Applications of dual quaternions to 2D geometry

In this article, we discuss certain applications of the dual quaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we will call the planar quaternions. The planar quaternions make up a four-dimensional algebra over the real numbers. Their primary application is in representing rigid body motions in 2D space. Unlike multiplication of dual numbers or of complex numbers, that of planar quaternions is non-commutative. In this article, the set of planar quaternions is denoted . A general element of has the form where , , and are real numbers; is a dual number that squares to zero; and , , and are the standard basis elements of the quaternions. Multiplication is done in the same way as with the quaternions, but with the additional rule that is nilpotent of index , i.e., , which in some circumstances makes comparable to an infinitesimal number. It follows that the multiplicative inverses of planar quaternions are given by The set forms a basis of the vector space of planar quaternions, where the scalars are real numbers. The magnitude of a planar quaternion is defined to be For applications in computer graphics, the number is commonly represented as the 4-tuple . A planar quaternion has the following representation as a 2x2 complex matrix: It can also be represented as a 2×2 dual number matrix: The above two matrix representations are related to the Möbius transformations and Laguerre transformations respectively. The algebra discussed in this article is sometimes called the dual complex numbers. This may be a misleading name because it suggests that the algebra should take the form of either: The dual numbers, but with complex-number entries The complex numbers, but with dual-number entries An algebra meeting either description exists. And both descriptions are equivalent. (This is due to the fact that the tensor product of algebras is commutative up to isomorphism). This algebra can be denoted as using ring quotienting.