Fixed-size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23× speedup for Clifford products and a 33× speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor.