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https://www.zhihu.com/tardis/zm/art/78987582
四元数和旋转 (Quaternion & rotation)
四元数 (quaternion)可以看作中学时学的复数的扩充，它有三个虚部。 形式如下： ，可以写成 具有如下性质： 设 ， ，则 3.2 共 轭四元数 一个四元数 的共轭 (用 表示)为 一个四元数和它的共轭的积等于该四元数与自身的点乘，也等于该四元数长度的平方。 即，
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How can one intuitively think about quaternions?
Here is the intuitive interpretation of this. Given a particular rotation axis $\omega$, if you restrict the 4D quaternion space to the 2D plane containing $ (1,0,0,0)$ and $ (0,\omega_x,\omega_y,\omega_z)$, the unit quaternions representing all possible rotations about the axis $\vec \omega$ form the unit circle in that plane.
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Real world uses of Quaternions? - Mathematics Stack Exchange
The quaternion algebra shows there as a way of disentangling two Alamouti coded signals transmitted by a pair of antennas. The advantages come from the fact that even if the signal from one antenna is lost for a particular receiver (due to sitting in a node for that particular radio wave), then the signal from the other antenna saves the day.
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Concise description of why rotation quaternions use half the angle
The idea of Hamilton was to find some generalization of this formula for three-dimensional rotations. The quaternions can do such a generalization identifying a $3D$ -vector with a pure imaginary quaternion $\mathbf {v}$ and using a pure imaginary versor $\mathbf {u}$ to identify the axis of rotation.
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Understanding quaternions - Mathematics Stack Exchange
Of course adding two quaternions gives a quaternion, so algebraically this is clear. I don't really think it's clear geometrically, however, and with good reason: this is a very exceptional accident that occurs in precisely four dimensions, and no other dimensions.
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linear algebra - Conversion of rotation matrix to quaternion ...
One of the quaternion elements is guaranteed to have a magnitude of greater than 0.5 and hence a squared value of 0.25. We can use this to determine the "best" set of parameters to use to calculate the quaternion from a rotation matrix
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rotations - How do you rotate a vector by a unit quaternion ...
Do one quaternion multiplication and you rotate the circular component just that far around, and the quaternion axis gives you the rest of the location, and the fourth dimension says how far ahead or behind you are in time relative to that fraction of a full orbit. All in one operation.
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complex numbers - What exactly does a quaternion represent ...
This is a slightly complicated question but put simply. C is the field over R such that it is of the form (a+bi) where (a,b) are in R. The quaternion is a further extension where numbers are of the form (a+bi+cj+dk) where (a,b,c,d) are over the Reals. It has the properties i^2 = -1, j^2 = -1 and k^2 = -1 and ijk = -1. From this it can be derived that IJ=K,JK=I,KI=J and JI=-k,KJ=-I,IK=-j. This ...
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Apply Quaternion Rotation to Vector - Mathematics Stack Exchange
A quaternion can be thought of as a scalar plus a 3D vector (also known as real and imaginary parts). The product of a scalar and a 3D vector is the usual scalar multiplication. The product of two vectors produces a quaternion with both scalar and vector components, given by (minus) the dot product and cross product respectively.
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How to convert a quaternion from one coordinate system to another
I am trying to find a way of converting a quaternion from an arbitrary coordinate system to a fixed coordinate system that is used in my application. I have two different coordinate systems, one is...