  We introduce a new presentation of the two dimensional rigid transformation
which is more concise and efficient than the standard matrix presentation. By
modifying the ordinary dual number construction for the complex numbers, we
define the ring of the anti-commutative dual complex numbers, which
parametrizes two dimensional rotation and translation all together. With this
presentation, one can easily interpolate or blend two or more rigid
transformations at a low computational cost. We developed a library for C++
with the MIT-licensed source code and demonstrate its facility by an
interactive deformation tool developed for iPad.
