Chapter 15 - Constructions in Euclidean Geometry

Abstract

The basic constructions in geometric algebra are spanning and intersection. The meet is greatly extended in its capabilities to intersect arbitrary flats and rounds, or to compute other incidences. The results can be real or imaginary, or even the infinitesimal tangents. The dual of the meet provides a novel operation: the plunge, which constructs the simplest element intersecting a given group of elements perpendicularly. The principle of the meet is that it constructs the largest subblade common to the blades A and B. The duality relative to the join constructs the complements within the smallest blade containing both; these complements do not contain common factors, so that their outer product is nonzero. Normal vector, position vector, free vector, line vector, and tangent vector now all automatically move in the correct way under the same Euclidean rotors. This demonstrates clearly that the conformal model performs both the geometrical computations and the “data type management” required in Euclidean geometry. The carrier of an element is defined as the smallest grade flat that contains it and a flat is therefore its own carrier.