Narrow Results

We present the use of mapping functions to automatically generate levels of detail with known error bounds for polygonal models. We develop a piece-wise linear mapping function for each simplification operation and use this function to measure deviation of the new surface from both the previous level of detail and from the original surface. In addition, we use the mapping function to compute appropriate texture coordinates if the original model has texture coordinates at its vertices. Our overall algorithm uses edge collapse operations. We present rigorous procedures for the generation of local orthogonal projections to the plane as well as for the selection of a new vertex position resulting from the edge collapse operation. The algorithm computes guaranteed error bounds on surface deviation and produces an entire continuum of levels of detail with mappings between them. We demonstrate the effectiveness of our algorithm on several models: a Ford Bronco consisting of over 300 parts and 70, 000 triangles, a textured lion model consisting of 49 parts and 86, 000 triangles, a textured, wrinkled torus consisting of 79, 000 triangles, a dragon model consisting of 871, 000 triangles, a Buddha model consisting of 1,000,000 triangles, and an armadillo model consisting of 2, 000, 000 triangles.

The bisector of two plane curve segments (other than lines and circles) has, in general, no simple — i.e., rational — parameterization, and must therefore be approximated by the interpolation of discrete data. A procedure for computing ordered sequences of point/tangent/curvature data along the bisectors of polynomial or rational plane curves is described, with special emphasis on (i) the identification of singularities (tangent–discontinuities) of the bisector; (ii) capturing the exact rational form of those portions of the bisector with a terminal footpoint on one curve; and (iii) geometrical criteria the characterize extrema of the distance error for interpolants to the discretely–sample data. G¹ piecewise– parabolic and G² piecewise–cubic approximations (with O(h⁴) and O(h⁶) convergence) are described which, used in adaptive schemes governed by the exact error measure, can be made to satisfy any prescribed geometrical tolerance.