Computer Graphics

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Basics of Computer Graphics

Homogeneous Coordinates

http://www.opengl-tutorial.org/beginners-tutorials/tutorial-3-matrices/

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
}

Perspective Projection Matrix

https://www.scratchapixel.com/lessons/3d-basic-rendering/perspective-and-orthographic-projection-matrix/building-basic-perspective-projection-matrix

Shading

Flat Shading

Gourard Shading

Phong Shading

More Terms

Resources