The Three-Dimensional Parametric Ellipse

Clark R. Andersen and William L. Buford, Jr.
Department of Orthopedics, University of Texas Medical Branch

This paper appears in issue Volume 6, Number 3.
Purchase this issue from the akpeters.com web site.

Abstract

The common approach to generating an arbitrarily oriented ellipse or circle in three dimensions is to generate it first in two dimensions, then transform the result into the desired position and orientation. We present a technique to model the ellipse directly as a three-dimensional parametric function, a constrained variation of a Lissajous curve, avoiding the need for subsequent transformations.

Author Information

Clark R. Andersen, Galveston, Texas clanders@utmb.edu

William L. Buford, Jr., Galveston, Texas wbuford@utmb.edu

Supplements

• Some related information in the form of mathematica worksheets:

  • Proof that the path described by the parametric function is indeed an ellipse: EllipseProof.nb

  • Proof that the path is coplanar: EllipseCoplanarProof.nb

  • The relation of rho and phi to the orientation and size of the ellipse: EllipseComponents.nb

  • Finding the vertices of the ellipse when rho and phi are known: EllipseVertices.nb

• A document discussing the determination of rho and phi to generate an ellipse with particular semimajor and semiminor axes and normal: SolvingForParametricEllipse.doc

@article{AndersenBuford01,
  author = "Clark R. Andersen and William L. Buford, Jr.",
  title = "The Three-Dimensional Parametric Ellipse",
  journal = "journal of graphics tools",
  volume = "6",
  number = "3",
  pages = "45-48",
  year = "2001",
}