Real world uses of Quaternions? - Mathematics Stack Exchange
Jul 20, 2010 at 22:09. The relationship between spinors and quaternions could be made stronger. The quaternions are the dual of spinors within a 3D geometric algebra. The quaternions are made up of linear combinations of the oriented planes and a scalar, while the spinors are linear combinations of a vector and the 3d oriented volume or ...
How to convert a quaternion from one coordinate system to another
Jun 24, 2022 · Once all quaternions are under the same convention they need to be rotated to which ever reference frame they are needed in. Given: q_ar = quaternion from the reference frame r to a q_ia = quaternion from a to i q_ir = quaternion from r to i. q_ir = q_ia X q_ar. where X is the quaternion Hamilton product.
How can one intuitively think about quaternions?
Oct 19, 2010 · Thinking about quaternions as 4D is misleading. Quaternions are the union of a scalar and a 3-vector. Think: time and space. Space is a 3-vector. You can point in directions in space. Time is a scalar. There is a past (negative time) and a future (positive time) and now (0), but no ability to point in the direction of time.
Combining rotation quaternions - Mathematics Stack Exchange
Basically the idea is this: every rotation in 3-space is specified by an axis of rotation and the angle you rotate about that axis. To find your customized u u, you first compute a unit quaternion h h which is normal to the plane of rotation, and then an expression like u = cos(θ/2) + h sin(θ/2) u = cos. . (θ / 2) + h sin (θ / 2) turns ...
matrices - Are there different conventions for representing …
Either recipe works, for the simple reason that the "real part" of any quaternion product $\beta \gamma$ is the same as the real part of $\gamma \beta.$ So, given that the real part of $\alpha$ is $0,$ the real part of $ ( \bar{\xi} \alpha ) \xi $ is the same as the real part of $ \xi ( \bar{\xi} \alpha ) $ is the same as the real part of ...
the logarithm of quaternion - Mathematics Stack Exchange
I'm reading 3D math primer for graphics and game development by Fletcher Dunn and Ian Parberry. On page 170, the logarithm of quaternion is defined as. logq = log([cos α n sin α]) ≡[0 αn] log q = log ([cos α n sin α]) ≡ [0 α n] I don't see how log([cos α n sin α]) log ([cos α n sin α]) is equal to [0 αn] [0 α n]. Can anyone help ...
Concise description of why rotation quaternions use half the angle
Aug 5, 2015 · When defining a rotation quaternion (with vector notation) representing a rotation of θ around a vector u = (ux, uy, uz) as: q = (cos(θ 2), uxsin(θ 2), uysin(θ 2), uzsin(θ 2)), I think it is necessary to explain why there is a 1 2 tied to the angle everywhere. I found some suggestions in the answer to this question, but I feel the answers ...
rotations - How do you rotate a vector by a unit quaternion ...
Turn your 3-vector into a quaternion by adding a zero in the extra dimension. [0,x,y,z]. Now if you multiply by a new quaternion, the vector part of that quaternion will be the axis of one complex rotation, and the scalar part is like the cosine of the rotation around that axis. This is the part you want, for a 3D rotation.
3d - Averaging quaternions - Mathematics Stack Exchange
There are at least two methods for averaging together more than two quaternions. 1. Direct averaging (fast/approximate) 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 ...
Exponential Function of Quaternion - Derivation
Nov 20, 2014 · 38. The equation for the exponential function of a quaternion q = a + bi + cj + dk is supposed to be eq = ea(cos(√b2 + c2 + d2) + (bi + cj + dk) √b2 + c2 + d2sin(√b2 + c2 + d2)) I'm having a difficult time finding a derivation of this formula. I keep trying to derive it, but I end up getting different results.