I'm asking this question out of curiosity, having first tried to implement such an algorithm before using GLU's for performance reasons.
I've looked into common algorithms (Delaunay, Ear Clipping are often mentioned), but I can't seem to understand how GLU does its job so well all the time.
Do any of you have interesting papers or articles on that subjects?
There are some notes alongside the source:
This is only a very brief overview. There is quite a bit of additional documentation in the source code itself.
Goals of robust tesselation
The tesselation algorithm is fundamentally a 2D algorithm. We initially project all data into a plane; our goal is to robustly tesselate the projected data. The same topological tesselation is then applied to the input data.
Topologically, the output should always be a tesselation. If the input is even slightly non-planar, then some triangles will necessarily be back-facing when viewed from some angles, but the goal is to minimize this effect.
The algorithm needs some capability of cleaning up the input data as well as the numerical errors in its own calculations. One way to do this is to specify a tolerance as defined above, and clean up the input and output during the line sweep process. At the very least, the algorithm must handle coincident vertices, vertices incident to an edge, and coincident edges.
Phases of the algorithm
- Find the polygon normal N.
- Project the vertex data onto a plane. It does not need to be perpendicular to the normal, eg. we can project onto the plane
perpendicular to the coordinate axis whose dot product with N is largest.- Using a line-sweep algorithm, partition the plane into x-monotone regions. Any vertical line intersects an x-monotone region in at most one interval.
- Triangulate the x-monotone regions.
- Group the triangles into strips and fans.
Finding the normal vector
A common way to find a polygon normal is to compute the signed area when the polygon is projected along the three coordinate axes. We can't do this, since contours can have zero area without being degenerate (eg. a bowtie).
We fit a plane to the vertex data, ignoring how they are connected into contours. Ideally this would be a least-squares fit; however for our purpose the accuracy of the normal is not important. Instead we find three vertices which are widely separated, and compute the normal to the triangle they form. The vertices are chosen so that the triangle has an area at least 1/sqrt(3) times the largest area of any triangle formed using the input vertices.
The contours do affect the orientation of the normal; after computing the normal, we check that the sum of the signed contour areas is non-negative, and reverse the normal if necessary.
Projecting the vertices
We project the vertices onto a plane perpendicular to one of the three coordinate axes. This helps numerical accuracy by removing a transformation step between the original input data and the data processed by the algorithm. The projection also compresses the input data; the 2D distance between vertices after projection may be smaller than the original 2D distance. However by choosing the coordinate axis whose dot product with the normal is greatest, the compression factor is at most 1/sqrt(3).
Even though the accuracy of the normal is not that important (since we are projecting perpendicular to a coordinate axis anyway), the robustness of the computation is important. For example, if there are many vertices which lie almost along a line, and one vertex V which is well-separated from the line, then our normal computation should involve V otherwise the results will be garbage.
The advantage of projecting perpendicular to the polygon normal is that computed intersection points will be as close as possible to their ideal locations. To get this behavior, define TRUE_PROJECT.
The Line Sweep
There are three data structures: the mesh, the event queue, and the edge dictionary.
The mesh is a "quad-edge" data structure which records the topology of the current decomposition; for details see the include file "mesh.h".
The event queue simply holds all vertices (both original and computed ones), organized so that we can quickly extract the vertex with the minimum x-coord (and among those, the one with the minimum y-coord).
The edge dictionary describes the current intersection of the sweep line with the regions of the polygon. This is just an ordering of the edges which intersect the sweep line, sorted by their current order of intersection. For each pair of edges, we store some information about the monotone region between them -- these are call "active regions" (since they are crossed by the current sweep line).
The basic algorithm is to sweep from left to right, processing each vertex. The processed portion of the mesh (left of the sweep line) is a planar decomposition. As we cross each vertex, we update the mesh and the edge dictionary, then we check any newly adjacent pairs of edges to see if they intersect.
