[Paper Review] Analytical Methods for Squaring the Disc

Chamberlain Fong|arXiv (Cornell University)|Sep 21, 2015

Computer Graphics and Visualization Techniques18 references30 citations

TL;DR

This paper presents analytical mappings that transform a circular disc into a square region with smooth, invertible, and geometrically desirable properties such as conformality, equiareality, and radial constraint. It introduces novel closed-form expressions for point-wise transformation, enabling applications in panoramic imaging, logo design, 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.

Motivation & Objective

• To develop smooth, invertible analytical mappings from a circular disc to a square region.
• To ensure mappings preserve key geometric properties such as conformality, equiareality, and radial constraints.
• To provide explicit, closed-form mathematical expressions for point-wise transformation.
• To explore practical applications in panoramic photography, logo design, and hyperbolic art.
• To extend prior work on rectangular image transformation, including the 'Elliptification of Rectangular Imagery' offshoot.

Proposed method

• Proposes a family of analytical mappings using piecewise-defined functions that map polar coordinates (u,v) in the disc to Cartesian coordinates (x,y) in the square.
• Employs radial scaling and angular correction techniques to maintain smoothness and invertibility across the domain.
• Introduces a conformal mapping variant that preserves angles locally, using complex analysis-inspired transformations.
• Develops an equiareal variant that preserves area elements, ensuring uniform distortion in area metrics.
• Applies radial constraint to maintain symmetry and avoid extreme stretching near the boundary.
• Validates mappings through visual and geometric analysis, with emphasis on continuity and differentiability.

Experimental results

Research questions

• RQ1 How can a smooth, invertible mapping be constructed from a circular disc to a square with minimal distortion?
• RQ2 What analytical formulations yield conformal or equiareal mappings between a disc and a square?
• RQ3 How can radial constraints be enforced to preserve symmetry and reduce boundary distortion?
• RQ4 What are the practical implications of such mappings in image processing and artistic visualization?
• RQ5 How do these mappings compare to existing numerical or heuristic approaches in terms of analytical tractability and geometric fidelity?

Key findings

• The paper successfully derives closed-form analytical expressions for mapping each point (u,v) in the disc to a corresponding point (x,y) in the square.
• The proposed mappings achieve smoothness and invertibility, with continuity and differentiability verified across the domain.
• Conformal variants preserve local angles, enabling applications where angular fidelity is critical.
• Equiareal variants maintain area ratios, making them suitable for texture and image mapping with uniform distortion.
• Radially-constrained mappings reduce edge stretching, improving visual quality in applications like panoramic imaging.
• The methods are demonstrated to be effective in practical domains, including logo design and hyperbolic art, with visual examples provided.