Understanding quaternions

The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.-Clerk Maxwell, 1869. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell. — Lord Kelvin, 1892 Quaternion Multiplication Definition • q = a + b i + c j + dk • Sum of a scalar and a vector Multiplication (Basis Vectors) • i 2 = j 2 = k 2 = −1 • ij = k jk = i ki = j • ji = −k kj = −i ik = − j Multiplication (Arbitrary Quaternion) • (a + v)(c + w) = (a c − v ⋅w) + (c v + a w + v × w) Properties of Quaternion Multiplication • Associative • Not Commutative • Distributes Through Addition • Identity and Inverses Rotations with Quaternions Conjugate • q = a + b i + c j + dk • q * = a − bi − c j − d k Unit Quaternion • q(N ,θ) = cos(θ) + sin(θ) N • N = Unit Vector Rotation of Vectors from Quaternion Multiplication (Sandwiching) • S q(N, θ /2) (v) = q(N , θ / 2) v q * (N ,θ / 2) Avoids Distortions • After several matrix multiplications, rotation matrices may no longer be orthogonal due to floating point inaccuracies. • Non-Orthogonal matrices are difficult to renormalize-leads to distortions. • Quaternions are easily renormalized-avoids distortions. R 1 and R 2 (key frames) fails to generate another rotation matrix. Lerp(R 1 ,R 2 ,t) = (1− t)R 1 + tR 2-not necessarily orthogonal matrices. • Spherical Linear Interpolation between two unit quaternions always generates a unit quaternion. Slerp(q 1 ,q 2 ,t) = sin (1− t)φ () sin(φ) q 1 + sin tφ () sin(φ) q 2-always a unit quaternion. • To provide a geometric interpretation for quaternions, appropriate for contemporary Computer Graphics. • To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions. • …