Stereohomology by Alston Chen, 2007
MEANING OF ELEMENTARY MATRICES IN PROJECTIVE
GEOMETRY AND ITS APPLICATIONS
How to represent some of the commonly used geometric transformations in
computer graphics with elementary matrices
Alston Zi-qiang CHEN
Department of Polymer Science, East China University of Science & Technology, Shanghai, 200237,P R China
stereohomology@126.com, stereohomology@gmail.com
Abstract:
By taking advantage of an extension of Desargues theorem, and the extended Desarguesian configuration
thereof, the analytical definition for the concept of stereohomology geometric transformation in projective
geometry is proposed in this work. A series of such commonly used geometric transformations in computer
graphics as central projection, parallel projection, centrosymmetry, reflection and translation transformation,
which can be included in stereohomology, are thus analytically defined in projective geometry. It has been
proved that, the transformation matrices of stereohomology are actually equivalent to the existing concept in
numerical analysis: elementary matrices. Consequently, the current work not only presents a new represen-
tation method for the geometric transformations mentioned here, but also depicts the meaning of elementary
matrices in projective geometry. Some of the concepts, speculations and rules are also proposed for stereoho-
mology’s applications in computer graphics, especially in 3D reconstructing objects from multiple views.
Keywords:
elementary matrix, homogeneous coordinates, Desargues theorem, perspective projection, 3D reconstruction,
projective geometry.
1 INTRODUCTION
As is well known that, such geometric transformations
as translation transformation, perspective projection
in computer graphics, can be represented as transfor-
mation matrices only through the homogeneous co-
ordinate representation. Since it is very convenient
for us to represent different geometric transformations
and their compound operations in matrices by this
method, homogeneous coordinate has been widely ac-
cepted in computer graphics for many years.
Yet it is surprising to see that presently there is
no rigorous definition for the geometric transforma-
tions represented by matrices based on homogeneous
coordinates. Though ideally, a geometric transforma-
tion, and therefore its transformation matrix, should
be defined and identifed reference coordinate system
independently, the current representing method and
the definitions in computer graphics, at least for geo-
metric transformation matrices, are not reference co-
ordinate system independent. Actually, the current
definitions for these commonly used geometric trans-
formations and their transformation matrices, are not
perfect, and even not theoretically rigorous, which can
be seen from the following aspects:
First, by using homogeneous coordinates, we as-
sume that the geometric transformations are defined
in projective space not in Euclidean space. As we
know, in most of the current computer graphics text-
books, the geometric transformations, for example,
central projection, and parallel projection, are actually
defined in Euclidean space by using the concepts of
measure, distance directly or implicitly in Euclidean
geometry, which will be inapplicable in projective ge-
ometry.
Second, conventionally, the geometric transfor-
mation matrices can also be established for per-
spective projection, translation transformation and so
forth, based on homogeneous coordinates representa-
tion and specially selected reference coordinate sys-
tems. While mainly due to the limitations of the tradi-
tional definitions of either the geometric transforma-
tions or their transformation matrices, there is no sim-
ple or straightforward method for establishing trans-
formation matrices reference-coordinate-systeminde-
pendently.
Since most of the geometric transformations
can be represented by transformation matrices only
through homogeneous coordinates, an improved rep-
resentation method for geometric transformations bet-
ter be based on homogeneous coordinates and projec-
tive geometry.