Rational Geometry in Space

The objects and motions of geometrical modelling, graphics and robotics are most often described by rational (or close to rational) data in R3. This leads naturally to a notion of rational geometry (as opposed to the classical notion of Euclidean geometry) where two objects are considered equal if there exists a rational rigid motion from one to the other. We show here that although the notion of equality in rational geometry differs from that of Euclidean geometry for lines and planes, the two notions coincide on collections of line segments, in particular for polyhedra. Next, by an explicit symbolic calculation and the use of a simple technique of classical number theory, we obtain a rational parameterization of the subgroup of all rational linear movements of the space which keep a given point fixed. Finally we study the equivalence relation of Q-equality among segments in the space Q3, and give a representation of the space of equivalence classes, i.e. of the quotient space Q3/O(3, Q). © 1991, Academic Press Limited. All rights reserved.