Reflection in more than 3 dimensionsĪnd the quantities Vb,Va and P are still vectors but the mirror can no longer be thought of as a plane, the equivalent in 4D for the parallel component would be a directed volume (say a cube). Translation and rotation which each have 3 degrees of freedom. Therefore reflection has only 2 degrees of freedom, unlike Since the vector to Px,Py,Pz is unit length (divided by Px 2 + Py 2 + Pz 2). See this page for more details of reflections using quaternion algebra. Since this only works in 3D and since it can't be combined with other types of transform it is more of a curiosity than a useful technique. Since, in 3D, vector and bivector algebras are virtually identical we can use quaternions to calculate reflections. Quaternions are considered to be a scalar value and a 3D bivector algebra value. See this page for more details of reflections using matrix algebra. See this page for more details of reflections using Clifford algebra.Īssumes orthogonal matrix t=1 and det M = -1 otherwise multiply by:
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