General Polygon Tesselation
---------------------------
This note describes a tesselator for polygons consisting of one or
more closed contours. It is backward-compatible with the current
OpenGL Utilities tesselator, and is intended to replace it. Here is
a summary of the major differences:
- input contours can be intersecting, self-intersecting, or degenerate.
- supports a choice of several winding rules for determining which parts
of the polygon are on the "interior". This makes it possible to do
CSG operations on polygons.
- boundary extraction: instead of tesselating the polygon, returns a
set of closed contours which separate the interior from the exterior.
- returns the output as a small number of triangle fans and strips,
rather than a list of independent triangles (when possible).
- output is available as an explicit mesh (a quad-edge structure),
in addition to the normal callback interface.
- the algorithm used is extremely robust.
Where to get it
---------------
rcp guest@bonnie:/proj/irix5.3/isms/oglsi/src/gfx/lib/glu/libtess .
See the Makefile for compilation instructions. The include file for
applications is "tesselator.h".
The tesselator library is for internal SGI use only.
The interface
-------------
The tesselator state is maintained in a "tesselator object".
These are allocated and destroyed using
GLUtesselator *gluNewTess( void );
void gluDeleteTess( GLUtesselator *tess );
Several tesselator objects may be used simultaneously.
Inputs
------
The input contours are specified with the following routines:
void gluTessBeginPolygon( GLUtesselator *tess );
void gluTessBeginContour( GLUtesselator *tess );
void gluTessVertex( GLUtesselator *tess, GLUcoord coords[3], void *data );
void gluTessEndContour( GLUtesselator *tess );
void gluTessEndPolygon( GLUtesselator *tess );
Within each BeginPolygon/EndPolygon pair, there can be zero or more
calls to BeginContour/EndContour. Within each contour, there are zero
or more calls to gluTessVertex(). The vertices specify a closed
contour (the last vertex of each contour is automatically linked to
the first).
"coords" give the coordinates of the vertex in 3-space. For useful
results, all vertices should lie in some plane, since the vertices
are projected onto a plane before tesselation. "data" is a pointer
to a user-defined vertex structure, which typically contains other
information such as color, texture coordinates, normal, etc. It is
used to refer to the vertex during rendering.
The library can be compiled in single- or double-precision; the type
GLUcoord represents either "float" or "double" accordingly. The GLU
version will be available in double-precision only. Compile with
GLU_TESS_API_FLOAT defined to get the single-precision version.
When EndPolygon is called, the tesselation algorithm determines
which regions are interior to the given contours, according to one
of several "winding rules" described below. The interior regions
are then tesselated, and the output is provided as callbacks.
Rendering Callbacks
-------------------
Callbacks are specified by the client using
void gluTessCallback( GLUtesselator *tess, GLenum which, void (*fn)());
If "fn" is NULL, any previously defined callback is discarded.
The callbacks used to provide output are: /* which == */
void begin( GLenum type ); /* GLU_TESS_BEGIN */
void edgeFlag( GLboolean flag ); /* GLU_TESS_EDGE_FLAG */
void vertex( void *data ); /* GLU_TESS_VERTEX */
void end( void ); /* GLU_TESS_END */
Any of the callbacks may be left undefined; if so, the corresponding
information will not be supplied during rendering.
The "begin" callback indicates the start of a primitive; type is one
of GL_TRIANGLE_STRIP, GL_TRIANGLE_FAN, or GL_TRIANGLES (but see the
notes on "boundary extraction" below).
It is followed by any number of "vertex" callbacks, which supply the
vertices in the same order as expected by the corresponding glBegin()
call. After the last vertex of a given primitive, there is a callback
to "end".
If the "edgeFlag" callback is provided, no triangle fans or strips
will be used. When edgeFlag is called, if "flag" is GL_TRUE then each
vertex which follows begins an edge which lies on the polygon boundary
(ie. an edge which separates an interior region from an exterior one).
If "flag" is GL_FALSE, each vertex which follows begins an edge which lies
in the polygon interior. "edgeFlag" will be called before the first
call to "vertex".
Other Callbacks
---------------
void mesh( GLUmesh *mesh ); /* GLU_TESS_MESH */
- Returns an explicit mesh, represented using the quad-edge structure
(Guibas/Stolfi '85). Other implementations of this interface might
use a different mesh structure, so this is available only only as an
SGI extension. When the mesh is no longer needed, it should be freed
using
void gluDeleteMesh( GLUmesh *mesh );
There is a brief description of this data structure in the include
file "mesh.h". For the full details, see L. Guibas and J. Stolfi,
Primitives for the manipulation of general subdivisions and the
computation of Voronoi diagrams, ACM Transactions on Graphics,
4(2):74-123, April 1985. For an introduction, see the course notes
for CS348a, "Mathematical Foundations of Computer Graphics",
available at the Stanford bookstore (and taught during the fall
quarter).
void error( GLenum errno ); /* GLU_TESS_ERROR */
- errno is one of GLU_TESS_MISSING_BEGIN_POLYGON,
GLU_TESS_MISSING_END_POLYGON,
GLU_TESS_MISSING_BEGIN_CONTOUR,
GLU_TESS_MISSING_END_CONTOUR,
GLU_TESS_COORD_TOO_LARGE,
GLU_TESS_NEED_COMBINE_CALLBACK
The first four are obvious. The interface recovers from these
errors by inserting the missing call(s).
