Anti-commutative Dual Complex Numbers and 2D Rigid Transformation

Abstract

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

A Concise Parametrization of Affine Transformation

“…More generally, using Clifford algebras one can parametrise various kinds of transformations. For example, the anti-commutative dual complex numbers ( [28]) parametrise SE(2) and conformal geometric algebra (CGA, for short) can be used to present Sim + (3). In fact, CGA deals with a larger class of transformation including non-linear ones ( [12,30,43,15,44,9,45]).…”

Directional Statistics and Filtering Using libDirectional

“…The second distribution on SE(2) is related to the Bingham distribution in the sense that it also arises by restricting a Gaussian random vector (Gilitschenski, Kurz, Julier, and Hanebeck 2014a). This is motivated by the fact that a multiplicative subgroup of dual quaternions can be used for representing elements of SE (2), which is reminiscent of the approach by Matsuda, Kaji, and Ochiai (2014). Similar to the Bingham case, our distribution needs to be antipodally symmetric in order to account for the fact that unit dual quaternions are a double cover of SE(2).…”

Dual-complex generalized k-Horadam numbers

“…In the 19 th century, Cli¤ord described the dual numbers with in the form A = a+"a , where a; a 2 R, " 2 = 0 and " 6 = 0 [4]. Up to this time, there are number of studies in the literature that concern about the dual numbers and dual complex numbers [1,3,5,[8][9][10][17][18][19][20]. For instance, Fjelstad and Gal examined the extensions of the hyperbolic complex numbers to n dimensions and they presented n dimensional dual complex numbers [9].…”

“…For instance, Fjelstad and Gal examined the extensions of the hyperbolic complex numbers to n dimensions and they presented n dimensional dual complex numbers [9]. Matsuda et al examine the ring of anti commutative dual complex numbers, which parametrizes two dimensional rotation and translation all together, by modifying the ordinary dual number construction for the complex numbers [18]. Majernik presented three types of the four component number Table 1.…”