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Characterization of polynomial surfaces of revolution and polynomial quadrics

Michal Bizzarri, Miroslav Lávička, J. Rafael Sendra, Jan Vršek

Abstract

In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial parametrization, over $\mathbb{C}$, of the surface. Furthermore, we perform the first steps towards the analysis of the existence, and actual computation, of real polynomial parametrizations of surfaces of revolution. As a consequence, we give a complete picture of the real polynomiality of quadrics and we formulate a conjecture for the general case.

Characterization of polynomial surfaces of revolution and polynomial quadrics

Abstract

In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial parametrization, over , of the surface. Furthermore, we perform the first steps towards the analysis of the existence, and actual computation, of real polynomial parametrizations of surfaces of revolution. As a consequence, we give a complete picture of the real polynomiality of quadrics and we formulate a conjecture for the general case.
Paper Structure (10 sections, 26 equations, 1 figure, 1 table)

This paper contains 10 sections, 26 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and preliminary results
  3. Polynomial Surfaces of Revolution
  4. Polynomial Surfaces of Revolution with Real Parameterizations
  5. $\boldsymbol{p(z)=-z^2-|\lambda|}$
  6. $\boldsymbol{p(z)=-z^2+|\lambda|}$
  7. $\boldsymbol{p(z)=z^2+|\lambda|}$
  8. $\boldsymbol{p(z)=z^2-|\lambda|}$
  9. Addendum -- (real) polynomial parametrizations of quadrics
  10. Summary

Figures (1)

  • Figure 1: Surface of revolution. (a) Rotating a curve (green) around an $z$-axis yields a surface of revolution $S$. (b) Profile curve $P \subset S$ (blue) corresponds to the section of $S$ with the $xz$-plane. (c) Mapping $\mathbb{A}^2\rightarrow\mathbb{A}^2$ given by $[x,z]\mapsto[x^2,z]$ yields the curve $P^2 \not\subset S$ (red).

Theorems & Definitions (2)

  • proof
  • proof