Fast and Efficient Calculations of Structural Invariants of Chirality

TL;DR

This paper introduces five new three-dimensional chirality invariants based on generating functions, which are simple, computationally efficient, and effective for symmetry detection and shape analysis in various scientific fields.

Contribution

The paper presents five novel 3D chirality invariants with low complexity, brief expressions, and proven effectiveness, including three invariants proposed for the first time.

Findings

The five invariants are effective for symmetry detection.

They have low computational complexity O(n).

Three invariants decode a universal chirality index.

Abstract

Chirality plays an important role in physics, chemistry, biology, and other fields. It describes an essential symmetry in structure. However, chirality invariants are usually complicated in expression or difficult to evaluate. In this paper, we present five general three-dimensional chirality invariants based on the generating functions. And the five chiral invariants have four characteristics:(1) They play an important role in the detection of symmetry, especially in the treatment of 'false zero' problem. (2) Three of the five chiral invariants decode an universal chirality index. (3) Three of them are proposed for the first time. (4) The five chiral invariants have low order no bigger than 4, brief expression, low time complexity O(n) and can act as descriptors of three-dimensional objects in shape analysis. The five chiral invariants give a geometric view to study the chiral invariants. And the experiments show that the five chirality invariants are effective and efficient, they can be used as a tool for symmetry detection or features in shape analysis.