Personal notes of graphics (2)

Tips ： Remember one 2×2 Multiply by a matrix of The result of two-dimensional vector can fully understand the following formula ：

  The form of transformation ： Multiply a matrix by a vector （ Multiply the matrix of the same dimension by the variables of the same dimension ）

linear transformation

a1 a2            x

            ride

b1 b2            y

a1x+a2y,b1x+b2y 

Transformation ：

Except for rotation

And from the three-dimensional world to the two-dimensional world

1. The zoom ：

 

s=0.5

Express in matrix

  Uneven scaling ：

 sx=0.5 sy=1

2. Reflection （ symmetry ）

 3. shear （ Pull a corner ）

 （ Ideas ： All points y It's all the same , The only change is all points x value ,x How does the value change ？

When y by 0 When x unchanged , When y by 1 When x Add 1, When y Greater than zero and less than one be x Add a*y）

Express in matrix

  An idea of writing transformation ： Find some rules before and after the transformation

4. rotate （ The default is around the origin , Default CCW ）

deduction

 

  The original coordinate is （1,0） After conversion, it is （cos,sin）

Get this formula

The formula is solved as

I got it AC Value The rest is to find a point to calculate the rest BD The value of

The inverse matrix of this rotation only needs to change the lower left corner to negative , Just change the upper right corner to positive
 

 //+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++//

Homogeneous coordinates ： Because translation transformation is special , So we have the concept of homogeneous coordinates

  But it can't be written in the form of a matrix multiplied by a vector just now

It can only be written like this （abcd The matrix is linear Don't do anything 1001）：

  Because the translation transformation is not a linear transformation

reflection ： Can the calculation methods of the above transformations be unified （ Including translation , Including zoom ...）

answer ： You can add a dimension

（ spot ： increase 1  vector ： increase 0 Because vectors have translation invariance ）

More significance of introducing homogeneous coordinates

  Point plus point equals midpoint

Affine transformation can be in the form of coprocess matrix multiplied by variables

  With translation without translation There is linear transformation, no linear transformation

inverse transformation ： Reverse the previous transformation

Combination of various transformations ：

  The order of transformation is very important

Translate first and then rotate

First rotate in Translation

【 From right to left ！！！！！！！！！】

It means that the rightmost matrix multiplies first That is to say, the rightmost one should rotate first The next matrix is the moving matrix

  Decomposition of transformation ：

  ask ： How to rotate the first picture along the lower left corner 45 degree ？

answer ：1. First subtract all points c 2. Rerotation 45 degree 3. Add all the points c

Matrix form （ From right to left ）：

Three dimensional space transformation

Pretty much , But there are several important properties

1. The inverse matrix of rotation is equal to the transpose matrix of rotation （ Here's the picture ） That is to say, the rotation matrix is an orthogonal matrix

 

2.

For us, a xyzw The point of stay 3 The point representation of dimensional space is Divide everything by w（w It must be 1 w by 1 Represents a dot ）

 

3.

about For the rotation of three-dimensional space , Looking at the rotation can raise our right hand for instance x to turn to z Just forgive y Shaft rotation ,y Constant so 0 1 0

But why is the edge here y Axis rotation is different from other matrices , Hurt to raise our right hand , Think about the cross product rule ,

In order to get z Axis must x Cross riding y Axis （ Anti-clockwise ）, In order to get x Axis must y Cross riding z（ Anti-clockwise ）, In order to get y Axis must z Cross riding x（ Clockwise ）

At this time, the smart little friend has seen it

We must remember For complex rotation, we must be able to decompose it into simple rotation That is to say, arbitrary 3d rotate

We can all write it around x Axial winding y Axial winding z Axis

 α β γ These three angles are also called Euler angles

How to prove ？ You can imagine three operations of the aircraft

pitch（ pitch ）x The axis does not change z turn x Axis

roll（ Roll ）z The axis does not change x To y Axis

yaw（ Turn around ）...