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Surface. Curve. Point. Surface. 2 : Surfaces overlap 1 : Surfaces are tangent along a curve 0 : Surfaces are tangent at a point. 1 : A curve lies on a surface 0 : A curve is tangent to a surface at a point. 0 : A point lies on a surface. Curve. 1 : Curves overlap

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Surface

Curve

Point

Surface

2: Surfaces overlap

1: Surfaces are tangent along a curve

0: Surfaces are tangent at a point

1: A curve lies on a surface

0: A curve is tangent to a surface at a point

0: A point lies on a surface

Curve

1: Curves overlap

0: Curves intersect tangentially

0: A point lies on a curve

Point

0: Points coincide

ACM Symposium on Solid Modelling and Applications SM’04

Handling Degeneracies in

Exact Boundary Evaluation

John Keyser Koji Ouchi

Texas A&M University

http://research.cs.tamu.edu/keyser/geom/

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

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 ofprimitives

  • 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-0220047 and NSF/DARPA CARGO award DMS-0138446

Bradley models provided courtesy of Army Research Lab


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