tess );
47 void gluTessBeginContour( GLUtesselator *tess );
48 void gluTessVertex( GLUtesselator *tess, GLUcoord coords[3], void *data );
49 void gluTessEndContour( GLUtesselator *tess );
50 void gluTessEndPolygon( GLUtesselator *tess );
51
52 Within each BeginPolygon/EndPolygon pair, there can be zero or more
53 calls to BeginContour/EndContour. Within each contour, there are zero
54 or more calls to gluTessVertex(). The vertices specify a closed
55 contour (the last vertex of each contour is automatically linked to
56 the first).
57
58 "coords" give the coordinates of the vertex in 3-space. For useful
59 results, all vertices should lie in some plane, since the vertices
60 are projected onto a plane before tesselation. "data" is a pointer
61 to a user-defined vertex structure, which typically contains other
62 information such as color, texture coordinates, normal, etc. It is
63 used to refer to the vertex during rendering.
64
65 The library can be compiled in single- or double-precision; the type
66 GLUcoord represents either "float" or "double" accordingly. The GLU
67 version will be available in double-precision only. Compile with
68 GLU_TESS_API_FLOAT defined to get the single-precision version.
69
70 When EndPolygon is called, the tesselation algorithm determines
71 which regions are interior to the given contours, according to one
72 of several "winding rules" described below. The interior regions
73 are then tesselated, and the output is provided as callbacks.
74
75
76 Rendering Callbacks
77 -------------------
78
79 Callbacks are specified by the client using
80
81 void gluTessCallback( GLUtesselator *tess, GLenum which, void (*fn)());
82
83 If "fn" is NULL, any previously defined callback is discarded.
84
85 The callbacks used to provide output are: /* which == */
86
87 void begin( GLenum type ); /* GLU_TESS_BEGIN */
88 void edgeFlag( GLboolean flag ); /* GLU_TESS_EDGE_FLAG */
89 void vertex( void *data ); /* GLU_TESS_VERTEX */
90 void end( void ); /* GLU_TESS_END */
91
92 Any of the callbacks may be left undefined; if so, the corresponding
93 information will not be supplied during rendering.
94
95 The "begin" callback indicates the start of a primitive; type is one
96 of GL_TRIANGLE_STRIP, GL_TRIANGLE_FAN, or GL_TRIANGLES (but see the
97 notes on "boundary extraction" below).
98
99 It is followed by any number of "vertex" callbacks, which supply the
100 vertices in the same order as expected by the corresponding glBegin()
101 call. After the last vertex of a given primitive, there is a callback
102 to "end".
103
104 If the "edgeFlag" callback is provided, no triangle fans or strips
105 will be used. When edgeFlag is called, if "flag" is GL_TRUE then each
106 vertex which follows begins an edge which lies on the polygon boundary
107 (ie. an edge which separates an interior region from an exterior one).
108 If "flag" is GL_FALSE, each vertex which follows begins an edge which lies
109 in the polygon interior. "edgeFlag" will be called before the first
110 call to "vertex".
111
112 Other Callbacks
113 ---------------
114
115 void mesh( GLUmesh *mesh ); /* GLU_TESS_MESH */
116
117 - Returns an explicit mesh, represented using the quad-edge structure
118 (Guibas/Stolfi '85). Other implementations of this interface might
119 use a different mesh structure, so this is available only only as an
120 SGI extension. When the mesh is no longer needed, it should be freed
121 using
122
123 void gluDeleteMesh( GLUmesh *mesh );
124
125 There is a brief description of this data structure in the include
126 file "mesh.h". For the full details, see L. Guibas and J. Stolfi,
127 Primitives for the manipulation of general subdivisions and the
128 computation of Voronoi diagrams, ACM Transactions on Graphics,
129 4(2):74-123, April 1985. For an introduction, see the course notes
130 for CS348a, "Mathematical Foundations of Computer Graphics",
131 available at the Stanford bookstore (and taught during the fall
132 quarter).
