Abstract. 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 ([13]) and demonstrate its facility by an interactive deformation tool developed for iPad. Rigid TransformationThe n-dimensional rigid transformation (or Euclidean) group E(n) consists of transformations of R n which preserves the standard metric. This group serves as an essential mathematical backend for many applications (see [2,9]). It is well-known (see [6], for example) that any element of E(n) can be written as a composition of a rotation, a reflection, and a translation, and hence, it is represented by (n + 1) × (n + 1)-homogeneous matrix;Here, we adopt the convention that a matrix acts on a (column) vector by the multiplication from the left. E(n) has two connected components. The identity component SE(n) consists of those without reflection. More precisely,The group SE(n) is widely used in computer graphics such as for expressing motion and attitude, displacement ([10]), deformation ([1, 5, 11]), skinning ([8]), and camera control ([3]). In some cases, the matrix representation of the group SE(n) is not convenient. In particular, a linear combination of two matrices in SE(n) does not necessarily belong to SE(n) and
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