# Eccentric pie charts and an unusual pie cutting

## TL;DR

This paper introduces eccentric pie charts, a generalization of traditional pie charts, and presents a method for calculating sector areas using polynomial approximations and homotopy continuation, demonstrated on a pie cutting problem.

## Contribution

It develops a novel approach for analyzing eccentric pie charts by transforming the problem into polynomial systems and solving with homotopy continuation and Newton iteration.

## Key findings

- Method effectively computes sector areas in eccentric pie charts.
- Approach applicable to various nonlinear, non-polynomial systems.
- Illustrated on a specific pie cutting problem.

## Abstract

The eccentric pie chart, a generalization of the traditional pie chart is introduced. An arbitrary point is fixed within the circle and rays are drawn from it. A sector is bounded by a pair of neighboring rays and the arc between them, The sector's area, aimed to be equal to a given proportion, is calculated from some well known equations in coordinate geometry. The resulting system of polynomial and trigonometric equations can be approximated by a fully polynomial system, once the non-polynomial functions are approximated by their Taylor series written up to the first few terms. The roots of the polynomial system have been found by the homotopy continuation method, then used as starting points of a Newton iteration for the original (non-polynomial) system. The method is illustrated on a special pie cutting problem, and is applicable to a wide class of nonlinear, non-polynomial systems.