133
134 void error( GLenum errno ); /* GLU_TESS_ERROR */
135
136 - errno is one of GLU_TESS_MISSING_BEGIN_POLYGON,
137 GLU_TESS_MISSING_END_POLYGON,
138 GLU_TESS_MISSING_BEGIN_CONTOUR,
139 GLU_TESS_MISSING_END_CONTOUR,
140 GLU_TESS_COORD_TOO_LARGE,
141 GLU_TESS_NEED_COMBINE_CALLBACK
142
143 The first four are obvious. The interface recovers from these
144 errors by inserting the missing call(s).
145
146 GLU_TESS_COORD_TOO_LARGE says that some vertex coordinate exceeded
147 the predefined constant GLU_TESS_MAX_COORD in absolute value, and
148 that the value has been clamped. (Coordinate values must be small
149 enough so that two can be multiplied together without overflow.)
150
151 GLU_TESS_NEED_COMBINE_CALLBACK says that the algorithm detected an
152 intersection between two edges in the input data, and the "combine"
153 callback (below) was not provided. No output will be generated.
154
155
156 void combine( GLUcoord coords[3], void *data[4], /* GLU_TESS_COMBINE */
157 GLUcoord weight[4], void **outData );
158
159 - When the algorithm detects an intersection, or wishes to merge
160 features, it needs to create a new vertex. The vertex is defined
161 as a linear combination of up to 4 existing vertices, referenced
162 by data[0..3]. The coefficients of the linear combination are
163 given by weight[0..3]; these weights always sum to 1.0. All vertex
164 pointers are valid even when some of the weights are zero.
165 "coords" gives the location of the new vertex.
166
167 The user must allocate another vertex, interpolate parameters
168 using "data" and "weights", and return the new vertex pointer in
169 "outData". This handle is supplied during rendering callbacks.
170 For example, if the polygon lies in an arbitrary plane in 3-space,
171 and we associate a color with each vertex, the combine callback might
172 look like this:
173
174 void myCombine( GLUcoord coords[3], VERTEX *d[4],
175 GLUcoord w[4], VERTEX **dataOut )
176 {
177 VERTEX *new = new_vertex();
178
179 new->x = coords[0];
180 new->y = coords[1];
181 new->z = coords[2];
182 new->r = w[0]*d[0]->r + w[1]*d[1]->r + w[2]*d[2]->r + w[3]*d[3]->r;
183 new->g = w[0]*d[0]->g + w[1]*d[1]->g + w[2]*d[2]->g + w[3]*d[3]->g;
184 new->b = w[0]*d[0]->b + w[1]*d[1]->b + w[2]*d[2]->b + w[3]*d[3]->b;
185 new->a = w[0]*d[0]->a + w[1]*d[1]->a + w[2]*d[2]->a + w[3]*d[3]->a;
186 *dataOut = new;
187 }
188
189 If the algorithm detects an intersection, then the "combine" callback
190 must be defined, and must write a non-NULL pointer into "dataOut".
191 Otherwise the GLU_TESS_NEED_COMBINE_CALLBACK error occurs, and no
192 output is generated. This is the only error that can occur during
193 tesselation and rendering.
194
195
196 Control over Tesselation
197 ------------------------
198
199 void gluTessProperty( GLUtesselator *tess, GLenum which, GLUcoord value );
200
201 Properties defined:
202
203 - GLU_TESS_WINDING_RULE. Possible values:
204
205 GLU_TESS_WINDING_ODD
206 GLU_TESS_WINDING_NONZERO
207 GLU_TESS_WINDING_POSITIVE
208 GLU_TESS_WINDING_NEGATIVE
209 GLU_TESS_WINDING_ABS_GEQ_TWO
210
211 The input contours parition the plane into regions. A winding
212 rule determines which of these regions are inside the polygon.
213
214 For a single contour C, the winding number of a point x is simply
215 the signed number of revolutions we make around x as we travel
216 once around C (where CCW is positive). When there are several
217 contours, the individual winding numbers are summed. This
218 procedure associates a signed integer value with each point x in
219 the plane. Note that the winding number is the same for all
220 points in a single region.
221
222 The winding rule classifies a region as "inside" if its winding
223 number belongs to the chosen category (odd, nonzero, positive,
224 negative, or absolute value of at least two). The current GLU
225 tesselator implements the "odd" rule. The "nonzero" rule is another
226 common way to define the interior. The other three rules are
227 useful for polygon CSG operations (see below).
228
229 - GLU_TESS_BOUNDARY_ONLY. Values: TRUE (non-zero) or FALSE (zero).
230
231 If TRUE, returns a set of closed contours which separate the
232 polygon interior and exterior (rather than a tesselation).
233 Exterior contours are oriented CCW with respect to the normal,
234 interior contours are oriented CW. The GLU_TESS_BEGIN callback
235 uses the type GL_LINE_LOOP for each contour.
236
237 - GLU_TESS_TOLERANCE. Value: a real number between 0.0 and 1.0.
238
239 This specifies a tolerance for merging features to reduce the size
240 of the output. For example, two vertices which are very close to
241 each other might be replaced by a single vertex. The tolerance
242 is multiplied by the largest coordinate magnitude of any input vertex;
243 this specifies the maximum distance that any feature can move as the
244 result of a single merge operation. If a single feature takes part
245 in several merge operations, the total distance moved could be larger.
246
247 Feature merging is completely optional; the tolerance is only a hint.
248 The implementation is free to merge in some cases and not in others,
249 or to never merge features at all. The default tolerance is zero.
250
251 The current implementation merges vertices only if they are exactly
252 coincident, regardless of the current tolerance. A vertex is
253 spliced into an edge only if the implementation is unable to
254 distinguish which side of the edge the vertex lies on.
255 Two edges are merged only when both endpoints are identical.
256
257
258 void gluTessNormal( GLUtesselator *tess,
259 GLUcoord x, GLUcoord y, GLUcoord z )
260
261 - Lets the user supply the polygon normal, if known. All input data
262 is projected into a plane perpendicular to the normal before
263 tesselation. All output triangles are oriented CCW with
264 respect to the normal (CW orientation can be obtained by
265 reversing the sign of the supplied normal). For example, if
266 you know that all polygons lie in the x-y plane, call
267 "gluTessNormal(tess, 0.0, 0.0, 1.0)" before rendering any polygons.
268
269 - If the supplied normal is (0,0,0) (the default value), the
270 normal is determined as follows. The direction of the normal,
271 up to its sign, is found by fitting a plane to the vertices,
272 without regard to how the vertices are connected. It is
273 expected that the input data lies approximately in plane;
274 otherwise projection perpendicular to the computed normal may
275 substantially change the geometry. The sign of the normal is
276 chosen so that the sum of the signed areas of all input contours
277 is non-negative (where a CCW contour has positive area).
278
279 - The supplied normal persists until it is changed by another
280 call to gluTessNormal.
281
282
283 Backward compatibility with the GLU tesselator
284 ----------------------------------------------
285
286 The preferred interface is the one described above. The following
287 routines are obsolete, and are provided only for backward compatibility:
288
289 typedef GLUtesselator GLUtriangulatorObj; /* obsolete name */
290
291 void gluBeginPolygon( GLUtesselator *tess );
292 void gluNextContour( GLUtesselator *tess, GLenum type );
293 void gluEndPolygon( GLUtesselator *tess );
294
295 "type" is one of GLU_EXTERIOR, GLU_INTERIOR, GLU_CCW, GLU_CW, or
296 GLU_UNKNOWN. It is ignored by the current GLU tesselator.
297
298 GLU_BEGIN, GLU_VERTEX, GLU_END, GLU_ERROR, and GLU_EDGE_FLAG are defined
299 as synonyms for GLU_TESS_BEGIN, GLU_TESS_VERTEX, GLU_TESS_END,
300 GLU_TESS_ERROR, and GLU_TESS_EDGE_FLAG.
301
302
303Polygon CSG operations
304----------------------
305
306 The features of the tesselator make it easy to find the union, difference,
307 or intersection of several polygons.