GLU_TESS_COORD_TOO_LARGE says that some vertex coordinate exceeded
the predefined constant GLU_TESS_MAX_COORD in absolute value, and
that the value has been clamped. (Coordinate values must be small
enough so that two can be multiplied together without overflow.)
GLU_TESS_NEED_COMBINE_CALLBACK says that the algorithm detected an
intersection between two edges in the input data, and the "combine"
callback (below) was not provided. No output will be generated.
void combine( GLUcoord coords[3], void *data[4], /* GLU_TESS_COMBINE */
GLUcoord weight[4], void **outData );
- When the algorithm detects an intersection, or wishes to merge
features, it needs to create a new vertex. The vertex is defined
as a linear combination of up to 4 existing vertices, referenced
by data[0..3]. The coefficients of the linear combination are
given by weight[0..3]; these weights always sum to 1.0. All vertex
pointers are valid even when some of the weights are zero.
"coords" gives the location of the new vertex.
The user must allocate another vertex, interpolate parameters
using "data" and "weights", and return the new vertex pointer in
"outData". This handle is supplied during rendering callbacks.
For example, if the polygon lies in an arbitrary plane in 3-space,
and we associate a color with each vertex, the combine callback might
look like this:
void myCombine( GLUcoord coords[3], VERTEX *d[4],
GLUcoord w[4], VERTEX **dataOut )
{
VERTEX *new = new_vertex();
new->x = coords[0];
new->y = coords[1];
new->z = coords[2];
new->r = w[0]*d[0]->r + w[1]*d[1]->r + w[2]*d[2]->r + w[3]*d[3]->r;
new->g = w[0]*d[0]->g + w[1]*d[1]->g + w[2]*d[2]->g + w[3]*d[3]->g;
new->b = w[0]*d[0]->b + w[1]*d[1]->b + w[2]*d[2]->b + w[3]*d[3]->b;
new->a = w[0]*d[0]->a + w[1]*d[1]->a + w[2]*d[2]->a + w[3]*d[3]->a;
*dataOut = new;
}
If the algorithm detects an intersection, then the "combine" callback
must be defined, and must write a non-NULL pointer into "dataOut".
Otherwise the GLU_TESS_NEED_COMBINE_CALLBACK error occurs, and no
output is generated. This is the only error that can occur during
tesselation and rendering.
Control over Tesselation
------------------------
void gluTessProperty( GLUtesselator *tess, GLenum which, GLUcoord value );
Properties defined:
- GLU_TESS_WINDING_RULE. Possible values:
GLU_TESS_WINDING_ODD
GLU_TESS_WINDING_NONZERO
GLU_TESS_WINDING_POSITIVE
GLU_TESS_WINDING_NEGATIVE
GLU_TESS_WINDING_ABS_GEQ_TWO
The input contours parition the plane into regions. A winding
rule determines which of these regions are inside the polygon.
For a single contour C, the winding number of a point x is simply
the signed number of revolutions we make around x as we travel
once around C (where CCW is positive). When there are several
contours, the individual winding numbers are summed. This
procedure associates a signed integer value with each point x in
the plane. Note that the winding number is the same for all
points in a single region.
The winding rule classifies a region as "inside" if its winding
number belongs to the chosen category (odd, nonzero, positive,
negative, or absolute value of at least two). The current GLU
tesselator implements the "odd" rule. The "nonzero" rule is another
common way to define the interior. The other three rules are
useful for polygon CSG operations (see below).
- GLU_TESS_BOUNDARY_ONLY. Values: TRUE (non-zero) or FALSE (zero).
If TRUE, returns a set of closed contours which separate the
polygon interior and exterior (rather than a tesselation).
Exterior contours are oriented CCW with respect to the normal,
interior contours are oriented CW. The GLU_TESS_BEGIN callback
uses the type GL_LINE_LOOP for each contour.
- GLU_TESS_TOLERANCE. Value: a real number between 0.0 and 1.0.
