1 . New Algebraic Tools for Classical Geometry †

Classical geometry has emerged from efforts to codify perception of space and motion. With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich properties of geometric objects were explored. Though fundamental ideas of classical geometry are permanently imbedded and broadly applied in mathematics and physics, the subject itself has practically disappeared from the modern mathematics curriculum. Many of its results have been consigned to the seldom-visited museum of mathematics history, in part, because they are expressed in splintered and arcane language. To make them readily accessible and useful, they need to be reexamined and integrated into a coherent mathematical system. Classical geometry has been making a comeback recently because it is useful in such fields as Computer-Aided Geometric Design (CAGD), CAD/CAM, computer graphics, computer vision and robotics. In all these fields there is a premium on computational efficiency in designing and manipulating geometric objects. Our purpose here is to introduce powerful new mathematical tools for meeting that objective and developing new insights within a unified algebraic framework. In this and subsequent chapters we show how classical geometry fits neatly into the broader mathematical system of Geometric Algebra (GA) and its extension to a complete Geometric Calculus (GC) that includes differential forms and much more. Over the last four decades GC has been developed as a universal geometric language for mathematics and physics. This can be regarded as the culmination of an R & D program innaugurated by Hermann Grassmann in 1844 [G44, H96]. Literature on the evolution of GC with extensive applications to math and physics can be accessed from the GC web site