308
309 First, assume that each polygon is defined so that the winding number
310 is 0 for each exterior region, and 1 for each interior region. Under
311 this model, CCW contours define the outer boundary of the polygon, and
312 CW contours define holes. Contours may be nested, but a nested
313 contour must be oriented oppositely from the contour that contains it.
314
315 If the original polygons do not satisfy this description, they can be
316 converted to this form by first running the tesselator with the
317 GLU_TESS_BOUNDARY_ONLY property turned on. This returns a list of
318 contours satisfying the restriction above. By allocating two
319 tesselator objects, the callbacks from one tesselator can be fed
320 directly to the input of another.
321
322 Given two or more polygons of the form above, CSG operations can be
323 implemented as follows:
324
325 Union
326 Draw all the input contours as a single polygon. The winding number
327 of each resulting region is the number of original polygons
328 which cover it. The union can be extracted using the
329 GLU_TESS_WINDING_NONZERO or GLU_TESS_WINDING_POSITIVE winding rules.
330 Note that with the nonzero rule, we would get the same result if
331 all contour orientations were reversed.
332
333 Intersection (two polygons at a time only)
334 Draw a single polygon using the contours from both input polygons.
335 Extract the result using GLU_TESS_WINDING_ABS_GEQ_TWO. (Since this
336 winding rule looks at the absolute value, reversing all contour
337 orientations does not change the result.)
338
339 Difference
340
341 Suppose we want to compute A \ (B union C union D). Draw a single
342 polygon consisting of the unmodified contours from A, followed by
343 the contours of B,C,D with the vertex order reversed (this changes
344 the winding number of the interior regions to -1). To extract the
345 result, use the GLU_TESS_WINDING_POSITIVE rule.
346
347 If B,C,D are the result of a GLU_TESS_BOUNDARY_ONLY call, an
348 alternative to reversing the vertex order is to reverse the sign of
349 the supplied normal. For example in the x-y plane, call
350 gluTessNormal( tess, 0.0, 0.0, -1.0 ).
351
352
353Performance
354-----------
355
356 The tesselator is not intended for immediate-mode rendering; when
357 possible the output should be cached in a user structure or display
358 list. General polygon tesselation is an inherently difficult problem,
359 especially given the goal of extreme robustness.
360
361 The implementation makes an effort to output a small number of fans
362 and strips; this should improve the rendering performance when the
363 output is used in a display list.
364
365 Single-contour input polygons are first tested to see whether they can
366 be rendered as a triangle fan with respect to the first vertex (to
367 avoid running the full decomposition algorithm on convex polygons).
368 Non-convex polygons may be rendered by this "fast path" as well, if
369 the algorithm gets lucky in its choice of a starting vertex.
370
371 For best performance follow these guidelines:
372
373 - supply the polygon normal, if available, using gluTessNormal().
374 This represents about 10% of the computation time. For example,
375 if all polygons lie in the x-y plane, use gluTessNormal(tess,0,0,1).
376
377 - render many polygons using the same tesselator object, rather than
378 allocating a new tesselator for each one. (In a multi-threaded,
379 multi-processor environment you may get better performance using
380 several tesselators.)
381
382
383Comparison with the GLU tesselator
384----------------------------------
385
386 On polygons which make it through the "fast path", the tesselator is
387 3 to 5 times faster than the GLU tesselator.
388
389 On polygons which don't make it through the fast path (but which don't
390 have self-intersections or degeneracies), it is about 2 times slower.
391
392 On polygons with self-intersections or degeneraces, there is nothing
393 to compare against.
