Lecture_004_Spatial_Transformations

Rotation

Orthogonal Transformation: the transformation that preserve distance and origin

• represented by matrices $Q^T Q = I$ (orthogonal matrix)

• Rotation: additionally preserve orientation ($\det Q > 0$)

• Reflection: reverse orientation ($\det Q < 0$)

Scaling

Scaling: preserve direction while change magnitude

Negative Scaling: a sequence of reflections (how many reflections depends on the dimension of the space)

• in 2D: Negative Scaling is Rotation

• in 3D: Negative Scaling is Reflection

Nonuniform Scaling: a symmetric matrix $A := R^TDR$

• rotate to new axes ($R$)

• apply a diagonal scaling ($D$)

• rotate back to original axes ($R^T$)

Symmetric Matrix: a matrix $A$ such that $A = A^T$.

• has orthonormal eigenvectors $e_1, ..., e_n \in \mathbb{R}^n$

• has real eigenvalues $\lambda_1, ..., \lambda_n \in \mathbb{R}$

• Spectral Theeorem: all symmetric matrix represent non-uniform scaling along some set of orthogonal axes (by $A = A^T \implies A = RDR^T$ where $R = \begin{bmatrix} e_1, ..., e_n\\ \end{bmatrix}$, $R = \begin{bmatrix} \lambda_1 & & \\ & ... & \\ & & \lambda_n\\ \end{bmatrix}$)

• if $\lambda_i > 0e$: then scaling is positive along corresponding $e_i$

Shear

Shear in direction $u$ according to its distance along a fixed vector $v$ is $f_{u, v}(x) = x + (x \cdot v)u$.

• $x$: a vector in space.

• $u$: Any vector to add. Displace point in the direction of this vector. Can be $u = (\cos t, 0, 0)$

• $v$: A unit vector represent dirrection, orthogonal to sheer's parallel dirrection (parallel dirrection is dirrection with eigenvalues is $1$). Displace those points according to its distance in. For example $v = (0, 1, 0)$ says the bigger $y$ component is, the more we move in direction $u$.

The transformation can be represented as $A_{u, v} = I + uv^t \implies A_{u, v}x = Ix + u(v \cdot x)$

// QUESTION: Can we linear time discompose a sheer to standard basis? Two sheers together is still a sheer or not?

Decomposition of Linear Transformation

No unique way to write a given linear transformation as a composition of basic transformations, but many useful decomposition:

• Singular Value Decomposition: for signal processing

• LU Factorization: solving linear systems

• Polar Decomposition: for spatial transformation

Polar Decomposition: decompose $A$ into orthogonal matrix $Q$ and symmetric positive-semidefinite matrix $P$. So that $A = QP$.

• $Q$: rotation and reflection

• $P$: non-negative, possibly non-uniform scaling.

Spectral Decomposition: Since $P$ is symmetirc, we can do spectral decomposition $P = VDV^T$ where $V$ is orthogonal and $D$ is diagonal. So that $A = QVDV^T$

Singular Value Decomposition: $A = QVDV^T = UDV^T$ where we combine two orthogonal matrix $U = QV$. (Every matrix has a singular value decomposition)

Interpolation Using Decomposition

Interpolating Transformations:

• Linear Decomposition: $A(t) = (1-t)A_0 + tA_1$ (awful in between frames)

• Polar Decomposition: separately interpolate components of polar decomposition, making $A_0 = Q_0P_0, A_1 = Q_1P_1$. $A(t) = Q(t)P(t)$

• where $P(t) = (1 - t)P_0 + tP_1$

• where $Q'(t) = (1-t)Q_0 + t Q_1$ and $Q'(t) = Q(t)X(t)$. $Q$ is rotational component of $Q'(t)$ after singular value decomposition.

Translation

Homogeneous Coordinates

Translation: it is not linear transformation (it is affine). We can make transformation linear using 4th dimension with Homogeneous Coordinates.

Idea of Homogeneous Coordinates: Fix origin $o$ and a point in 2D space specified by plane $z = 1$ $p \in \mathbb{R}^2$ can be represented by any points (except the origin) along the line that contains $o$ and $p$.

• $\hat{p} = (a, b, c)$ such that $(\frac{a}{c}, \frac{b}{c}) = (x, y) = p$ are homogeneous coordinates for $p$.

• $\hat{p}, \hat{q} \in \mathbb{R}^3 - \{o\}$ describe the same point in 2D (and line in 3D) if $\hat{p} = \lambda \hat{q}$ for some $\lambda \neq 0$

Proof it is a sheer ing 3D:

• For 2D Coordinate: $(p_1, p_2) + (u_1, u_2) = (p_1 + u_1, p_2 + i_2)$

• For Homogeneous Coordinates: $(cp_1, cp_2, c) + (u_1, u_2) = (cp_1 + cu_1, cp_2 + cu_2, c)$ (it is a shear because we are shifting $x$ and $y$ coordinates by amount that is proportional to $z$.)

A sheer in matrix form: $f_{u, v}(x) = (I + uv^T)x$. For the use in homogeneous coordinates to represent 2D coordinates, it is always the case $v = (0, 0, 1)$. So we have matrix $\begin{bmatrix}1 & 0 & u_1\\ 0 & 1 & u_2\\ 0 & 0 & 1\\ \end{bmatrix}$. Observe the following:

Advantages:

• We can do rotation in 2D as rotation in 3D in homogeneous coordinates.

• We can do scaling in 2D as scaling with only 2D coordinates in 3D.

• We can do 2D translation as shear in 3D.

Dealing with Unit Vectors (like Normal)

In general:

• A point in 3D will have a non-zero homogeneous coordinate ($c \neq 0$)

• A vector in 3D (like normal) will have a zero homogeneous coordinate ($c = 0$)

The problem is, when we want to convert vector in homogeneous coordinate back to 3D coordinate, we might encounter division by $0$ (since $c = 0$)

Mathematical intuition: as $c \rightarrow 0$, we get vectors that represent a line (or a direction towards infinity), which is intuitive for unit vectors since they suppose to only give us a direction.

Practical solution: branch if to check for $0$.

Perspective Projection

Scene Graph

Note: Keep transformation on a stack to reduce redundant multiplication

1. you might push() a transformation (from body position to leg position)
2. and then draw something (for example a leg)
3. and then pop() a transformation (from leg position to body position)

Instancing: Used to draw many copies of the same object in a scene. It can save memory so that the same model only get loaded once in the tree.