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

Jul 16, 2026

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This paper classifies a general quadric into an axisymmetric quadric (AQ) and addresses the proximity of a given point to an AQ. The proximity problem in R3 is transformed into an equivalent problem in R2, enabling a geometrically driven solution. The R2 method exploits conic properties—sub-normal, semi-major axis length, eccentricity, slope, and radius—then splits the point casework for parabola and for ellipse/hyperbola. The approach is implementable in common programming languages and is shown to be faster than Bullet for the described task.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages35

Reading Time1 h 28 min

Introduction

Problem of point-to-quadric proximity

Reduction from R3 to R2 via axis symmetry

Conic-based formulation

Motivation from existing literature

arXiv :2510 .08973v2 [ cs .RO] 11 Jul 2026 A geometrical approach to determine the proximity of a point to an axisymmetric quadric in space Bibekananda Patra, Aditya Mahesh Kolte, and Sandipan Bandyopadhyay ∗ Department of Engineering Design Indian Institute of Technology Madras, Chennai 600036, India 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 R3 is reduced to the same in R2 , which is not found in the literature. A new method to solve the problem in R2 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 R2 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. Keywords: Axisymmetric quadric, Central axisymmetric quadric, Proximity, Conics | Nomenclature AQS x I λ p0 pc v 3 N (·)∥ · ∥ηi = 0 pprmin | Axisymmetric quadric A quadratic surface or a quadric [x, y, z]⊤ , a point in R3 An identity matrix of appropriate dimension Eigenvalue of a matrix A given point in R3 which does not lie on S Centre or vertex of an AQ Axis of symmetry of an AQ Nullspace of a matrix Euclidean norm of any vector Equation of a conic Point corresponding to p0 represented in a plane Shortest distance between p0 and an AQ | | --- | --- | 1 Introduction The shortest distance proximity of a quadratic surface in R3 (also known as quadric) to a given point which does not lie on the given surface has a wide range of applications in engineering and technology, ∗ Corresponding author Email addresses: [bibeka.patra2@gmail.com](bibeka.patra2@gmail.com) (Bibekananda Patra), [adityakolte72@gmail.com](adityakolte72@gmail.com) (Aditya Mahesh Kolte), [sandipan@iitm.ac.in](sandipan@iitm.ac.in) (Sandipan Bandyopadhyay) Figure 1: Intersecting plane, which contains the axis v3 , and the points p0 , and pc, that intersects the AQ, e.g., a spheroid in this case. such as robotics, computer graphics, computer vision, computer-aided design, manufacturing, and the global positioning system. In general, quadrics are classified into 17 types of surfaces [1] (see, pp. 605-606) including imaginary ones. The real surfaces, such as ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, elliptic paraboloids, elliptic cones, elliptic cylinders, and so on, are discovered to be used in applications for all practical reasons. Many researchers have tried to compute the least distance between a given point and a quadric. The analytical derivation of the proximity of a point to an ellipsoid was derived in [2], and [3] . The most common quadric used in engineering applications is perhaps the ellipsoid, as mentioned in [4], [5], [6], [7] . There are a few research articles which focus on the proximity of a point to an ellipsoid [8] (see, pp. 403-405),[9], and [10] . The problem of proximity was solved as the projection of a point onto an ellipsoid in an optimisation framework and a comparison is made among the seven fast numerical algorithms in [11] . The shortest distance between a hyperboloid and a point was solved numerically in [12] . The applications related to the proximity of a point and other quadrics are not found in the literature. Moreover, the research on the computation of the proximity of a point to an axisymmetric quadric (AQ) is not found in the literature according to authors’ knowledge. This paper focuses only on the identification of an AQ and the computation of the normal distance from a point to the surface. The researchers hav

FAQs

What problem does the paper solve?

It computes the shortest distance (proximity) from a point not lying on an axisymmetric quadric (AQ) to the AQ surface, by identifying the AQ and deriving the normal distance.

How is the proximity problem reduced from R3 to R2?

Using the axis of symmetry of the AQ, a plane containing both the axis and the given point is constructed; the intersection with the quadric forms a conic, making the R3 proximity equivalent to a point-to-conic normal distance problem in that plane.

What geometric properties are used to solve the R2 problem?

The method uses geometric properties of conics such as sub-normal, semi-major axis length, eccentricity, slope, and radius, and further categorizes cases for parabola and ellipse/hyperbola based on the point location.

Jul 16, 2026

Upload by Patrick