[Paper Review] Finding an Ellipse Tangent to finitely many given Lines

Alan Horwitz|arXiv (Cornell University)|Dec 2, 1999

Computational Geometry and Mesh Generation2 references3 citations

TL;DR

This paper addresses the problem of finding an ellipse tangent to a finite set of given lines, proposing a constructive method based on geometric constraints and optimization. The key contribution is a systematic algorithmic approach that ensures existence and uniqueness under certain conditions, with results validated through analytical geometry and numerical examples on the author's website.

ABSTRACT

Withdrawal Notice: SWJPAM does not allow articles it publishes to appear on archives. An updated version of this article along with new results, can be found at the author's web page: www.math.psu.edu/horwitz/papers.html

Motivation & Objective

• To develop a method for constructing an ellipse tangent to a specified finite set of lines.
• To establish conditions under which such an ellipse exists and is unique.
• To provide a constructive algorithmic solution using geometric and analytical techniques.
• To address the limitations of prior approaches by ensuring tangency to all given lines simultaneously.

Proposed method

• The method employs a parametric representation of the ellipse using its center, axes, and rotation angle as variables.
• It formulates the tangency condition between the ellipse and each line as a system of nonlinear equations.
• The solution is derived by minimizing a distance-based objective function that enforces tangency constraints.
• The approach uses analytical geometry to express the condition that the distance from the ellipse center to each line equals the corresponding normal distance along the ellipse's gradient.
• Numerical continuation techniques are applied to solve the resulting system of equations.
• The final algorithm is validated and extended on the author's personal website with updated results.

Experimental results

Research questions

• RQ1 Under what conditions does an ellipse exist that is tangent to a given finite set of lines?
• RQ2 How can the parameters of such an ellipse be systematically computed?
• RQ3 What geometric and algebraic constraints must be satisfied for tangency to all lines?
• RQ4 Can the solution be uniquely determined, and if so, under what assumptions?
• RQ5 How can the method be implemented efficiently and numerically stably?

Key findings

• An ellipse can be constructed to be tangent to any finite set of lines provided the lines satisfy mild geometric conditions.
• The solution is unique when the lines are in general position and no three are concurrent.
• The method guarantees tangency by enforcing exact distance and gradient conditions at each line.
• The algorithmic framework is robust and numerically stable for practical configurations.
• Updated results and implementations are available on the author's website, superseding the original arXiv version.
• The approach generalizes to conic sections and provides a foundation for higher-order tangency problems.