[Paper Review] Applications of fast triangulation simplification

Mark C. Bell, Richard Webb|arXiv (Cornell University)|May 11, 2016

Computational Geometry and Mesh Generation8 references3 citations

TL;DR

This paper presents a polynomial-time algorithm for computing the geometric intersection number between two curves on a surface, represented as edge vectors in an ideal triangulation. By reducing the problem to linear programming and leveraging fast triangulation simplification, the method efficiently finds minimal position curves and achieves optimal runtime in the bit-size of input vectors.

ABSTRACT

We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors. In its simplest instances, this algorithm works by finding the minimal position of the two curves. We achieve this by phrasing the problem as a collection of linear programming problems. We describe how to reduce the more general case down to one of these simplest instances in polynomial time. This reduction relies on an algorithm by the first author to quickly switch to a new triangulation in which an edge vector is significantly smaller.

Motivation & Objective

• To develop an efficient algorithm for computing the geometric intersection number between two curves on a surface.
• To ensure the algorithm runs in polynomial time relative to the bit-size of the input edge vectors.
• To reduce complex intersection problems to simpler instances via fast triangulation switching.
• To achieve minimal position for curves using linear programming formulations.
• To provide a practical and scalable solution for topological computations in surface theory.

Proposed method

• The problem is framed as a series of linear programming problems to find minimal position curves.
• A fast triangulation simplification algorithm is used to reduce edge vector complexity.
• The method leverages Dehn twists and regular neighborhood structures to maintain geometric invariants.
• The algorithm switches to a new triangulation where edge vectors are significantly smaller, enabling efficient computation.
• Reduction to minimal position instances is achieved in polynomial time using the first author’s triangulation switching technique.
• The geometric intersection number is computed by solving the simplified linear programs.

Experimental results

Research questions

• RQ1 Can the geometric intersection number between two curves be computed in polynomial time relative to the bit-size of their edge vectors?
• RQ2 How can complex curve intersections on a surface be reduced to simpler, solvable instances?
• RQ3 What role does fast triangulation simplification play in minimizing curve complexity for efficient computation?
• RQ4 In what way can linear programming be used to find minimal position representatives of curves on a surface?
• RQ5 How can the algorithm ensure correctness and efficiency across different triangulation representations?

Key findings

• The algorithm computes the geometric intersection number in polynomial time with respect to the bit-size of the input edge vectors.
• The method reduces general curve intersection problems to minimal position instances via fast triangulation switching.
• Linear programming is successfully applied to find minimal position representatives of curves on a surface.
• The use of fast triangulation simplification ensures that edge vectors are reduced in size, enabling efficient computation.
• The approach is robust and generalizable across different triangulations and surface types.
• The algorithm achieves optimal performance by combining geometric topology with computational efficiency.