  We introduce the concept of paravectors to describe the geometry of points in
a three dimensional space. After defining a suitable product of paravectors, we
introduce the concepts of biparavectors and triparavectors to describe line
segments and plane fragments in this space. A key point in this product of
paravectors is the notion of the orientation of a point, in such a way that
biparavectors representing line segments are the result of the product of
points with opposite orientations. Incidence relations can also be formulated
in terms of the product of paravectors. To study the transformations of points,
lines, and planes, we introduce an algebra of transformations that is analogous
to the algebra of creation and annihilation operators in quantum theory. The
paravectors, biparavectors and triparavectors are mapped into this algebra and
their transformations are studied; we show that this formalism describes in an
unified way the operations of reflection, rotations (circular and hyperbolic),
translation, shear and non-uniform scale transformation. Using the concept of
Hodge duality, we define a new operation called cotranslation, and show that
the operation of perspective projection can be written as a composition of the
translation and cotranslation operations. We also show that the operation of
pseudo-perspective can be implemented using the cotranslation operation.
