Conics Quadrics Mapping & Cones
Paul Zsombor-Murray, Martin Pfurner
TL;DR
This paper presents an efficient method for deriving implicit equations of conics and quadrics using pencils and explores cone intersections and symmetries with potential extensions to nine-point quadrics.
Contribution
It introduces a novel approach to compute implicit equations of conics and quadrics via pencils and cone mappings, including a kinematic mapping with dual quaternions.
Findings
Parallel axis right cones intersect on a conic.
Eight real solutions belong to a unique cone pair intersection.
A method to place five coplanar points on a cone using kinematic mapping is demonstrated.
Abstract
An efficient way to get implicit equations of conics on five points and quadrics on nine, using pencils of conics and quadrics, is revealed. Parallel axis right cones intersect on a conic. An example, to show how to place five coplanar points on a cone, using kinematic mapping with dual quaternions is presented. A second congruent cone is found as a translation of the first. Cone symmetry helps to explain how mapping produces eight real solutions, apparently all different, belong to a unique cone pair intersection. Future extension of this, pertaining to a nine point quadric, can be contrived if a way to map planar points to parallel axis cones is formulated.
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