**Conformal Geometric Algebra by Extended Vahlen Matrices**

Leo Dorst, University of Amsterdam, The Netherlands

### Abstract

In this paper, conformal geometric algebra and the
Clifford-valued Vahlen matrix representation of Möbius transformations meet, to
lay the foundation for a fruitful interaction.

To embed the full conformal
model we need to define the matrices that represent the blades in geometric
algebra -- this gives us an extension of the matrix representation from
transformations to geometric primitives such as circles, lines and tangents (and
more). This is done by translating the outer product and contraction product
constructions of such elements into matrix operations, through the intermediate
step of writing them as linear combinations of geometric products. We provide a
useful dictionary for this translation. Using the dictionary, we show some
examples of how the two representations can interact to study properties of
conformal transformations, with some emphasis on Euclidean similarities and

motions.