# IJPAM: Volume 31, No. 3 (2006)

**ON THE ARC LENGTH PARAMETRIZATION PROBLEM**

Department of Computer Science

Technion - Israel Institute of Technology

Technion City, Haifa, 32000, ISRAEL

e-mail: yogi@cs.technion.ac.il

**Abstract.**Arc length parametrization can be thought of as the most
natural among all possible parameterizations of a given curve.
Beyond having several nice mathematical properties, this parametrization
is useful for computer graphics applications:
drawing a curve given in this form and computing
its length, are particularly easy.
Unfortunately, for the curves used mostly in computer graphics,
namely cubic splines, the arc length parametrization
cannot, in general, be expressed as any elementary function.
As a result, practitioners are forced into employing approximate numerical
methods which are in many cases complex, computationally intensive,
and susceptible to error accumulation due to their iterative
nature.

This paper explores the following new direction
to this problem.
Instead of the traditional classes of curves, such as polynomials and
rational functions, we promote the usage of a new class of
curves which are all given in an *explicit*
arc length parametrization form.
On a par with polynomials, it is possible to select a subclass of
curves which has any desired number of degrees of freedom.
Our results show that several important settings of the general
interpolation problem can be solved using curves from this
class.

**Received: **August 13, 2006

**AMS Subject Classification: **26B15, 51M25

**Key Words and Phrases: **arc length, length parametrization

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2006

**Volume:** 31

**Issue:** 3