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By De Simone A., Mundici D.

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A complex bivector C E Qn,n+l has the form C = A + iB for A, B E Q~,n' The complex bivectors make up the complex Lie algebra spini(n) under the Lie bracket operation. The null cone N should be thought of as a home base. If we wish to do a rotation in a pseudo euclidean space IRp,q where p + q = n, then we project the null cone N onto IRp,q, perform the rotation using the bivectors of spinor algebra spin(p, q), and then project back to the null cone. Suppose that x = {e }X{e} = I:~=1 Xiei E N. The projection Pp,q : N -+ IRp,q is defined by x' = Pp,q(x) = Ip,q .

The latter was employed in [1], but we will stick with the former. The multiplicative split n 4 ,1 = n3 ® n1,1 has significant applications to computational geometry, robotics, computer vision, crytallography and molecular geometry. At a more sophisticated level, the split n4,2 = n3,1 ® n1,1 defines a conformal split of spacetime with potential applications to twist or theory and cosmological models in gauge gravity. 4 Models of Euclidean Space We can model En as a set of points with algebraic properties.

48) where the displacement versor D = TaR specifies a rotation around an axis with direction n = RnRt through the origin, followed by a translation Ta = 1 + ~ea. David Hestenes 14 According to Chasles' Theorem: Any rigid displacement can be expressed as a screw displacement. 50) is a rotation that leaves b fixed. 49) can be solved directly for b -2 -1 = a-L (1 - R ) 1 - R2 1 = 2 a -L 1 _ (R2 ) . 51 ) This illustrates the computational power of geometric algebra. 53) where i = E is the pseudoscalar for E 3 , and S is called a screw.

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