From a2f3162504ea229530c67a93ab36ed992c512240 Mon Sep 17 00:00:00 2001 From: lecoanet Date: Mon, 17 Mar 2003 16:10:19 +0000 Subject: *** empty log message *** --- libtess/alg-outline | 229 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 229 insertions(+) create mode 100644 libtess/alg-outline (limited to 'libtess/alg-outline') diff --git a/libtess/alg-outline b/libtess/alg-outline new file mode 100644 index 0000000..fd0a8bf --- /dev/null +++ b/libtess/alg-outline @@ -0,0 +1,229 @@ +/* +** $Header$ +*/ + +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 +----------------------- + +1. Find the polygon normal N. +2. 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. +3. 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. +4. Triangulate the x-monotone regions. +5. 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: + + 1. Writing the numerical computations for maximum precision rather + than maximum speed. + + 2. 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. + + 3. 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. -- cgit v1.1