[Computer Animation] Geometric Transformations

Geometric Transformations

Geometric Transformation, $\mathbb{T}$

• $\mathbb{T}$ is a function
  • Geometric structure A → Geometric structure B
• A geo. structure is represented by a set of points or vectors in 2D o 3D in the general case.

 

2D Examples

Scaling

● Scale by 2 in the y-axis. → $S(1, 2)$ ▶ $x^{'} = x$ ▶ $y^{'} = 2y$ ● Scale by 0.5 in the x-axis. → $S(0.5, 1)$ ▶ $x^{'} = 0.5x$ ▶ $y^{'} = y$ ● Scale by $s_{x}$ and $s_{y}$ in the x and y-axis respectively. → $S(s_{x}, s_{y})$ ▶ $x^{'} = s_{x}x$ ▶ $y^{'} = s_{y}y$

 

Rotation

● Rotate by 45˚ about the origin. → $R(45˚)$ ▶ $x^{'} = cos(45˚)x - sin(45˚)y$ ▶ $y^{'} = sin(45˚)x + cos(45˚)y$ ※ Because of the Law of Cosine. ● Rotate by θ˚ about the origin. → $R(\theta)$ ▶ $x^{'} = cos(θ˚)x - sin(θ˚)y$ ▶ $y^{'} = sin(θ˚)x + cos(θ˚)y$

 

Translation

● Translate by 2 in the x-axis. → $T(2, 0)$ ▶ $x^{'} = x + 2$ ▶ $y^{'} = y$ ● Translate by -1 in the y-axis. → $T(0, 1)$ ▶ $x^{'} = x$ ▶ $y^{'} = y - 1$ ● Translate by $t_{x}$ and $t_{y}$ in the x and y-axis, respectively. → $T(t_{x}, t_{y})$ ▶ $x^{'} = x + t_{x}$ ▶ $y^{'} = y + t_{y}$

 

Composite Transformations(합성 변환)

Example 1

• Rotate by 45˚ about the origin, then scale by 2 in the y-axis.
  • $R(45˚) → S(1, 2)$

 

● $R(45˚)$
$x' = cos45˚x - sin45˚y$
$y' = sin45˚x + cos45˚y$

● $S(1, 2)$
$x'' = 1x'$
$y'' = 2y'$

▼

● $R(45˚) → S(1, 2)$
$x''' = 1·(cos45˚x - sin45˚y)$
$y''' = 2·(sin45˚x + cos45˚y)$

 

Example 2

• $R(45˚) → S(1, 2) → R(-45˚) → T(-1, 0)$

 

● $R(45˚)$
$x' = cos45˚x - sin45˚y$
$y' = sin45˚x + cos45˚y$

● $S(1, 2)$
$x'' = 1x'$
$y'' = 2y'$

● $R(-45˚)$
$x''' = cos(-45˚)x'' - sin(-45˚)y''$
$y''' = sin(-45˚)x'' + cos(-45˚)y''$

● $T(-1, 0)$
$x'''' = x''' - 1$
$y'''' = y'''$

▼

● $R(45˚) → S(1, 2) → R(-45˚) → T(-1, 0)$
$x'''' = cos(-45˚)·(1·(cos(-45˚)x - sin(-45˚)y) - sin(-45˚)·(sin(-45˚)x'' + cos(-45˚)y'') - 1$
$y'''' = sin(-45˚)·1·(cos45˚x - sin45˚y) + cos(-45˚)·2·(sin45˚x + cos45˚y)$

 

2×2 Matrix Form

• Scale by $s_{x}$ and $s_{y}$.

$x' = s_{x}x$ $y' = s_{y}y$

=>

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} s_{x} & 0 \\ 0 & s_{y} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

• Rotate by $\theta˚$ about the origin.

$x' = cos(θ˚)x - sin(θ˚)y$ $y' = sin(θ˚)x + cos(θ˚)y$

=>

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} cosθ & -sinθ \\ sinθ & cosθ \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

 

• Translate by $t_{x}$ and $t_{y}$.

