Brought a hundred and sixty years in the past as an try and generalize complicated numbers to better dimensions, quaternions at the moment are famous as the most vital options in smooth special effects. they give a strong option to symbolize rotations and in comparison to rotation matrices they use much less reminiscence, compose speedier, and are evidently fitted to effective interpolation of rotations. regardless of this, many practitioners have kept away from quaternions due to the arithmetic used to appreciate them, hoping that a few day a extra intuitive description should be available.
The wait is over. Andrew Hanson's new ebook is a clean point of view on quaternions. the 1st a part of the booklet specializes in visualizing quaternions to supply the instinct essential to use them, and comprises many illustrative examples to inspire why they're important―a attractive advent to these eager to discover quaternions unencumbered via their mathematical features. the second one half covers the all-important complicated functions, together with quaternion curves, surfaces, and volumes. eventually, for these short of the total tale of the math in the back of quaternions, there's a mild advent to their 4-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions.
* Richly illustrated creation for the developer, scientist, engineer, or pupil in special effects, visualization, or leisure computing.
* Covers either non-mathematical and mathematical methods to quaternions.
* significant other site with an collection of quaternion utilities and pattern code, information units for the book's illustrations, and Mathematica notebooks with crucial algebraic utilities.
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Additional info for Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology)
1) 92 bankruptcy 09. VISUALIZING LOGARITHMS AND EXPONENTIALS ordinary operations comparable to taking derivatives continue within the traditional manner. simply because log q = (0, nˆ θ2 ) looks in each time period of the exponential strength sequence ∞ q at = exp(at log q) = k=0 1 (at)k okay! okay zero, nˆ j =0 θ , 2 the place the goods are quaternion items, we will pull out an element of log q at both the start or the top and acquire a similar outcome. utilizing the ability sequence to compute the by-product phrases, we could resume with the issue of log q at both finish, yielding the spinoff formulation dq at = a(log q)q at = aq at (log q). dt Polar (logarithmic) model of the multiplication rule: simply because the complex-number multiplication rule shows a few simplification in polar shape, we will use quaternion logarithms to hunt an analogous quaternion expression. simply as lets rewrite the advanced multiplication algebra suggestively as (r1 , θ1 ) (r2 , θ2 ) = (r1 r2 , θ1 + θ2 ), we will be able to do an identical for quaternions in a polar shape within which the “angle” is a natural quaternion vector: θ1 θ2 θ12 ˆ p q = r1 , nˆ 1 r2 , nˆ 2 = r1 r2 , m . 2 2 2 the following, p0 q + q zero p + p × q ˆ = , m p0 q + q zero p + p × q and cos(θ12 /2) = p0 q0 − p · q. Noncommutativity makes this type less ˆ ≡ r and (r1 )(r2 ) = (r1 r2 ) it invaluable than the advanced case, yet simply because (r, θ n) does exhibit that the norm-preserving algebra holds at once. Visualizing Interpolation equipment during this bankruptcy we entire our set of primary visualization tools through learning interpolation within the context of spheres, and at last within the context of quaternion issues. The interpolation from one quaternion to a different has profound analogies with ordinary polynomial interpolation equipment in Euclidean area. we are going to see that geodesic curves on spheres give you the place to begin for a wealthy family members of interpolation equipment and their graphical depiction. 10 10. 1 B A S I C S O F I N T E R P O L AT I O N we'll commence with the main primary object—the interpolation that creates an outstanding circle on a sphere of any size. This interpolation is in reality a bit of nontrivial to derive even for S1 . The derivation we current is the vintage technique used through the mathematical and crew thought literature yet much less usually visible within the special effects literature. 1 zero . 1 . 1 I N T E R P O L AT I O N I S S U E S In Euclidean area, linear interpolation (sometimes abbreviated LERP) takes the shape x(t) = x0 + t (x1 − x0 ) = (1 − t)x0 + tx1 , ninety three 94 bankruptcy 10. VISUALIZING INTERPOLATION equipment determine 10. 1 The 1D linear interpolation generating issues on a immediately line. where—with x(0) ≡ x0 and x(1) ≡ x1 —taking zero t 1 restricts the parametric curve to the instantly line phase among x0 and x1 . The LERP is proven in determine 10. 1. despite the fact that, it might be foolish to use the LERP to issues q on a sphere, simply because whether q0 = q1 = 1 the linearly interpolated aspect p(t) = (1 − t)q0 + tq1 do not need the specified homes. we will see simply from determine 10. 2 and the direct computation p(t) · p(t) = 1 − 2t + 2t 2 + 2t (1 − t) cos φ, the place q0 · q1 = cos φ, that p(t) should be anyplace from zero to one and therefore won't describe some extent at the related circle as q0 and q1 .