1. Handling Degeneracies in Exact Boundary Evaluation
ACM Symposium on Solid Modelling and Applications SM’04
Handling Degeneracies in
Exact Boundary Evaluation
John Keyser Koji Ouchi
Texas A&M University
Goal
Develop robust boundary evaluation
Input: A CSG tree of solids in B-rep + a tolerance
Output: A degeneracy-free solid in B-rep
Robust = exact + degeneracy-free
We are free to adjust the surfaces of input solids within a tolerance
Maintain the designer’s intent
Use exact computation in order to eliminate numerical errors
No intentional degeneracy occurs
Input degeneracies are removed by numerical perturbation
Every surface of the output solid is parallel to some surface of the degenerate solid (or artificially added)
Detecting Degeneracies
Degeneracies are detected by checking irregular intersections
Use exact computation of surfaces/curves and points
(polynomials and algebraic numbers)
The library MAPC for manipulate algebraic points and curves
using Sylvester resultant and the box-hit algorithm
The library ERUR for computing
the rational univariate reduction of a polynomial system
using the toric resultants
0: Points coincide
Point
0: A point lies on a surface
1: A curve lies on a surface
0: A curve is tangent to a surface at a point
2: Surfaces overlap
1: Surfaces are tangent along a curve
0: Surfaces are tangent at a point
Surface
0: A point lies on a curve
1: Curves overlap
0: Curves intersect tangentially
Curve
Table: Types of degeneracies
The entries show degeneracies between surfaces, curves and points. The order of each degenerate intersections involved in bold: (points 0, curves 1, surfaces 2)
Removing Degeneracies
Degeneracies are removed by expansion / contraction of primitives
Perturb the surfaces of primitives inward / outward depending on the operations applied to them
The perturbation information at the root is propagated to the leaves
A  B: expand both A and B
A  B: contract both A and B
A  B: expand A, contract B
Every surface of the output solid is parallel to some surface of the degenerate solid
in order to catch the designer’s intent
Thus, exact computation is required
But, there are some shortcomings:
An example of degeneracy
that cannot be removed
Undesirable small faces are created
Real World Example
Examples from a Bradley Fighting Vehicle developed by the Army Research Lab
Cargo hatch is a join of 11 solids, 4 of them have degeneracies
Commander hatch is a join of 5 solids, 1 of them has a degeneracy
Engine is a join of 13 solids, 3 of them have degeneracies
The perturbed versions run slower than the unperturbed versions, and require more exact computation than floating-point filters
For Cargo hatch, the run-time increases 38% and the ratio of exact computation increases from 6.4% to 7.4%
For Commander hatch, the run-time increases 633% and the ratio of exact computation increases from 3.7% to 7.9%
For Engine, the run-time increases 24% and the ratio of exact computation increases from 3.6% to 4.5%
Cargo hatch
Commander hatch
Engine
Work supported in part by NSF ITR award CCR and NSF/DARPA CARGO award DMS
Bradley models provided courtesy of Army Research Lab