This specifies a tolerance for merging features to reduce the size
of the output. For example, two vertices which are very close to
each other might be replaced by a single vertex. The tolerance
is multiplied by the largest coordinate magnitude of any input vertex;
this specifies the maximum distance that any feature can move as the
result of a single merge operation. If a single feature takes part
in several merge operations, the total distance moved could be larger.
Feature merging is completely optional; the tolerance is only a hint.
The implementation is free to merge in some cases and not in others,
or to never merge features at all. The default tolerance is zero.
The current implementation merges vertices only if they are exactly
coincident, regardless of the current tolerance. A vertex is
spliced into an edge only if the implementation is unable to
distinguish which side of the edge the vertex lies on.
Two edges are merged only when both endpoints are identical.
void gluTessNormal( GLUtesselator *tess,
GLUcoord x, GLUcoord y, GLUcoord z )
- Lets the user supply the polygon normal, if known. All input data
is projected into a plane perpendicular to the normal before
tesselation. All output triangles are oriented CCW with
respect to the normal (CW orientation can be obtained by
reversing the sign of the supplied normal). For example, if
you know that all polygons lie in the x-y plane, call
"gluTessNormal(tess, 0.0, 0.0, 1.0)" before rendering any polygons.
- If the supplied normal is (0,0,0) (the default value), the
normal is determined as follows. The direction of the normal,
up to its sign, is found by fitting a plane to the vertices,
without regard to how the vertices are connected. It is
expected that the input data lies approximately in plane;
otherwise projection perpendicular to the computed normal may
substantially change the geometry. The sign of the normal is
chosen so that the sum of the signed areas of all input contours
is non-negative (where a CCW contour has positive area).
- The supplied normal persists until it is changed by another
call to gluTessNormal.
Backward compatibility with the GLU tesselator
----------------------------------------------
The preferred interface is the one described above. The following
routines are obsolete, and are provided only for backward compatibility:
typedef GLUtesselator GLUtriangulatorObj; /* obsolete name */
void gluBeginPolygon( GLUtesselator *tess );
void gluNextContour( GLUtesselator *tess, GLenum type );
void gluEndPolygon( GLUtesselator *tess );
"type" is one of GLU_EXTERIOR, GLU_INTERIOR, GLU_CCW, GLU_CW, or
GLU_UNKNOWN. It is ignored by the current GLU tesselator.
GLU_BEGIN, GLU_VERTEX, GLU_END, GLU_ERROR, and GLU_EDGE_FLAG are defined
as synonyms for GLU_TESS_BEGIN, GLU_TESS_VERTEX, GLU_TESS_END,
GLU_TESS_ERROR, and GLU_TESS_EDGE_FLAG.
Polygon CSG operations
----------------------
The features of the tesselator make it easy to find the union, difference,
or intersection of several polygons.
First, assume that each polygon is defined so that the winding number
is 0 for each exterior region, and 1 for each interior region. Under
this model, CCW contours define the outer boundary of the polygon, and
CW contours define holes. Contours may be nested, but a nested
contour must be oriented oppositely from the contour that contains it.
If the original polygons do not satisfy this description, they can be
converted to this form by first running the tesselator with the
GLU_TESS_BOUNDARY_ONLY property turned on. This returns a list of
contours satisfying the restriction above. By allocating two
tesselator objects, the callbacks from one tesselator can be fed
directly to the input of another.
Given two or more polygons of the form above, CSG operations can be
implemented as follows:
Union
Draw all the input contours as a single polygon. The winding number
of each resulting region is the number of original polygons
which cover it. The union can be extracted using the
GLU_TESS_WINDING_NONZERO or GLU_TESS_WINDING_POSITIVE winding rules.
Note that with the nonzero rule, we would get the same result if
all contour orientations were reversed.
Intersection (two polygons at a time only)
Draw a single polygon using the contours from both input polygons.
Extract the result using GLU_TESS_WINDING_ABS_GEQ_TWO. (Since this
winding rule looks at the absolute value, reversing all contour
orientations does not change the result.)
Difference
Suppose we want to compute A \ (B union C union D). Draw a single
polygon consisting of the unmodified contours from A, followed by
the contours of B,C,D with the vertex order reversed (this changes
the winding number of the interior regions to -1). To extract the
result, use the GLU_TESS_WINDING_POSITIVE rule.
If B,C,D are the result of a GLU_TESS_BOUNDARY_ONLY call, an
alternative to reversing the vertex order is to reverse the sign of
the supplied normal. For example in the x-y plane, call
gluTessNormal( tess, 0.0, 0.0, -1.0 ).
Performance
-----------
The tesselator is not intended for immediate-mode rendering; when
possible the output should be cached in a user structure or display
list. General polygon tesselation is an inherently difficult problem,
especially given the goal of extreme robustness.
The implementation makes an effort to output a small number of fans
and strips; this should improve the rendering performance when the
output is used in a display list.
