Q2

9/10

half hour

Solution

Q1

$\hat u_x=[1,0,0]$

$\hat u_z=\frac{\vec a - \vec c}{|\vec a-\vec c|}$

$\hat u_y=\hat u_x\times\hat u_z$

期中也会考

Q2

• $\theta$: law of cosine
• $\hat u$: cross product

$\vec q = [\cos\frac{\theta}{2},\sin\frac{\theta}{2}\hat u]$

$\theta=\arccos\frac{\vec a\cdot\vec b}{||\vec a||||\vec b||}$

$\hat u=\frac{(\vec a-\vec c)\times(\vec b-\vec c)}{\|(\vec a-\vec c)\times(\vec b-\vec c)\|}$

$\|a\times b\|=\|a\|\cdot\|b\|\sin\theta$

Q3

-0.5

左乘右乘

local=>post

global=>pre

$q_2=q\cdot q_1$

Q4

-0.5

$slerp(p_1,p_2,u)$

$u=\frac{t-t_0}{t_1-t_0}=\frac{t}{2}$

Key Points

• Quaternions
• Bezier Curve

Rotation Matrix

$$R^0_1=[\hat u_x^0|\hat u_y^0|\hat u_z^0]$$

What is $R_1^0$

hint

• 看新的每根坐标轴在w.r.t的里面怎么表示出来

Quaternions

Axis/Angle

$$\vec q=[\cos\frac{\theta}{2},\sin \frac{\theta}{2} \hat u]^T$$

Bezier

What the value of coeeffients

B0, b3: points

Interpolating Quaternion

$$\hat u_x=\frac{\vec a - \vec b}{||\vec a-\vec b||}\\ \hat u_y=\frac{\vec a - \vec b}{||\vec a-\vec b||}\\$$