• Rotations 1. rotation of a vector around an arbitrary axis Given a vector~b, an axis~a, and an angle α the goal is to ﬁnd a formula (or function or op-erator) that produces the vector~c created by rotating~b an angle α around~a counterclockwise (in a right-handed system). The expression will be found with brute force transformations
• 3. The Justification for the Quaternion Rotation Formula In the first set of notes I stated that to rotate a point P by θ around an arbitrary unit axis, u, through the origin one can use the formula P’ = qPq-1, where q = (cos(2!), sin(2!)u). Obviously I need to justify this.
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• The quaternion that computes this rotation is We will represent a point p in space by the quaternion P=(0,p) We compute the desired rotation of that point by this formula: You may want to confirm that q is a unit quaternion, since that will allow us to use the fact that the inverse of q is q' if q is a unit quaternion. Concatenating Rotations
• Inverting unit quaternions is especially easy: if |z| = 1, then z-1 = z * (the conjugate z * of the quaternion z = a + v is defined as z * = a - v) and this makes our rotation formula even easier. It turns out that the angle of rotation α is also easy to read off if we are dealing with a unit quaternion z = a + v: we have cos(α/2) = a. To summarize:
• Apr 06, 2017 · Quaternion rotation = Quaternion.identity; for (int i = 1; i < Joints.Length; i++) {// Rotates around a new axis rotation *= Quaternion.AngleAxis(angles[i – 1], JointScript[i-1].Axis); Vector3 nextPoint = prevPoint + rotation * JointScript[i].StartOffset; prevPoint = nextPoint;} return prevPoint;} [/code]
1 27-750 Texture, Microstructure & Anisotropy A.D. (Tony) Rollett, S. R. Wilson Rodrigues vectors, unit Quaternions Lastrevised:%2 ndJan.2015
Apr 14, 2014 · In this post, I will show some examples of quaternion implementations. The Interface. We want our quaternion class to handle basic arithmetics (addition, subtraction, and scalar multiplciation), as well as various operations (normalization, re-normalization, inversion, dot product, projection, quaternion product, vector rotation, and slerp).
The plane of rotation r equals cosine theta multiplied by the elevation plus sine theta multiplied by the axis a crossed with projection p. The point where the elevation (lime green) joins the...v' = ∑ wi (v) (Qi * v + ti) where Qi is the absolute bone (or joint) rotation (e.g. stored as a quaternion) of bone i. These absolute rotations are just a concatenation down from the root to bone i of the relative rotations we’re storing for each bone. These relative rotations are what we’re changing during an animation.
A very good way to understand quaternions in 3D graphics when applied to simple rotations is as "a vector representing the axis of rotation plus the angle of rotation". Most of math libraries with quaternions allow you to create a quaternion like that which is very easy to grasp.
The four major representations of 3D rotations are rotation matrix, Euler angle (e.g., roll-pitch-yaw), axis-angle (which is very similar to the rotation vector representation), and quaternion. All representations are somewhat equivalent in that they can be converted to a rotation matrix and back again. Feb 05, 2018 · Quaternions were invented in 1843 by William Hamilton, who was so inspired by his discovery that he carved the formula into a bridge nearby. Just as a complex number in 2D contains a real and imaginary component, a quaternion in 4D contains three imaginary components, x , y and z (sometimes a , b and c ), and a real component, w (sometimes d ).
Sep 02, 2010 · Formula<br /> 5. Example<br />7 + 9i + 5j + 1k<br /> 6. 7. Matrix to Quaternion<br /> 8. Trace<br />sum of the main diagonal of a matrix<br /> 9. Due to the tangent, the rotation vector is indeterminate when the rotation angle equals ±pi radians or ±180 deg. Values can be negative or positive. References [1] Dai, J.S. "Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections."