Fig. 1. Collapsing a triangle: The shaded triangle is selected from the mesh in (a), collapsed toward the centroid of the triangle in (b), creating < new mesh in (c) which has four fewer triangles than the original mesh. Fig. 3. Edge swapping to remove cycles in the stencil. (a) The selected triangle contains a cycle in the stencil. (b) The cycle is removed by swapping the common edge between the triangle and the neighboring triangle that belongs to the cycle. Fig. 2. Stencils of triangles. (a) The shaded triangle has a connected acyclic stencil. (b) The shaded triangle has a disconnected acyclic stencil. (c) The shaded triangle has a connected cyclic stencil. (d) The shaded triangle has a complete stencil. In the last case, the stencil boundary polygon is outlined in bold. We note that the boundary polygon of the triangle in (a) contains a vertex of the triangle, and therefore the stencil is not com- plete. Fig. 4. A single point-at-infinity is added to the mesh and connected to each vertex on the boundary. This enables us to associate complete stencils to triangles on the boundary. Fig. 5. A triangle and its stencil undergoing the collapse operation. As the triangle is collapsed, a new mesh is created containing four fewer trian- gles. Geometrically, this transition is smooth; topologically, itis “discontinuous.” Fig. 6. Topology of the triangulation before the collapse (a); and after the collapse (b). In the process, the triangles TT y Ty, , and Ty, are elimi nated from the triangulation. The new triangulation has a vertex atc, and the valence of each of the vertices p, ,p, , and p, is reduced by one. Fig. 6. Topology of the triangulation before the collapse (a); and after the collapse (b). In the process, the triangles Ty Tyga Ty. , and Ty are eli Fig. 8. Folds can occur in the modified triangles of the stencil if a star-shaped property does not hold. In this planar example, the triangle in (a) ha: a non-star-shaped stencil. As we collapse in (b), the potential conflict can be seen, and, in (c), the fold can be seen. Fig. 7. Introducing cycles into the triangulation. (a) The vertex Py, has valence four, and when triangle T is collapsed in (b), a cycle is introduced Fig. 9. Establishing a coordinate system in the plane P. Use the cen- troid ¢ of the triangle 7 as the origin and use any two orthonormal vectors U and Vv in the plane P as basis vectors. The vectors u,v, and the unit normal n form an orthonormal coordinate system. Fig. 10. To establish coordinates of the stencil points, each point is projected onto P. The coordinates u; and v, along with the distance d; from the plane, define the coordinates of the point in the local coordi- nate system. 4.1 Curvature Estimates Fig. 11. On the approximating surface, the image of a triangle edge is a quadratic polynomial passing through po and pj. Fig. 12. The Maximum error e between an edge of a triangle and th approximating surface can occur at either at a local maximum betwee 0 and 1 or at the endpoints. Fig. 13. To establish the point to which a triangle will collapse, we find the intersection of the line through the origin of the local coordinate system in the direction of the normal vector to the triangle, and the approximating surface. Fig. 14. Four types of boundary triangles can be identified in the mesh: 1) Triangles that have an edge on the boundary; 2) triangles that have a single vertex on the boundary; 3) triangles that have two edges on the boundary (representing corners); and 4) triangles that have two vertices on the boundary. Each triangle has stencil triangles that con- tain the pointatinfinity po. Fig. 15. Type-1 triangles are collapsed to a point that represents the edge of the mesh. This point can be found by finding the line that passes through the midpoint of the edge in the direction of the normal and intersecting this line with the approximating surface. Fig. 16. Type-2 triangles are collapsed to the point on the approximat- ing surface that corresponds to the triangle boundary vertex. Fig. 17. Several triangles of the mesh can be collapsed simultaneously. The triangle stencils of 7; and Tz must not intersect. Fig. 18. Smooth transition between meshes. J. 19. Producing vertices of high valence. If the dark-shaded triangle in (a) is collapsed, the mesh in (b) is produced. If the dark triangle in (b) is llapsed, the mesh in (c) is produced, which contains a vertex of valence 12. SUMMARIES OF THE MESH REDUCTION PROCESS FOR THE FOUR DATA SETS Fig. 20. Curvature and error weights on a portion of the Crater Lake data set. Fig. 21. The triangle T is the result of triangle T’ after the collapse of T¢. The ancestral points of T are the union of the vertices of T’ and Tc. els, and the triangles selected for collapse at level 1 and at level 10. Due to a user-defined limit on the number of trian- gles selected for each level, the algorithm initially selects tri- angles close to the saddle point. However, as the algorithm proceeds, the triangles selected for collapse are nearly uni- formly distributed over the mesh. Even though the mesh at level 30 has only 2.4 percent of the original number of trian- gles, it still gives an adequate representation. This model originally has 303,454 triangles. Fig. 23b shows the triangles and their stencils chosen for collapse in the initial level. We note that these are distributed in the nearly flat areas of the data set. At level 10, most of the triangles in the flat areas have been collapsed, and the area weight begins to force the algorithm to choose triangles in the highly curved areas of the mesh (Fig. 23d). At level 30, the reduced data set has less than 2.5 percent of the original number of triangles. While most of the detail has disappeared, this still is a very good representa- tion of the Mount St. Helens region. gles there fit the resulting surface very well. Eventually, th area weight forces the algorithm to consider these triangles