Single-contour input polygons are first tested to see whether they can
be rendered as a triangle fan with respect to the first vertex (to
avoid running the full decomposition algorithm on convex polygons).
Non-convex polygons may be rendered by this "fast path" as well, if
the algorithm gets lucky in its choice of a starting vertex.
For best performance follow these guidelines:
- supply the polygon normal, if available, using gluTessNormal().
This represents about 10% of the computation time. For example,
if all polygons lie in the x-y plane, use gluTessNormal(tess,0,0,1).
- render many polygons using the same tesselator object, rather than
allocating a new tesselator for each one. (In a multi-threaded,
multi-processor environment you may get better performance using
several tesselators.)
Comparison with the GLU tesselator
----------------------------------
On polygons which make it through the "fast path", the tesselator is
3 to 5 times faster than the GLU tesselator.
On polygons which don't make it through the fast path (but which don't
have self-intersections or degeneracies), it is about 2 times slower.
On polygons with self-intersections or degeneraces, there is nothing
to compare against.
The new tesselator generates many more fans and strips, reducing the
number of vertices that need to be sent to the hardware.
Key to the statistics:
vert number of input vertices on all contours
cntr number of input contours
tri number of triangles in all output primitives
strip number of triangle strips
fan number of triangle fans
ind number of independent triangles
ms number of milliseconds for tesselation
(on a 150MHz R4400 Indy)
Convex polygon examples:
New: 3 vert, 1 cntr, 1 tri, 0 strip, 0 fan, 1 ind, 0.0459 ms
Old: 3 vert, 1 cntr, 1 tri, 0 strip, 0 fan, 1 ind, 0.149 ms
New: 4 vert, 1 cntr, 2 tri, 0 strip, 1 fan, 0 ind, 0.0459 ms
Old: 4 vert, 1 cntr, 2 tri, 0 strip, 0 fan, 2 ind, 0.161 ms
New: 36 vert, 1 cntr, 34 tri, 0 strip, 1 fan, 0 ind, 0.153 ms
Old: 36 vert, 1 cntr, 34 tri, 0 strip, 0 fan, 34 ind, 0.621 ms
Concave single-contour polygons:
New: 5 vert, 1 cntr, 3 tri, 0 strip, 1 fan, 0 ind, 0.052 ms
Old: 5 vert, 1 cntr, 3 tri, 0 strip, 0 fan, 3 ind, 0.252 ms
New: 19 vert, 1 cntr, 17 tri, 2 strip, 2 fan, 1 ind, 0.911 ms
Old: 19 vert, 1 cntr, 17 tri, 0 strip, 0 fan, 17 ind, 0.529 ms
New: 151 vert, 1 cntr, 149 tri, 13 strip, 18 fan, 3 ind, 6.82 ms
Old: 151 vert, 1 cntr, 149 tri, 0 strip, 3 fan, 143 ind, 2.7 ms
New: 574 vert, 1 cntr, 572 tri, 59 strip, 54 fan, 11 ind, 26.6 ms
Old: 574 vert, 1 cntr, 572 tri, 0 strip, 31 fan, 499 ind, 12.4 ms
Multiple contours, but no intersections:
New: 7 vert, 2 cntr, 7 tri, 1 strip, 0 fan, 0 ind, 0.527 ms
Old: 7 vert, 2 cntr, 7 tri, 0 strip, 0 fan, 7 ind, 0.274 ms
New: 81 vert, 6 cntr, 89 tri, 9 strip, 7 fan, 6 ind, 3.88 ms
Old: 81 vert, 6 cntr, 89 tri, 0 strip, 13 fan, 61 ind, 2.2 ms
New: 391 vert, 19 cntr, 413 tri, 37 strip, 32 fan, 26 ind, 20.2 ms
Old: 391 vert, 19 cntr, 413 tri, 0 strip, 25 fan, 363 ind, 8.68 ms
Self-intersecting and degenerate examples:
Bowtie: 4 vert, 1 cntr, 2 tri, 0 strip, 0 fan, 2 ind, 0.483 ms
Star: 5 vert, 1 cntr, 5 tri, 0 strip, 0 fan, 5 ind, 0.91 ms
Random: 24 vert, 7 cntr, 46 tri, 2 strip, 12 fan, 7 ind, 5.32 ms
Font: 333 vert, 2 cntr, 331 tri, 32 strip, 16 fan, 3 ind, 14.1 ms
: 167 vert, 35 cntr, 254 tri, 8 strip, 56 fan, 52 ind, 46.3 ms
: 78 vert, 1 cntr, 2675 tri, 148 strip, 207 fan, 180 ind, 243 ms
: 12480 vert, 2 cntr, 12478 tri, 736 strip,1275 fan, 5 ind, 1010 ms
