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A geometrical approach to solve the proximity of a point to an axisymmetric quadric in space

Bibekananda Patra, Aditya Mahesh Kolte, Sandipan Bandyopadhyay

TL;DR

The paper addresses the problem of finding the shortest distance from a point to an axisymmetric quadric (AQ) in $\mathbb{R}^3$ by reducing the 3D proximity problem to a 2D conic problem via a plane that contains the AQ's symmetry axis. It introduces an invariant-based identification of AQ surfaces and a novel 2D proximity approach that uses conic geometry, including cubic and quartic univariate equations with special-case handling. The method yields a fast, implementable solution with demonstrations across spheroids, hyperboloids, cones, paraboloids, cylinders, and spheres, and it reports favorable performance against the Bullet library in selected tests. The work provides a practical, geometry-driven framework suitable for applications in robotics, graphics, and collision avoidance, with a C implementation and extensive numerical examples.

Abstract

This paper presents the classification of a general quadric into an axisymmetric quadric (AQ) and the solution to the problem of the proximity of a given point to an AQ. The problem of proximity in $R^3$ is reduced to the same in $R^2$, which is not found in the literature. A new method to solve the problem in $R^2$ is used based on the geometrical properties of the conics, such as sub-normal, length of the semi-major axis, eccentricity, slope and radius. Furthermore, the problem in $R^2$ is categorised into two and three more sub-cases for parabola and ellipse/hyperbola, respectively, depending on the location of the point, which is a novel approach as per the authors' knowledge. The proposed method is suitable for implementation in a common programming language, such as C and proved to be faster than a commercial library, namely, Bullet.

A geometrical approach to solve the proximity of a point to an axisymmetric quadric in space

TL;DR

The paper addresses the problem of finding the shortest distance from a point to an axisymmetric quadric (AQ) in by reducing the 3D proximity problem to a 2D conic problem via a plane that contains the AQ's symmetry axis. It introduces an invariant-based identification of AQ surfaces and a novel 2D proximity approach that uses conic geometry, including cubic and quartic univariate equations with special-case handling. The method yields a fast, implementable solution with demonstrations across spheroids, hyperboloids, cones, paraboloids, cylinders, and spheres, and it reports favorable performance against the Bullet library in selected tests. The work provides a practical, geometry-driven framework suitable for applications in robotics, graphics, and collision avoidance, with a C implementation and extensive numerical examples.

Abstract

This paper presents the classification of a general quadric into an axisymmetric quadric (AQ) and the solution to the problem of the proximity of a given point to an AQ. The problem of proximity in is reduced to the same in , which is not found in the literature. A new method to solve the problem in is used based on the geometrical properties of the conics, such as sub-normal, length of the semi-major axis, eccentricity, slope and radius. Furthermore, the problem in is categorised into two and three more sub-cases for parabola and ellipse/hyperbola, respectively, depending on the location of the point, which is a novel approach as per the authors' knowledge. The proposed method is suitable for implementation in a common programming language, such as C and proved to be faster than a commercial library, namely, Bullet.

Paper Structure

This paper contains 24 sections, 44 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Identification of an axisymmetric quadric from a generic quadric
  4. Computation of the proximity of a given point to an axisymmetric quadric
  5. Determination of intersecting plane $\mathcal{P}$
  6. Intersection of $\mathcal{P}\xspace$ and a spheroid
  7. Intersection of $\mathcal{P}\xspace$ with hyperboloid of one sheet and hyperboloid of two sheets
  8. Intersection of $\mathcal{P}\xspace$ with a cone
  9. Intersection of $\mathcal{P}\xspace$ with a sphere
  10. Intersection of $\mathcal{P}\xspace$ with a paraboloid
  11. Intersection of $\mathcal{P}\xspace$ with a cylinder
  12. Computation of $\boldsymbol{p}\xspace_\text{p}$ on $\mathcal{P}\xspace$
  13. Proximity of $\boldsymbol{p}\xspace_{\text{p}}\xspace$ and conics in $\mathbb{R}\xspace^2$
  14. Proximity of $\boldsymbol{p}\xspace_{\text{p}}\xspace$ to a circle $(e=0)$
  15. Proximity of $\boldsymbol{p}\xspace_{\text{p}}\xspace$ to a parabola $(e=1)$
  16. ...and 9 more sections

Figures (17)

  • Figure 1: Intersecting plane, which contains the axis $\boldsymbol{v}\xspace_3$, and the points $\boldsymbol{p}\xspace_0$, and $\boldsymbol{p}\xspace_{\text{c}}\xspace$, that intersects the AQ, e.g., a spheroid in this case.
  • Figure 2: Examples of proximity of the point $\boldsymbol{p}\xspace_{\text{p}}\xspace$ to an ellipse and a hyperbola in $\mathbb{R}\xspace^2$
  • Figure 3: An example of the proximity of the point $\boldsymbol{p}\xspace_{\text{p}}\xspace$ to a parabola in $\mathbb{R}\xspace^2$
  • Figure 4: Classification of a quadric into the AQs
  • Figure 5: Three planes resulting from Eq. (\ref{['eq:gradients']}) intersecting at the centre, $\boldsymbol{p}\xspace_{\text{c}}\xspace$, of a central quadric
  • ...and 12 more figures