[Paper Review] Finding an ordinary conic and an ordinary hyperplane

Olivier Devillers, Asish Mukhopadhyay|ArXiv.org|Sep 27, 1999

Computational Geometry and Mesh Generation11 references3 citations

TL;DR

This paper extends the classical concept of ordinary lines to ordinary conics and ordinary hyperplanes in higher dimensions. By adapting an existing algorithm for finding ordinary lines, the authors present efficient methods to compute a conic passing through exactly five points (ordinary conic) when not all points lie on a single conic, and an ordinary hyperplane in 3D and higher, where only one point lies on the hyperplane. The approach simplifies prior proofs and algorithms, offering a unified and efficient computational framework.

ABSTRACT

Given a finite set of non-collinear points in the plane, there exists a line that passes through exactly two points. Such a line is called an ordinary line. An efficient algorithm for computing such a line was proposed by Mukhopadhyay et al. In this note we extend this result in two directions. We first show how to use this algorithm to compute an ordinary conic, that is, a conic passing through exactly five points, assuming that all the points do not lie on the same conic. Both our proofs of existence and the consequent algorithms are simpler than previous ones. We next show how to compute an ordinary hyperplane in three and higher dimensions.

Motivation & Objective

• To extend the concept of ordinary lines to ordinary conics in the plane.
• To develop an efficient algorithm for computing an ordinary conic passing through exactly five points.
• To generalize the ordinary line concept to ordinary hyperplanes in three and higher dimensions.
• To simplify existing proofs and algorithms for ordinary conics and hyperplanes.
• To provide a unified computational framework for finding such geometric objects using a core algorithmic approach.

Proposed method

• Adapt the existing algorithm for finding ordinary lines to detect ordinary conics by leveraging point sets not lying on a single conic.
• Use combinatorial and geometric properties to ensure that a conic passes through exactly five points from the input set.
• Apply duality and incidence geometry techniques to extend the method to higher dimensions for hyperplane computation.
• Construct a hyperplane that contains exactly one point from the given point set in 3D and higher dimensions.
• Use the same core algorithmic structure as in the ordinary line case, generalized to higher-dimensional incidence structures.
• Ensure correctness and efficiency through careful selection of point configurations and geometric invariants.

Experimental results

Research questions

• RQ1 Can the algorithm for finding ordinary lines be extended to compute an ordinary conic in the plane?
• RQ2 What conditions ensure the existence of a conic passing through exactly five points from a non-conic point set?
• RQ3 How can the concept of ordinary lines be generalized to hyperplanes in three and higher dimensions?
• RQ4 What is the computational complexity of finding an ordinary hyperplane in higher-dimensional spaces?
• RQ5 Can the proofs and algorithms for ordinary conics and hyperplanes be simplified compared to prior work?

Key findings

• An ordinary conic—passing through exactly five points—can be computed efficiently when all points are not concyclic or co-conic.
• The algorithm for finding ordinary conics is simpler and more efficient than previous approaches, relying on the same foundational logic as the ordinary line algorithm.
• An ordinary hyperplane in d-dimensional space (d ≥ 3) can be computed such that it contains exactly one point from the input set.
• The method generalizes naturally from 2D to higher dimensions using duality and incidence geometry.
• The existence proofs and algorithmic constructions are both simpler and more intuitive than earlier works in the field.
• The approach maintains computational efficiency while extending the scope of the ordinary object problem to higher-degree curves and higher dimensions.