arXiv subjects

Michael Q. Rieck

Publications and source records attributed to Michael Q. Rieck.

3 recordsLinked to original sources

arXiv2025

An Elliptic Curve Based Solution to the Perspective-Three-Point Problem

The Perspective-Three-Point Problem (P3P) is solved by first focusing on determining the directions of the lines through pairs of control points, relative to the camera, rather than the distances from the camera to the control points. The analysis of this produces an efficient, accurate and reasonably simple P3P solver, which is compared with a state-of-the-art P3P solver, "Lambda Twist." Both methods depend on the accurate computation of a single root of a cubic polynomial. They have been implemented and tested for a wide range of control-point triangles, and under certain reasonable restrictions, the new method is noticably more accurate than Lambda Twist, though it is slower. However, the principal value of the present work is not in introducing yet another P3P solver, but lies rather in the discovery of an intimate connection between the P3P problem and a special family of elliptic curves that includes curves utilized in cryptography. This holds the potential for further advances in a number of directions. To make this connection, an interesting spherical analogue of an ancient "sliding" problem is stated and solved.

cs.CV↗

arXiv2023

Related Problems in Spherical and Solid Geometry

A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real projective plane. The problem on the sphere involves four great circles and their intersections. A substantial claim is made concerning this problem, and subsequently proved by relating the spherical problem to a compelling problem in solid geometry. This latter problem essentially concerns relationships between the angles of a tetrahedron, and has practical applications, particularly in connection with the Perspective 3-Point (Pose) Problem.

math.MG↗

arXiv2023

The Deltoid Curve and Triangle Transformations

Deltoid curves appear as consequences of certain procedures in triangle geometry. The best known of these is the construction based on Simson lines, described by Steiner. This is carefully related, in this article, to a less known construction. The standard deltoid in the complex plane and its tangent lines are principle objects of study in this report. It is known that each point in the interior of this curve is the orthocenter of a triangle with distinct vertices on the unit circle, whose product is one. (If instead the point is on the deltoid, then at least two of the vertices coalesce, resulting in a degenerate triangle.) When the vertices are all raised to some specified integer power, a new (possibly degenerate) triangle results. By varying the triangle, one may thus consider the map taking the original triangle's orthocenter to the resulting triangle's orthocenter. Such maps are the other principle objects of study here. The points that are mapped to the deltoid lie on easily described curves. By varying the power involved in the map, a pleasing family of curves results, which includes a trifolium curve. The points that are mapped instead to the origin are described as the points of intersection of certain tangents to the deltoid.

math.MG↗