GAMES101-04空间变换 Cont.

一、3D transformations

//在数学上，如果一个矩阵的逆等于它的转置，则叫做正交矩阵。


万向锁

Rodrigues’ Rotation Formula
罗德里格斯旋转公式

定义了一个旋转轴和一个旋转角度（默认原点起点）

找到四元数并做矩阵变换

二、Viewing (观测) transformation

1.View(视图)/ Camera transformation

. Think about how to take a photo

• Find a good place and arrange people ( m odel transformation)
• • Find a good “angle” to put the camera ( v iew transformation)
• • Cheese! ( p rojection transformation)
    模型-识别-投影变换（MVP变换）

首先定义相机的放置位置，朝向，角度（顶朝向）

原本相机可以在任意位置，但为了简化，可以将相机放在标准位置（0，0）。

先平移，再旋转
Translate e to origin

Rotate g to -Z, t to Y,(g x t) To x
反过来写，再求逆
Consider its inverse rotation: × to (g x t),Y to t,Z to -g

Summary

• Transform objects together with the camera
• Until camera’s at the origin, up at Y, look at -Z
  ModelView Transformation模型+视图变换

2.Projection (投影) transformation

• Orthographic (正交) projection
  不会带来近大远小的视觉误差
  （道理我都懂，但是鸽子为什么这莫大？）
  we want to map a cuboid [1,r]x [b, t]x [f,n] to the "canonical (正则、规范、标准)"cube [-1，1]3
  正则立方体
  
  先将x，y，z方向移动到以原点中心
  再将立方体正则化
  Translate (center to origin) first,then scale (length/width/height to 2)

Caveat

• Looking at / along -Z is making near and far not intuitive (n > f)

• FYl: that’s why OpenGL (a Graphics APl) uses left hand coords.

• Perspective(透视)projection
  Most common in Computer Graphics, art, visual system
  Further objects are smaller
  Parallel lines not parallel; converge to single point

透视投影到别的平面上后，平行线会不平行。

Recall:
property of homogeneous coordinates

• (x，y,z，1)，(kx， ky,kz,k != O), (xz, yz,z2,z !=0)all represent the same point (x, y, z) in 3D
• e.g.(1,0,0,1) and (2,0,0,2) both represent (1,0, 0)
  透视投影的平面大小会改变，而正交投影是一样的。

规定：
近平面永远不变。
在平面内向里收缩，远平面不变位置。
Frustum to Cuboid
相似三角形

ln order to find a transformation

• Find the relationship between transformed points (x’,y’,z’)and the original points (x, y,z)
  In homogeneous coordinates
  
  继续推导得：
  
  Observation: the third row is responsible for z’
• Any point on the near plane will not change
• Any point’s z on the far plane will not change

Solve for A and B
An＋B = 2→4A =n＋ f
Af + B = f2
B=—n f
Finally, every entry in Mpersp->ortho is known!What’s next?

• Do orthographic projection(Mortho) to finish-Mpersp = Mortho Mpersp→ortho