3 - Linear Algebra for Graphics

ucla | CS 174A | 2024-01-16 18:29

Table of Contents

Point Arithmetic

  • $P_1-P_2=\vec V$
  • $\alpha P_1=P$
  • idea: scaling a point makes sense for affine transformations but not for linear combinations

    Affine Transformations

    \(P=\alpha_1 P_1+\alpha_2 P_2\) where \(\alpha_1+\alpha_2=1\)

    Parametric Form of a Line

    \(L_2=(1-t)P_1+tP_2\)

  • where $L_2$ denotes a 2-D line
  • $0\le t\le 1$ -> finite edge
  • $\lambda \le t$ -> semi-infinite ray
  • $t\in \mathbb R$ -> infinite line

    Constraints

  • given a convex region defined by some $n$ points $P_i$ and $n$ parameters $\alpha_i$ on the same plane (a planar polygon)
  • Affine constraint
    • given a point as a parametric sum of points \(P=\sum_i^n\alpha_i P_i\)
    • the resulting point will lie on the plane \(\sum_i^n \alpha_i=1\)
  • Convex constraint
    • given a point as a parametric sum of points: \(P=\sum_i^n\alpha_i P_i\)
    • the resulting point will lie within the convex region/hull iff.\(\alpha_i\gt 0 \quad \forall i\)

      Vector Representations

  • homogeneous representation of points and vectors for graphics: add a 4th dimension, 0 = vector, 1+ = point e.g., \(\vec v = (v_1,v_2,v_3,0)\quad P=(p_1,p_2,p_3,1)\)

    Affine Transformation of Homogeneous Tuples

  • math works the same, but now point arithmetic makes sense \(\alpha X +\beta Y = \big(\alpha X_i+\beta Y_i,...,\alpha+\beta\big)\)

    Interpolation

    Linear Interpolation of 2 points (parametric line)

    \(P_{\text{interp}}=(1-\alpha)P_1 + \alpha P_2\)

  • values of $\alpha\in(0,1)$ interpolate a point within the edge made by the 2 points
  • values of the parameter outside of this bounds will interpolate points on the line made by these 2 points
  • treating the parameter as a unit of time can animate the line as the parameter changes

    Planar Interpolation of 3 points (parametric polygon)

  • given a planar polygon of 3 vertices PRQ triangle, we can interpolate dimensionally
  • linear interpolation bw 2 points \(S(\alpha) = (1-\alpha)P + \alpha Q\)
  • planar interpolation (point by point) \(T(\beta) = (1-\beta)S+\beta R\) \(T(\alpha,\beta)=(1-\beta)\cdot\bigg((1-\alpha)P+\alpha Q\bigg) + \beta R\)

    3-D Rendering

  • use triangles to represent shapes bc closed, convex, planar, simple, and optimal in vertices
  • interpolate triangles to fill and tesselate shapes to make structures
  • create 3-d objects using indexed face/data sets
    • give an indexed list of vertices coordinates
    • then give an indexed list of faces using prev list in counter-clockwise order (to make normals outward pointing)
  • these point lists can be loaded into a matrix to parallelize then load onto GPU for fewer communication calls

    Matrix Arithmetic

  • we make homogenous representations using quaternions
  • all except scale and shear are orthonormal matrices i.e., rigid body transformations
  • all are affine transformations
    • they maintain collinearity
    • maintains planarity
    • maintain parallelism
    • maintain edge-length ratios
  • the final transformation matrix made of a series of products of transformations can be seen to be composed only of rigid body transformations IF the smaller 3x3 (top left) of the final matrix is orthonormal
  • the inverse of orthonormal matrices is the transpose of the matrix \(A^{-1}=A^T\quad\text{iff.}\quad \text{$A$ is orthogonal}\)
  • pure translation is commutative
  • pure rotation is commutative and additive
  • a linear combination of them is not commutative

    Scaling

  • scaling wrt to origin, use diagonal scaling matrix to scale unit vectors \(\begin{pmatrix}x' \\ y'\end{pmatrix}=\begin{pmatrix}S_x & 0 \\ 0 & _y\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}\)

  • Inverse

  • Rotation

  • use rotation matrices to rotate a vector by an angle $\theta_x$ off the axis of relevance, e.g. in 2-D about to axis $x$
  • 2d representation\(\begin{pmatrix}x' \\ y'\end{pmatrix}= \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}\)
  • 3-D representation \(\begin{alignat}{1} R_x(\theta) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\[3pt] 0 & \sin \theta & \cos \theta \\[3pt] \end{bmatrix} \\[6pt] R_y(\theta) &= \begin{bmatrix} \cos \theta & 0 & \sin \theta \\[3pt] 0 & 1 & 0 \\[3pt] -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} \\[6pt] R_z(\theta) &= \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\[3pt] \sin \theta & \cos \theta & 0 \\[3pt] 0 & 0 & 1 \\ \end{bmatrix} \end{alignat}\)
  • Quaternion representation

    Inverse

  • pure rotations are also orthonormal, so simply taking the transpose of the transformed/rotated points will give the inverse

  • Translation

  • use quaternions
  • such that the translation matrix is

    • Inverse

  • Shear

  • along x-axis
  • along y-axis \(\begin{pmatrix}x' \\ y' \\ z' \\ 1\end{pmatrix}= \begin{pmatrix}1&0&0&0\\b&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}\begin{pmatrix}x \\ y\\z\\1\end{pmatrix}\)

    Inverse

  • inverse in x-axis

    Inverse of Transformations

Composition of Rotation Matrices

  • Euler Angle

    Gimbal Lock - due to Euler Angle Representation

  • when the rotation angle wrt to y-axis is 90 deg -> the remaining two angles collapse to one parameter which is the difference of the 2
  • this reduces the degree of freedom of rotation