Computer Graphics

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Basics of Computer Graphics

Homogeneous Coordinates

Points and vectors are represented using homogeneous coordinates in computer graphics.
Points are $$\displaystyle (x,y,z,1)$$ and vectors are $$\displaystyle (x,y,z,0)$$.
The last coordinate in points allow for translations to be represented as matrix multiplications.

Notes
• The point $$\displaystyle (kx, ky, kz, k)$$ is equivalent to $$\displaystyle (x, y, z, 1)$$.

Transformations consists of translations, rotations, and scaling

Translation Matrix

$$\displaystyle T = \begin{bmatrix} 1 & 0 & 0 & X\\ 0 & 1 & 0 & Y\\ 0 & 0 & 1 & Z\\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Rotation Matrix

Rotations can be about the X, Y, and Z axis.
Below is a rotation about the Z axis by angle $$\displaystyle \theta$$.
$$\displaystyle R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 & 0\\ \sin(\theta) & \cos(\theta) & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$$

To formulate a rotation about a specific axis, we use Wikipedia:Rodrigues' rotation formula.
Suppose we want to rotate by angle $$\displaystyle \theta$$ around axis $$\displaystyle \mathbf{k}=(k_x, k_y, k_z)$$.
Let $$\displaystyle \mathbf{K} = [\mathbf{k}]_{\times} = \begin{bmatrix} 0 & -k_z & k_y\\ k_z & 0 & -k_x\\ -k_y & k_x & 0 \end{bmatrix}$$
Then the rotation matrix is $$\displaystyle \mathbf{R} = \mathbf{I}_{3} + (\sin \theta)\mathbf{K} + (1 - \cos \theta)\mathbf{K}^2$$
Here the 4x4 form is: $$\displaystyle R = \begin{bmatrix} \mathbf{R} & \mathbf{0}\\ \mathbf{0}^T & 1 \end{bmatrix}$$

Scaling Matrix

$$\displaystyle S = \begin{bmatrix} X & 0 & 0 & 0\\ 0 & Y & 0 & 0\\ 0 & 0 & Z & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$$

MVP Matrices

To convert from model coordinates $$\displaystyle v$$ to screen coordinates $$\displaystyle w$$, you do multiply by the MVP matrices $$\displaystyle w=P*V*M*v$$

• The model matrix $$\displaystyle M$$ applies the transform of your object. This includes the position and rotation. $$\displaystyle M*v$$ is in world coordinates.
• The view matrix $$\displaystyle V$$ applies the transform of your camera.
• The projection matrix $$\displaystyle P$$ applies the projection of your camera, typically an orthographic or a perspective camera. The perspective camera shrinks objects in the distance.

Model Matrix

Order of matrices
The model matrix is the product of the element's scale, rotation, and translation matrices.
$$\displaystyle M = T * R * S$$

View Matrix

Reference
Lookat function
The view matrix is a 4x4 matrix which encodes the position and rotation of the camera.
Given a camera at position $$\displaystyle \mathbf p$$ looking at target $$\displaystyle \mathbf t$$ and up vector $$\displaystyle \mathbf u$$.
We can calculate the forward vector (from target to position) as $$\displaystyle \mathbf{f}=\mathbf{p} - \mathbf{t}$$.
We can calculate the right vector as $$\displaystyle \mathbf u \times \mathbf f$$.
Then the view matrix is written as:

r_x r_y r_z 0
u_x u_y u_z 0
f_x f_y f_z 0
p_x p_y p_z 1

Matrix lookAt(camera_pos, target, up) {
forward = normalize(camera - target)
up_normalized = normalize(up)
right = normalize(cross(up, forward)
// Make sure up is perpendicular to forward
up = normalize(cross(forward, right)
m = stack([right, up, forward, camera], 0)
return m
}