Planar shape manipulation using approximate geometric primitives

TL;DR

This paper introduces robust algorithms for set operations and Euclidean transformations of curved planar shapes using approximate primitives, ensuring consistency and accuracy even with degenerate inputs.

Contribution

It presents a refinement algorithm that guarantees consistency in geometric computations with approximate primitives, with proven complexity and validation on various curved shapes.

Findings

Algorithms are robust and accurate for shapes with algebraic degree 1 to 6.

Computational complexity is O(n log n + k), with k = O(n^2) for violations.

Validated on sequences of set operations and transformations with both generic and degenerate inputs.

Abstract

We present robust algorithms for set operations and Euclidean transformations of curved shapes in the plane using approximate geometric primitives. We use a refinement algorithm to ensure consistency. Its computational complexity is $\bigo(n\log n+k)$ for an input of size $n$ with $k=\bigo(n^2)$ consistency violations. The output is as accurate as the geometric primitives. We validate our algorithms in floating point using sequences of six set operations and Euclidean transforms on shapes bounded by curves of algebraic degree~1 to~6. We test generic and degenerate inputs. Keywords: robust computational geometry, plane subdivisions, set operations.