Conics Quadrics Mapping & Cones

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.

Figures (7)

• Figure 1: Five Points, Four of Which Are Spanned by Six Line Segments
• Figure 2: Verification of Eq. \ref{['Ceq']}
• Figure 3: Point & Plane Topology of a Regular Octahedron
• Figure 4: Hyperboloid of One Sheet Defined by the Nine Given Points
• Figure 5: Cones Points & Planes