A vertex can have any number of edges. Vertices with many edges can be created as vertices are merged and intersection points are computed. For unprocessed vertices (right of the sweep line), these edges are in no particular order around the vertex; for processed vertices, the topological ordering should match the geometric ordering.
The vertex processing happens in two phases: first we process are the left-going edges (all these edges are currently in the edge dictionary). This involves:
- deleting the left-going edges from the dictionary;
- relinking the mesh if necessary, so that the order of these edges around the event vertex matches the order in the dictionary;
- marking any terminated regions (regions which lie between two left-going edges) as either "inside" or "outside" according to their winding number.
When there are no left-going edges, and the event vertex is in an "interior" region, we need to add an edge (to split the region into monotone pieces). To do this we simply join the event vertex to the rightmost left endpoint of the upper or lower edge of the containing region.
Then we process the right-going edges. This involves:
- inserting the edges in the edge dictionary;
- computing the winding number of any newly created active regions. We can compute this incrementally using the winding of each edge
that we cross as we walk through the dictionary.- relinking the mesh if necessary, so that the order of these edges around the event vertex matches the order in the dictionary;
- checking any newly adjacent edges for intersection and/or merging.
If there are no right-going edges, again we need to add one to split the containing region into monotone pieces. In our case it is most convenient to add an edge to the leftmost right endpoint of either containing edge; however we may need to change this later (see the code for details).
Invariants
These are the most important invariants maintained during the sweep. We define a function VertLeq(v1,v2) which defines the order in which vertices cross the sweep line, and a function EdgeLeq(e1,e2; loc) which says whether e1 is below e2 at the sweep event location "loc". This function is defined only at sweep event locations which lie between the rightmost left endpoint of {e1,e2}, and the leftmost right endpoint of {e1,e2}.
Invariants for the Edge Dictionary.
Each pair of adjacent edges e2=Succ(e1) satisfies EdgeLeq(e1,e2) at any valid location of the sweep event.
If EdgeLeq(e2,e1) as well (at any valid sweep event), then e1 and e2 share a common endpoint.
For each e in the dictionary, e->Dst has been processed but not e->Org.
Each edge e satisfies VertLeq(e->Dst,event) && VertLeq(event,e->Org) where "event" is the current sweep line event.
No edge e has zero length.
No two edges have identical left and right endpoints. Invariants for the Mesh (the processed portion).
The portion of the mesh left of the sweep line is a planar graph, ie. there is some way to embed it in the plane.
No processed edge has zero length.
No two processed vertices have identical coordinates.
Each "inside" region is monotone, ie. can be broken into two chains of monotonically increasing vertices according to VertLeq(v1,v2)
- a non-invariant: these chains may intersect (slightly) due to numerical errors, but this does not affect the algorithm's operation.
Invariants for the Sweep.
- If a vertex has any left-going edges, then these must be in the edge dictionary at the time the vertex is processed.
- If an edge is marked "fixUpperEdge" (it is a temporary edge introduced by ConnectRightVertex), then it is the only right-going edge from its associated vertex. (This says that these edges exist only when it is necessary.)
Robustness
The key to the robustness of the algorithm is maintaining the invariants above, especially the correct ordering of the edge dictionary. We achieve this by:
Writing the numerical computations for maximum precision rather than maximum speed.
Making no assumptions at all about the results of the edge intersection calculations -- for sufficiently degenerate inputs, the computed location is not much better than a random number.
When numerical errors violate the invariants, restore them by making topological changes when necessary (ie. relinking the mesh structure).
Triangulation and Grouping
We finish the line sweep before doing any triangulation. This is because even after a monotone region is complete, there can be further changes to its vertex data because of further vertex merging.
After triangulating all monotone regions, we want to group the triangles into fans and strips. We do this using a greedy approach. The triangulation itself is not optimized to reduce the number of primitives; we just try to get a reasonable decomposition of the computed triangulation.