Analytical Methods for Squaring the Disc

TL;DR

This paper reviews and introduces various analytical methods for mapping a circular disc onto a square, emphasizing properties like smoothness, invertibility, and conformality, with applications in design and art.

Contribution

It provides new analytical expressions for disc-to-square mappings with desirable geometric properties and discusses their applications.

Findings

Presented analytical expressions for mapping points inside a disc to a square.

Highlighted mappings with conformal, equiareal, and radial properties.

Demonstrated applications in logo design, panoramic photography, and hyperbolic art.

Abstract

We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphasis on mappings with desirable properties. These include conformal, equiareal, and radially-constrained mappings. Finally, we present applications to logo design, panoramic photography, and hyperbolic art.