Fast and efficient calculations of structural invariants of chirality

Abstract

Chirality describes the structural properties or configuration of an object, and it plays an important role in fields such as physics, chemistry and biology. It is critical to identify achiral objects and determine symmetry plane of them in shape analysis, and chiral invariants play essential role in above tasks. However, chiral invariants are often complex and time-consuming to calculate in practical applications. In this paper, we present five chiral invariants based on the concept of generating functions. Three of the five chiral invariants decode the expression of a chiral index, and they are the core part of this chiral index. In addition, this paper provides three new chiral invariants. These five chiral invariants have three characteristics: (1) Their results on the shape before and after the reflection transformation are opposite to each other. (2) The five chiral invariants have brief form and low time complexity (O(n)), the order and degree of them are not more than four. (3) They can be used to determine the global symmetry plane of reflection symmetry objects, and they help to deal with false zeros problem. The five chiral invariants give the invariants under reflection transformation from the geometrical point of view, and the experimental results show that they are effective and efficient. In conclusion, our invariants can be used for determining archiral objects and mutual symmetry shape, as well as for determining symmetry plane of reflection symmetry objects.