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- linear algebra - How can one intuitively think about quaternions . . .
The unit quaternions also act via left and right multiplication as rotations of the 4d space of all quaternions This gives a homomorphism from SU(2) × SU(2) onto the 4d rotation group SO(4) The kernel of this homomorphism is {±(1, 1)}, so we see SU(2) × SU(2) is a double cover of SO(4)
- complex numbers - What exactly does a quaternion represent . . .
Unit quaternions can be identified with rotations of three-dimensional space, which is often the best way to think about them Specifically, take a point in the three-dimensional sphere If it's either the origin of three p-space or the extra point, it represents the trivial rotation
- Understanding quaternions - Mathematics Stack Exchange
Quaternions have real and imaginary parts, or one may call them a scalar and vector part That is, we can interpret $\mathbb{H}$ (named after Hamilton) as $\mathbb{R}\oplus\mathbb{R}^3$ We already know how to multiply a scalar by a scalar, and a vector by a scalar, so it remains to describe how to multiply two 3D vectors
- 3d - Averaging quaternions - Mathematics Stack Exchange
If quaternions represent similar rotations, and the quaternions are normalized, and a correction has been applied for the "double-cover problem", then the quaternions can be directly averaged and then the result normalized again, treating them as 4-dimensional vectors, to produce a quaternion representing a roughly-average rotation
- Quaternions +Geometric (Clifford) Algebra: What Is the Proper . . .
IV Historical Fun Facts About Quaternions and the Truth About Maxwell Theory Oliver Heavside and his side-kick Gibbs back in the day called Quaternions, “pure evil”, and “the work of the devil”… no joke! True story! I reference Grant Sanderson and Ben Eater’s YouTube video The reason why vector calculus won the day back in the
- Super confused by SQUAD algorithm for quaternion interpolation
The demo generates 10 random unit quaternions and then interpolates between them indefinitely It shows 12 WebGL canvas instances, 2 per algorithm The top canvas displays the quaternions in 4d space and the current interpolated quaternion, the bottom canvas displays a cube that is being rotated by the current quaternion
- why are negative quaternions the same as positive quaternions?
From what I understand, quaternions are a way to represent a rotation In this formula, n is the axis of rotation and theta is the angle So if I'm trying to represent the following rotation The
- Quaternions: why does ijk = -1 and ij=k and -ji=k
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