Hybrid Trigonometric Varieties

TL;DR

This paper introduces hybrid trigonometric parametrizations combining rational, circular, and hyperbolic functions, characterizes the resulting varieties as real unirational, and provides algorithms for their implicitization and conversion.

Contribution

It defines hybrid trigonometric parametrizations, proves their varieties are exactly the real unirational varieties, and develops algorithms for implicitization and conversion.

Findings

Hybrid trigonometric varieties are exactly the real unirational varieties.

Algorithms for implicitization of hybrid trigonometric parametrizations.

Methods to convert between hybrid trigonometric and unirational parametrizations.

Abstract

In this paper we introduce the notion of hybrid trigonometric parametrization as a tuple of real rational expressions involving circular and hyperbolic trigonometric functions as well as monomials, with the restriction that variables in each block of functions are different. We analyze the main properties of the varieties defined by these parametrizations and we prove that they are exactly the class of real unirational varieties. In addition, we provide algorithms to implicitize and to convert a hybrid trigonometric parametrization into a unirational one, and viceversa.