[Paper Review] Simple and Robust Boolean Operations for Triangulated Surfaces

Gang Mei, John C. Tipper|arXiv (Cornell University)|Aug 20, 2013

Advanced Numerical Analysis Techniques25 references18 citations

TL;DR

This paper presents a simple and robust method for Boolean operations on closed and open triangulated surfaces using an Octree-based spatial partitioning to efficiently identify intersecting triangle pairs, followed by parallel computation of intersection lines. The approach forms closed or open intersection loops and sub-surfaces purely from topological updates without coordinate re-computation, enabling accurate and efficient union, intersection, and subtraction operations through a novel topology-driven classification technique instead of traditional inside/outside point testing.

ABSTRACT

Boolean operations of geometric models is an essential issue in computational geometry. In this paper, we develop a simple and robust approach to perform Boolean operations on closed and open triangulated surfaces. Our method mainly has two stages: (1) We firstly find out candidate intersected-triangles pairs based on Octree and then compute the inter-section lines for all pairs of triangles with parallel algorithm; (2) We form closed or open intersection-loops, sub-surfaces and sub-blocks quite robustly only according to the cleared and updated topology of meshes while without coordinate computations for geometric enti-ties. A novel technique instead of inside/outside classification is also proposed to distinguish the resulting union, subtraction and intersection. Several examples have been given to illus-trate the effectiveness of our approach.

Motivation & Objective

• To develop a simple and robust algorithm for Boolean operations on manifold triangulated surfaces, whether open or closed.
• To reduce computational cost by efficiently identifying potentially intersecting triangle pairs using spatial decomposition via Octree.
• To eliminate reliance on coordinate-based inside/outside classification by introducing a topology-driven method for distinguishing union, intersection, and subtraction results.
• To ensure robustness and efficiency in handling complex surface intersections without requiring exact arithmetic or volumetric representations.

Proposed method

• Employing an Octree data structure to accelerate the search for candidate intersected triangle pairs, reducing the number of pairwise intersection tests.
• Using a parallel algorithm to compute intersection lines between all candidate triangle pairs identified by the Octree.
• Classifying intersection loops as open, hard closed, or soft closed based on topological connectivity and vertex sharing patterns.
• Growing sub-surfaces from closed intersection loops by treating them as initial advancing fronts, using only topological mesh updates without geometric coordinate recalculations.
• Assembling sub-blocks (union, intersection, subtraction) based on topological relationships between sub-surfaces and original input blocks.
• Distinguishing result types by comparing min and max coordinates of updated vertices, avoiding traditional point-in-volume classification.

Experimental results

Research questions

• RQ1 How can Boolean operations on triangulated surfaces be made both efficient and robust without relying on coordinate-based inside/outside classification?
• RQ2 What spatial data structure can most effectively reduce the number of pairwise triangle intersection tests in Boolean operations?
• RQ3 How can intersection loops be topologically classified and used to robustly form sub-surfaces and sub-blocks?
• RQ4 Can sub-blocks for union, intersection, and subtraction be assembled and distinguished purely from topological mesh updates, without geometric coordinate re-computation?

Key findings

• The method successfully performs Boolean operations on open-and-open, open-and-closed, and closed-and-closed triangulated surfaces with consistent topological robustness.
• Intersection loops are correctly classified as open, hard closed, or soft closed, enabling reliable sub-surface formation.
• Sub-blocks for union, intersection, and subtraction are assembled and distinguished using only topological information and vertex coordinate bounds, avoiding complex point-in-volume tests.
• The use of Octree and parallel processing significantly reduces the number of expensive triangle-triangle intersection computations.
• The approach maintains manifold topology throughout, even after complex intersections, by relying on cleared and updated mesh topology rather than geometric coordinates.