transformations 变换集合关系 仿射变换

http://groups.csail.mit.edu/graphics/classes/6.837/F03/lectures/04_transformations.ppt

https://groups.csail.mit.edu/graphics/classes/6.837/F03/lectures/

Maps points (x, y) in one coordinate system to points (x', y') in another coordinate system

x' = ax + by + c

y' = dx + ey + f

For example, IFS:

Can be combined

Are these operations invertible?

Yes, except scale = 0

恒等 平移 旋转 等比缩放

可逆，除非等比缩放系数为0

Classes of Transformations 变换分类

Rigid Body / Euclidean Transforms 刚体、欧式变换

Similitudes / Similarity Transforms 相似性变换

Linear 线性变换

Affine 放射

Projective 投影

保持不变量的对象
点点之间
距离
线线之间
角度
平行关系

保距变换
保角变换
平行变换

Rigid-Body / Euclidean Transforms

Preserves distances

Preserves angles

Rigid / Euclidean

Translation Identity Rotation

Similitudes / Similarity Transforms

Linear Transformations

L(p + q) = L(p) + L(q)

L(ap) = a L(p)

shear

vt. 剪；修剪；剥夺

vi. 剪；剪切；修剪

切力 切变

Affine Transformations

Projective Transformations

preserves lines

Representing Transformations 变换的表示

Combining Transformations 变换的联合

Change of Orthonormal Basis 改变正交基

How are Transforms Represented?

Homogeneous Coordinates 齐次坐标

Add an extra dimension

in 2D, we use 3 x 3 matrices

in 3D, we use 4 x 4 matrices

Each point has an extra value, w

Most of the time w = 1, and we can ignore it

If we multiply a homogeneous coordinate by an affine matrix, w is unchanged

如果通过仿射矩阵来乘齐次坐标系，则w不变

Divide by w to normalize (homogenize)

W = 0? Point at infinity (direction)

https://en.wikipedia.org/wiki/Affine_transformation

Translate (tx, ty, tz)

Why bother with the extra dimension? Because now translations can be encoded in the matrix!

Translate(c,0,0)

Scale (sx, sy, sz)

Isotropic (uniform) scaling: sx = sy = sz

扩展

旋转

关于不同坐标轴旋转

About (kx, ky, kz), a unit vector on an arbitrary axis(Rodrigues Formula)

How are transforms combined?

Scale then Translate

Use matrix multiplication:   p'  =  T ( S p )  =  TS p

Caution: matrix multiplication is NOT commutative!

矩阵相乘不可以交换

Non-commutative Composition

Scale then Translate: p' = T ( S p ) = TS p

Translate then Scale:   p'  =  S ( T p )  =  ST p

Review of Dot Product

点乘

Change of Orthonormal Basis

Given: coordinate frames

xyz and uvn

point p = (x,y,z)

Find: p = (u,v,n)

Substitute into equation for p:

Rewrite:

p = (u,v,n) = u u + v v + n n

Expressed in uvn basis:

In matrix form: