第七讲计算机图形学—二维三维几何变换

Here t = <tx,ty,tz> is the direction and magnitude of move
t
3D TRANSLATION

1 0 0 0
0 1 0 0
0 2 4 2 0 - 1 1 3 0 0 1 1 6 1 3 1

3D TRANSLATION

The translation can be represented in 4 x 4 matrix T in homogeneous coordinates:
Identity Matrix
T=
1 0 0 0
0 1 0 0
0 tx 0 ty 1 tz 0 1
S
M= SR
R
3D SCALING
T

To scale according to object center, we need to multiply with translation matrix T, then scale translation matrix S and finally inverse of T (i.e. T-1 ) T-1ST

SUMMARY
An overview of computer graphics topics Basic 2D geometry: Vertex, Line, Triangle and etc. Mathematics of Vector, Matrix using 2D examples 2D Transformations: scaling, rotation and translation Relation between transformation and matrix multiplication

When multiplying with a vertex
二维放缩
一致放缩和差值放缩 放缩但保持原来的位置 方缩+平移

二维放缩

An example to shrink half the size:

So, every vertices are multiplying this matrix

二维旋转

So, the Rotation Matrix

A rotation with 45 degrees (PI/4) in anticlockwise
二维旋转
绕任意基准点旋转 旋转+平移

二维平移

Move horizontally (in x axis) or vertically (in y axis)
t
4 2 2 1 0 3 1
3D SCALING
Scale Matrix is a direct extension to its 2D version, Now, we have sx, sy and sz as the scaling factor in x, y and z directions
S
Note the translation is to move the object center to origin
T-1
INVERSE MATRIX

By definition, inverse matrix A-1of a matrix A should satisfy: A-1 A =I

为什么要用齐次坐标
按顺序多次几何变换 P’=M1P+M2 消除矩阵加法

齐次坐标下的二维几何变换表示
按列表示 二维平移矩阵 二维旋转矩阵 二维放缩矩阵 复合变换矩阵

MULTIPLE 2D TRANSFORMATIONS
How about the case we want to do more than one transformation the same time? For example,

3D SCALING

0 2.0 0 0 0.5 0 0 0 1.0 0 0 0 0 0 0 1 1 1 1 1 2 0 .5 1 1
3D SCALING
Notice that the scaling is in reference to the origin (0,0,0) If object’s center is not at origin, a direct multiply with the scale matrix will look like moving the object at the same time A simple solution is to first translate the object to the origin, scale and then translate back to the original center

First,
shrink in Y direction for 0.5 (Matrix S) Then, rotate 45 degree in anticlockwise (Matrix R)

If we represent in Matrix form, the above complex operation can also be represented in a matrix M

First, shrink in Y direction for 0.5 (Matrix S) Then, rotate 45 degree in anticlockwise (Matrix R)

The standard way is do it stepwise, but it is the same as we multiply RS to the vertex

MULTIPLE 3D TRANSFORMATIONS


Before discussing the solution, recall the topic about multiple transformation in last lesson To apply more than one transformation, e.g.

SR = RS So be-careful, Order DOES Matter!!
ORDER OF APPLYING TRANSFORMATIONS

We can also show the related Matrices formed:
RS
= M1
SR
= M2
They are different!!!

放射变换 （9.7节）
平移 放缩 旋转
反射
Байду номын сангаас错切
DOES MATRIX REALLY NECESSARY?
The answer is Yes or No No: You can do computation in sets of equations Yes: It sometimes make things more clear and easy Yes: We can think any kind of transformation (including S,R,T or etc) to be represented with matrix and performed by a multiplication in matrix Yes: It is already a standard and common language in Graphics

Transpose
Change
E.g.
row to column, column to row
Transpose
here matrix MT is the transpose of matrix M
二维放缩

Scale Matrix, sx and sy are the scaling factor in x and y directions
二维三维几何变换
2D BEFORE 3D
Although
everything in real world is 3D, when we have to display something, everything reduce to a 2D image / screen space Drawing in 2D is therefore a basic needs Some concepts and maths are easier to understand in 2D before going into 3D
S
M= SR R
ORDER OF APPLYING TRANSFORMATIONS
Another question, does M = SR = RS ? The order of multiplication a matter? From the example on right, we know that
二维旋转

2D Rotation commonly involves
Reference
point Angle of rotation
Usually, it is easiest to use the origin (0,0) as anti-clockwise the reference point The related equations:
MULTIPLE 2D TRANSFORMATIONS