c# - 3D rotation, given two specified directions -

i have 2 perpendicular unit vectors, p , q; think of being embedded in rigid object in 3d space. wish rotate object p points along positive x-axis , q points along positive y-axis.

i programming in c# , using quaternions compute rotations. how create quaternion perform desired rotation?

if imagine inverse operation of quaternion, know point (1,0,0) goes p, , (0,1,0) goes q. similarly, point (0,0,1) go point z equal cross product of p , q.

therefore have matrix representation of inverse rotation. first column p, second q, third p cross q.

so matrix rotation want given transpose of matrix, i.e. first row p, second row q, third row p cross q.

so recommend call library routine convert matrix quaternion form.

you may interested in document doom3 explaining algorithm id software used converting matrix quaternion - including optimised x86 assembler code.

orthogonal matrices

if have matrix m columns (a,b,c) forming orthonormal set, know a'a = b'b = c'c = 1, a'b = ... = c'a = 0.

therefore m'm = i.

if multiply on left m, , on right inv(m) find:

m.m'.m.inv(m) = m.inv(m) => m.m'.(m.inv(m)) = m.inv(m) = => m.m' = 

and so, if have orthonormal columns, must have orthonormal rows, , matrix m called orthogonal matrix.