modelMatrix: Rotation, Translation and Scaling

Transformations

There are three kind of basic linear transformation, we call this affine transformation.

Translation

For every vertex coordinate of shape SS, point p(x,y,z)p(x, y, z) apply a transition and get point p(x′,y′,z′)p(x', y', z')

So we have

Equation 1.0

x′=x+Txy′=y+Tyz′=z+Tzx' = x + T_x \\ y' = y + T_y \\ z' = z + T_z

In this case Tz is 00, very intuitive!

Rotation

First, let’s assume we are rotating around z-axis. Pick any point pp. point pp move to point p′p' after the rotation.

We have

Equation 1.1

y=rsin⁡αy = r\sin\alpha
Similarly
x′=rcos⁡(α+β)y′=rsin⁡(α+β)x'= r\cos(\alpha + \beta) \\\\ y'= r\sin(\alpha + \beta)
Use the addition theorem of trigonometric functions
x′=r(cos⁡αcos⁡β−sin⁡αsin⁡β)y′=r(sin⁡αcos⁡β+cos⁡αsin⁡β)x'= r(\cos\alpha\cos\beta - \sin\alpha\sin\beta) \\\\ y'= r(\sin\alpha\cos\beta + \cos\alpha\sin\beta)
By assigning Equation 1.1, you get the following expressions

Equation 1.2

x′=xcos⁡β−ysin⁡βy′=xsin⁡β+ycos⁡βz=z′x'= x\cos\beta-y\sin\beta \\\\ y'= x\sin\beta + y\cos\beta \\\\ z = z'

Scaling

Clearly, We have

Equation 1.3

x′=Sx×xy′=Sy×yz=Sz×zx'= S_x \times x \\\\ y'= S_y \times y \\\\ z = S_z \times z

Matrices

You will find it time consuming to write a mathematical expression every time you need a new set of transformation. To solve this problem, we introduce a new tool: Matrix.

Rotation Matrix

First, there is transformation

[x′y′z′]=[abcdefghi]×[xyz]\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z \end{bmatrix}

For now, you just hard code how matrix multiplication works, you could learn linear algebra later.

x′=ax+by+czy′=dx+ey+fzz′=gx+hy+izx'= ax + by + cz \\\\ y'= dx + ey + fz \\\\ z' = gx + hy + iz

comparing with Equation 1.2, we know that a=cos⁡βa=\cos\beta, b=−sin⁡βb=-\sin\beta and c=0c=0. Also, d=sin⁡βd=\sin\beta, e=cos⁡βe=\cos\beta f=0f=0 g=0g=0 h=0h=0 i=1i=1, then we have matrix

[x′y′z′]=[cos⁡β−sin⁡β0sin⁡βcos⁡β0001]×[xyz]\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \cos\beta & -\sin\beta & 0 \\ \sin\beta & \cos\beta & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z \end{bmatrix}

We call this a transformation matrix

Translation Matrix

What about translation? You can’t product x=x+Txx = x + T_x by 3x3 matrix. Again, yuo have to hard code the resolution. the answer is to use to 4x4 matrix called Homogeneous Transformation Matrix

[x′y′z′1]=[100Tx010Ty001Tx0001]×[xyz1]\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & T_x \\ 0 & 1 & 0 & T_y \\ 0 & 0 & 1 & T_x \\ 0 & 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}

You might wonder why there are four coordinates for p(x,y,z)p(x, y, z), this is called Homogeneous coordinates and you can ignore it.

Scaling Matrix

We can get a scaling matrix:

[x′y′z′1]=[Sx0000Sy0000Sz00001]×[xyz1]\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \begin{bmatrix} S_x & 0 & 0 & 0 \\ 0 & S_y & 0 & 0 \\ 0 & 0 & S_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}

modelMatrix

Say we want to do a translation TT, then rotation RR followed by scaling SS. First we make a translation matrix multiply origin Matrix AA

step1:

T⋅AT \cdot A

step2:

R⋅T⋅AR \cdot T \cdot A

step3:

S⋅R⋅T⋅AS \cdot R \cdot T \cdot A

An important thing you need to know is that

S⋅R⋅A≠R⋅S⋅A(S⋅R)⋅T⋅A=S⋅(R⋅T)⋅AS \cdot R \cdot A \neq R \cdot S \cdot A \\ (S \cdot R) \cdot T \cdot A = S \cdot (R \cdot T) \cdot A

means that matrix multiplication is associative but not commutative. We call this kind of matrix M=S⋅R⋅TM = S \cdot R \cdot T modelMatrix