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A linear space of geometric transformations

: Alexa, M.
: TU Darmstadt, Fachgebiet Graphisch-Interaktive Systeme

Darmstadt, 2001, 10 pp.
Reportnr.: GRIS-2001-03
Reportnr.: 01i030-GRIS
Fraunhofer IGD ()
computational geometry; linear space; geometric transformation

A commutative matrix product is defined by applying transforms associated to the matrices concurrently, not one after the other. In addition, it is shown that powers of matrices are natural scalar multiples of transforms. Together, the operations allow the construction of a linear space of matrices representing geometric transformations. A relation to matrix exponentials and logarithms is explained, which leads to compact expressions for both operations. Conditions for the existence of real results of these operations are discussed and simple implementations are presented in detail.
The concepts of commutativity and scalar multiples of transformations are useful in all areas of computer graphics. Specifically, they allow to seamlessly blend several transforms.