04-transformation Cont

viewing transformation

View transformation:

think about how to take a photo?

• Finda good place arrange people(model transformation)
• Find a good "angle" to put the camera(view transformation)
• Cheese(projection transformation)

how to perform view transformation?

1. define the camera

• camera position
• gaize direction
• up direction

2. key observation

• If the camera and objects move together,the picture will be the same

3. how to transform the camera?

• the origin,up at Y,look at -Z
• transform the object along with the camera

4. transform the camera by $M_view$

• $M_view=R_{view}T_{view}$
• 即先进行平移再旋转

$T_{view}={\begin{bmatrix} a & 0 & 0 & -x \\ 0& 1 & 0 & -y \\ 0 & 0 & 1 & -z \\ 0 & 0&0& 1 \end{bmatrix}}$ 对于矩阵$R_view$直接写出来较为困难，but它的逆矩阵容易写出来即

对于旋转矩阵来讲它的逆=它的转置

Summary:

• transform objects together with the camera
• Until camera's at origin up at Y,look at -Z

Projection transformation:

Orthographic Projection:

A simple way of understanding:

1. Drop the Z coordinate
2. Translate and scale the resulting rectangle to$[-1,1]^2$

Perspective Projection:

• Most common in Computer Graphics
• Further objects are smaller
• Parallel lines not parallel, converge to a point

How to do perspective projection?

1. “squish to a cuboid ”
2. Do orthographic projection

$M_{perspective}=M_{orthographic}M{perspective{\Rightarrow}{orthographic}}$