394
395 The new tesselator generates many more fans and strips, reducing the
396 number of vertices that need to be sent to the hardware.
397
398 Key to the statistics:
399
400 vert number of input vertices on all contours
401 cntr number of input contours
402 tri number of triangles in all output primitives
403 strip number of triangle strips
404 fan number of triangle fans
405 ind number of independent triangles
406 ms number of milliseconds for tesselation
407 (on a 150MHz R4400 Indy)
408
409 Convex polygon examples:
410
411New: 3 vert, 1 cntr, 1 tri, 0 strip, 0 fan, 1 ind, 0.0459 ms
412Old: 3 vert, 1 cntr, 1 tri, 0 strip, 0 fan, 1 ind, 0.149 ms
413New: 4 vert, 1 cntr, 2 tri, 0 strip, 1 fan, 0 ind, 0.0459 ms
414Old: 4 vert, 1 cntr, 2 tri, 0 strip, 0 fan, 2 ind, 0.161 ms
415New: 36 vert, 1 cntr, 34 tri, 0 strip, 1 fan, 0 ind, 0.153 ms
416Old: 36 vert, 1 cntr, 34 tri, 0 strip, 0 fan, 34 ind, 0.621 ms
417
418 Concave single-contour polygons:
419
420New: 5 vert, 1 cntr, 3 tri, 0 strip, 1 fan, 0 ind, 0.052 ms
421Old: 5 vert, 1 cntr, 3 tri, 0 strip, 0 fan, 3 ind, 0.252 ms
422New: 19 vert, 1 cntr, 17 tri, 2 strip, 2 fan, 1 ind, 0.911 ms
423Old: 19 vert, 1 cntr, 17 tri, 0 strip, 0 fan, 17 ind, 0.529 ms
424New: 151 vert, 1 cntr, 149 tri, 13 strip, 18 fan, 3 ind, 6.82 ms
425Old: 151 vert, 1 cntr, 149 tri, 0 strip, 3 fan, 143 ind, 2.7 ms
426New: 574 vert, 1 cntr, 572 tri, 59 strip, 54 fan, 11 ind, 26.6 ms
427Old: 574 vert, 1 cntr, 572 tri, 0 strip, 31 fan, 499 ind, 12.4 ms
428
429 Multiple contours, but no intersections:
430
431New: 7 vert, 2 cntr, 7 tri, 1 strip, 0 fan, 0 ind, 0.527 ms
432Old: 7 vert, 2 cntr, 7 tri, 0 strip, 0 fan, 7 ind, 0.274 ms
433New: 81 vert, 6 cntr, 89 tri, 9 strip, 7 fan, 6 ind, 3.88 ms
434Old: 81 vert, 6 cntr, 89 tri, 0 strip, 13 fan, 61 ind, 2.2 ms
435New: 391 vert, 19 cntr, 413 tri, 37 strip, 32 fan, 26 ind, 20.2 ms
436Old: 391 vert, 19 cntr, 413 tri, 0 strip, 25 fan, 363 ind, 8.68 ms
437
438 Self-intersecting and degenerate examples:
439
440Bowtie: 4 vert, 1 cntr, 2 tri, 0 strip, 0 fan, 2 ind, 0.483 ms
441Star: 5 vert, 1 cntr, 5 tri, 0 strip, 0 fan, 5 ind, 0.91 ms
442Random: 24 vert, 7 cntr, 46 tri, 2 strip, 12 fan, 7 ind, 5.32 ms
443Font: 333 vert, 2 cntr, 331 tri, 32 strip, 16 fan, 3 ind, 14.1 ms
444: 167 vert, 35 cntr, 254 tri, 8 strip, 56 fan, 52 ind, 46.3 ms
445: 78 vert, 1 cntr, 2675 tri, 148 strip, 207 fan, 180 ind, 243 ms
446: 12480 vert, 2 cntr, 12478 tri, 736 strip,1275 fan, 5 ind, 1010 ms
447