"Primitives for Computational Geometry."
    Jorge Stolfi.
    January 27, 1989.  228 pages.

Author's Abstract

    Many geometric algorithms become simpler, more general, and more
    efficient when recast in the language of projective geometry.
    Some reasons for this are the uniform handling of points at
    infinity, the attendant reduction in the number of special cases,
    and the perfect duality between points and hyperplanes that are
    possible in the projective model.  In fact, the homogeneous
    coordinates so widely used in computer graphics are essentially
    an analytical model of classical projective geometry.

    However, projective space is topologically quite different from
    Euclidean space.  For example, in the projective plane lines have
    only one side, all triangles have the same handedness, and there
    are two distinct segments with any given pair of endpoints.  These
    differences are a serious practical problem, since many geometric
    algorithms depend on orientation, ordering and separation tests
    that make sense only in the Euclidean model.

    This dissertation describes a slightly modified form of projective
    geometry which is free from this problem.  Analytically, the change
    consists in making the signs of homogeneous coordinates more
    significant.  Geometrically, the change consists in adopting
    oriented lines and planes as the elementary objects of the model,
    and redefining the basic geometric operation of meet and join
    so as to produce results with a definite orientation.
    Topologically, this is equivalent to working with a double covering
    projective space, which is equivalent to an  n-dimensional sphere.

    The resulting framework, here called oriented projective
    geometry, combines the elegance of classical projective geometry
    with the ability to talk about oriented lines and planes, signed
    angles, line segments, convex figures, and many other concepts
    that cannot be conveniently defined within that model.  The goals
    of this dissertation are: (1) to develop an intuitive understanding
    of oriented projective geometry in two and three dimensions; (2)
    to describe a formal geometric calculus for handling oriented
    lines, planes, and flat spaces of arbitrary dimension; and (3)
    to investigate the efficient representation of such objects in
    computers.