# Rational Curves with Polynomial Parameterization

### Dinesh Manocha and John F. Canny

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-90-560

February 1990

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/CSD-90-560.pdf

Rational curves and splines are one of the building blocks of computer graphics and geometric modeling. Although a rational curve is more flexible than its polynomial counterpart, many properties of polynomial curves are not applicable to it. For this reason it is very useful to know if a curve presented as a rational space curve has a polynomial parametrization. In this paper, we present an algorithm to decide if a polynomial parametrization exists, and to compute the parametrization.

In algebraic geometry it is known that a rational algebraic curve is polynomially parametrizable if it has one place at infinity. This criterion has been used in earlier methods to test polynomial parametrizability of space curves. These methods project the curve into the plane and test parametrizability there. But this gives only a sufficient condition for the original curve. In this paper we give a simple condition which is both necessary and sufficient for polynomial parametrizability. The calculation of the polynomial parametrization is simple, and involves only a rational reparametrization of the curve.

BibTeX citation:

@techreport{Manocha:CSD-90-560, Author = {Manocha, Dinesh and Canny, John F.}, Title = {Rational Curves with Polynomial Parameterization}, Institution = {EECS Department, University of California, Berkeley}, Year = {1990}, Month = {Feb}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/6181.html}, Number = {UCB/CSD-90-560}, Abstract = {Rational curves and splines are one of the building blocks of computer graphics and geometric modeling. Although a rational curve is more flexible than its polynomial counterpart, many properties of polynomial curves are not applicable to it. For this reason it is very useful to know if a curve presented as a rational space curve has a polynomial parametrization. In this paper, we present an algorithm to decide if a polynomial parametrization exists, and to compute the parametrization. <p>In algebraic geometry it is known that a rational algebraic curve is polynomially parametrizable if it has one place at infinity. This criterion has been used in earlier methods to test polynomial parametrizability of space curves. These methods project the curve into the plane and test parametrizability there. But this gives only a sufficient condition for the original curve. In this paper we give a simple condition which is both necessary and sufficient for polynomial parametrizability. The calculation of the polynomial parametrization is simple, and involves only a rational reparametrization of the curve.} }

EndNote citation:

%0 Report %A Manocha, Dinesh %A Canny, John F. %T Rational Curves with Polynomial Parameterization %I EECS Department, University of California, Berkeley %D 1990 %@ UCB/CSD-90-560 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/6181.html %F Manocha:CSD-90-560