Computer Graphics

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Basics of Computer Graphics

Homogeneous Coordinates

Points and vectors are represented using homogeneous coordinates in computer graphics.
This allows affine transformations in 3D (i.e. rotation and translation) to be represented as a matrix multiplication.
While rotations can typically be represented in a 3x3 matrix multiplication, a translation requires a shear in 4D.

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.

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

Affine transformations consist 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. \(\displaystyle V*M*v\) is in camera or view coordinates.
  • 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

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

Perspective Projection Matrix

Notes: In computer vision, this is called the calibration matrix \(\displaystyle K\). It contains the intrinsic parameters of your pinhole camera such as field of view and focal length (which determines the resolution of your output).

Inverting the projection

If you have the depth (either z-depth or euclidean depth), you can invert the projection operation.
The idea is to construct a ray from the camera to the pixel on a plane of the viewing frustrum and scale the distance accordingly.

See stackexchange.


Flat Shading

Gourard Shading

Phong Shading

More Terms