MA 423 Topics in Geometry
Applied Projective Geometry
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"All geometry is projective geometry",
Arthur Cayley
Projective geometry is the geometry of straight lines. It may be
defined either axiomatically in the manner of Euclidean geometry
or by building upon linear algebra. Axiomatically, projective geometry
pertains to the geometric constructions that can be accomplished
with a straight-edge alone (no measurement is possible). This is
entirely different than the standard Euclidean geometry, you are
familiar with, and at first glance it may seem that this is too
restrictive to generate any structure of interest. However, there
is a great deal of structure. It is just different. In fact, projective
geometry is the correct geometry to study and apply to optical systems,
computer vision, and computer graphics.
In this course, we will examine the axiomatic approach but focus
on building the structure of projective geometry from linear algebra.
We will focus our attention on the use of projective geometry to
transfer 3D objects to 2D images, and the use of projective geometry
as a tool for describing a 3D object as a 2D image. The ultimate
goal in this second objective is to use projectuve geometry as a
tool in CAGD (computer aided geometric design) to develop the language
to describe NURBS (non-uniform rational B-splines). |
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