Ashish Myles
Project Proposal:
Affine-Invariant Edge-Flip for Triangulation
Abstract:
The project will introduce a variation on Delaunay triangulation that is affine invariant.
This is useful in storing scatter data that is affine-transformable without explicitly
needing to store an explicit triangulation. I will employ barycentric coordinates to
develop an edge-flip criterion that is affine-invariant.
Introduction:
The “optimal” triangulation by many standards is the Delaunay triangulation. Its key
property is that it maximizes the minimum angle in the triangulation, resulting in “nicer”
looking triangles that are less susceptible to round-off errors in applications like finite
elements. However this triangulation is neither continuous nor affine invariant. Minor
changes to one of the vertices can change a few edges in the graph. Additionally, scaling
in any direction can completely change the triangulation. Notice that scaling the left quad
differently in the vertical and horizontal direction results in an edge-flip to maximize the
minimum angle.
I would like to address the latter issue. Affine-invariant triangulation is useful
representing such a triangulation by storing only the vertices, which are allowed to
undergo global affine transforms throughout their lifetime. My aim is to develop an edgeflip criterion that is affine-invariant based on barycentric coordinates.
To the best of my knowledge (after numerous online searches), no one else has an affineinvariant triangulation of point sets, and the standard triangulation used for most
applications is the Delaunay.
Method:
Barycentric coordinates in R2 are with reference to triangles. Given a, b, c, and p in R2,
the barycentric coordinates (u,v,w) of p with respect to the triangle abc, is uniquely
determined by the formula ua+vb+wc = p with the restriction that u+v+w=1.
Ashish Myles
(u,v,w) are invariant under global affine transformations, making them ideal for
determining an affine invariant edge-flip. In the case that p lies within the triangle, u, v,
and w are positive. In the case above, u and v are positive, but w is negative. This is the
case for all points that lie on the left half plane of ab, and hence will be the case for all
quads apbc where the edge is flip-able.
Given a quad apbc (as in the figure above above), the following are true:
ua + vb + wc = p
(u > 0, v > 0, w < 0)
(1/w)p – (u/w)a – (v/w)b = c
(1/w < 0, -u/w > 0, -u/v > 0)
–(v/u)b + (1/u)p – (w/u)c = a
(-v/u < 0, 1/u > 0, -w/u > 0)
–(u/v)a + (1/v)p – (w/v)c = b
(-u/v < 0, 1/v > 0, -w/v > 0)
Hence, once we have the barycentric coordinates of one point with respect to the other
three, all the barycentric coordinates are immediately available. Notice the relation
between opposite points: if c’s weight of p is w, then p’s weight of c is 1/w. We could
assign each diagonal ab and pc the values max(v/u, u/v) and max(-1/w, -w), respectively,
and consistently choose the edge whose value is either larger or smaller for the edge-flip
criterion.
Given edge-flip above, we need a proof of convergence. Each Delauney edge-flip only
further minimizes a particular objective function and hence eventually converges. I need
to similarly characterize the properties of the proposed edge-flip algorithm or come up
with another affine-invariant version that works.
References:
[1] Boris, N. Delaunay, (1934) "Sur la Sphere" Vide. Izvestia Akademia Nauk SSSR, VII
Seria, Otdelenie Matematicheskii i Estestvennyka Nauk Vol 7 pp.793-800
[2] Jonathan Richard Shewchuk, (Sep 20, 1999) Lecture Notes on Delaunay Mesh
Generation. http://citeseer.ist.psu.edu/shewchuk99lecture.html