quaternions

quaternions - Some Notes on Unit Quaternions and Rotation...

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Unformatted text preview: Some Notes on Unit Quaternions and Rotation Berthold K.P. Horn Copyright © 2001 A quaternion can be conveniently thought of as either (i) a vector with four components; or (ii) a scalar plus a vector with three components; or (iii) a complex number with three different “imaginary” parts. Here are three quaternions written in the “hyper complex” number form: ˚p = p + ip 1 + jp 2 + kp 3 ˚ q = q + iq 1 + jq 2 + kq 3 ˚ r = r + ir 1 + jr 2 + kr 3 The rules for multiplication are i 2 = j 2 = k 2 = − 1 (reminiscent of the square of the imaginary unit of complex numbers) and ij = − ji = k jk = − kj = i ki = − ik = j (reminiscent of pairwise cross products of unit vectors in the directions of orthogonal coordinate system axes). Then, if ˚ = p˚ r ˚q, we obtain from the rules above ⎡ ⎤ ⎡ ⎤⎡ ⎤ r p − p 1 − p 2 − p 3 q ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ r 1 ⎥ ⎢ p 1 p − p 3 p 2 ⎥⎢ q 1 ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎣ r 2 ⎦ ⎣ p 2 p 3 p − p 1 ⎦⎣ q 2 ⎦ r 3 p 3 − p 2 p 1 p q 3 or ˚ q r = P ˚ which can also be written in another way ⎡ ⎤ ⎡ ⎤⎡ ⎤ r q − q 1 − q 2 − q 3 p ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ r 1 ⎥ ⎢ q 1 q q 3 − q 2 ⎥⎢ p 1 ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎣ r 2 ⎦ ⎣ q 2 − q 3 q q 1 ⎦⎣ p 2 ⎦ r 3 q 3 q 2 − q 1 q p 3 or ∗ r = Q p ˚ ˚ These equations spell out in detail how to multiply two quaternions. Im- portantly, multiplication is not commutative. ( ) ( ) 2 Note that the matrices appearing above are orthogonal, in fact PP T = p · p ) I where I is the 4 × 4 identity matrix. The above gives two use- ( ˚ ˚ ful isomorphisms between quaternions (˚ q ) with orthogonal 4 × 4 p and ˚ matrices ( P and Q ∗ ) — one for “pre-multiplication”...
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This note was uploaded on 10/06/2009 for the course ECE Vision taught by Professor Bertholdhorn during the Spring '04 term at MIT.

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quaternions - Some Notes on Unit Quaternions and Rotation...

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