P = Plane represented by normal vector.This is the main result on these pages and it applies when we are working in any number of dimensions (although in a number of dimensions, other than 3, the mirror is not a plane). The result also applies regardless of whether the dimensions square to +ve or -ve, in fact, if the dimensions all square to +ve we can simplify to: The result does require that the dimensions all anti-commute. This is a very important result as it allows us to calculate rotations in any number of dimensions, on this page, by generating rotations from an even number of reflections.
N = unit vector which is normal to the mirror (plane in which p is reflected).Or if all dimensions square to +ve and n is unit length then we can simplify to: So the final result, expressed in different algebras is: Algebra Parallel + perpendicular (to check if =1) So to reverse the normal component in different types of algebras we can do the following: Vb = -(P * Va)/P Representing Reflection using Different Algebra Types Since commutes and anticommutes reversing the order gives: P = plane, represented by bivector which is normal to plane.We can calculate this result by more explicitly calculating the perpendicularĪnd parallel components, which were derived On this page so we can combine them as follows, So all that remains to do here is derive the result using different types of algebras and to look at the implications, especially in higher dimensional spaces. See this page for more details of reflections using Clifford algebra.Īssumes orthogonal matrix t=1 and det M = -1 otherwise multiply by: See this page for more details of reflections using matrix algebra. Quaternions are considered to be a scalar value and a 3D bivector algebra value. Since, in 3D, vector and bivector algebras are virtually identical we can use quaternions to calculate reflections.
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. See this page for more details of reflections using quaternion algebra. Since the vector to Px,Py,Pz is unit length (divided by Px 2 + Py 2 + Pz 2).
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