第三讲
Recall
Axis-angle → Rotation Matrix
Rotation Matrix → Axia-angle
函数的定义域是 ，值域是 ，单调递减。
故上述公式给出的 。
Quaternion
vector form:
conjugate:
norm:
Unit Quaternion as Rotation
A unit quaternion can represent a rotation.
rotate a vector by a quaternion :
- define
Composing rotations is as simple as multiplying quaternions!
and represent the same rotation! (double-covering)
Conversion between different rotation representations
We can conduct the “Axis-angle → rotation matrix” through quaternion.
- Axis-angle → Quaternion
- Quaternion → Rotation Matrix
Make respectively, we can get the roation matrix.
Distance between Quaternions
Define the dot-product of two quaternions:
The distance between the corresponding rotations:
加绝对值的含义：
首先，由于 和 表示的是相同的旋转，所以加绝对值是合理的。
如果不加绝对值的话， 的值域为 。×
加上绝对值， ，则 的值域为 。√
Interpolating between Quaternions
if the angle between and is less than , slerp between them.
else, slerp between and .
Uniform Sampling in
- Randomly sample a variable from
- Normalize it to unit length.
Why?
Standard normal distribution is isotropic.