# Dynamic evaluation of exponential polynomial curves and surfaces via basis transformation

## TL;DR

This paper introduces a robust and efficient basis transformation algorithm for the dynamic evaluation of exponential polynomial curves and surfaces, improving accuracy and reducing computational costs compared to traditional differential system methods.

## Contribution

It presents an explicit basis transformation approach that enables robust, accurate, and computationally efficient dynamic evaluation of exponential polynomial curves and surfaces with changing parameters.

## Key findings

- The new method provides robust and accurate evaluations for any parameter steps.
- It reduces computational time by using a constant matrix for polynomial curves.
- The approach outperforms conventional differential system solutions in efficiency.

## Abstract

It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can be evaluated dynamically by solving the differential systems numerically. In this paper we present an even more robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transform of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a pre-computed matrix. Like former dynamic evaluation algorithms, the newly proposed approach needs only arithmetic operations for evaluating exponential polynomial curves and surfaces. Unlike conventional numerical methods that solve a linear differential system, the new method can give robust and accurate evaluation results for any chosen parameter steps. Basis transformation technique also enables dynamic evaluation of polynomial curves with changing parameter steps using a constant matrix, which reduces time costs significantly than computing each point individually by classical algorithms.