$x' = x +t_{x}$ $y' = y + t_{y}$

=>

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} t_{x} \\ t_{y} \end{bmatrix} $$

 

Questions

Calculate new points of a, b and c after applying following transformations. Scale by 3 along the x-axis. Rotate by 90 degree. Translate by 2 along the x-axis and -2 along the y-axis.

 

3×3 Matrix Form (Homogeneous Coordinates(동차 좌표))

• Scale by $s_{x}$ and $s_{y}$.

$x' = s_{x}x$ $y' = s_{y}y$

=>

$$ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} s_{x} & 0 & 0 \\ 0 & s_{y} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$

• Rotate by $\theta˚$ about the origin.

$x' = cos(θ˚)x - sin(θ˚)y$ $y' = sin(θ˚)x + cos(θ˚)y$

=>

$$ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} cosθ & -sinθ  & 0 \\ sinθ & cosθ & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$

 

• Translate by $t_{x}$ and $t_{y}$.

$x' = x +t_{x}$ $y' = y + t_{y}$

=>

$$ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_{x} \\ 0 & 1 & t_{y} \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$

 

Composite Transformations

Example 1 

• $R(45˚) → S(1, 2)$

 

Example 2

• $R(45˚) → S(1, 2) → R(-45˚) → T(-1, 0)$

 

$$ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} =
\begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} cos(-45˚) & -sin(-45˚) & 0 \\ sin(-45˚) & cos(-45˚) & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix} 
\begin{bmatrix} cos45˚ & -sin45˚ & 0 \\ sin45˚ & cos45˚ & 0 \\ 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$

$ \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} =\begin{bmatrix} 1.5 & 0.5 & -1 \\ 0.5 & 1.5 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $

 

Ignoring the translation

• Translation시키지 않고, Scaling, Rotation을 바꾸고 싶을 경우, 끝 부분을 0으로 바꿔주면 된다.

 

Summary

• 2D Transformation
  • Scaling
  • Rotation
  • Translation
• Homogenous Coordinate(동차좌표)
  • Allows to represent scaling, rotation and translation in a single matrix.

 

 

In 3D Space

Transformation in 3D Space (Homogeneous Coordinates)

Transformation	Equations
3D Translation
3D Scaling
Rotation

3D Rotation around z-axis, x-axis and y-axis

 

Transformations

① Linear Transformation(선형 변환)

2D Spaces

3D Spaces

 

• A Linear Transformation $\mathbb{T}$ is a mapping between vector spaces.
  • $\mathbb{T}$ is a function that maps vectors to vectors.
    • $\mathbb{T}: v → v', $        $v, v' \in \mathbb{R}^{n}$
  • Linear Combination is invariant(불변한) under $\mathbb{T}$
    • $\mathbb{T}(\sum_{i=0}^{n} c_{i}v_{i}) = c_{0}\mathbb{T}(v_{0}) + c_{1}\mathbb{T}(v_{1}) + \cdots + c_{n}\mathbb{T}(v_{n})$

 

Examples of Linear Transformations

2D Rotation
2D Scaling
2D Shear
Along X-axis	Along Y-axis
2D Reflection
Along X-axis	Along Y-axis

 

Properties of Linear Transformations

• Any Linear Transformation between 3D spaces can be represented by a 3×3 matrix.
• Any Linear Transformation between 3D spaces can be represented by as a combination of rotation, shear and scaling.
• Rotation can be represented as a combination of scaling and shear.
• A Linear Transformation maps lines to lines.
• A Linear Transformation maps parallel lines to parallel lines.
• A Linear Transformation preserves ratios of distance along a line.
• A Linear Transformation does not preserve absolute distances and angles.

 

 

② Affine Transformation(아핀 변환)

2D Spaces

3D Spaces

 

• An Affine Transformation $\mathbb{T}$ is a mapping between affine spaces.
  • $\mathbb{T}$ is a function that maps vectors to vectors, and points to points.
    • $\mathbb{T}: v → v', $        $v, v' \in \mathbb{R}^{n}$
    • $\mathbb{T}:p → p', $        $p, p' \in \mathbb{R}^{n}$
  • Affine Combination is invariant(불변한) under $\mathbb{T}$
    • $\mathbb{T}(\sum_{i=0}^{n} c_{i}p_{i}) = c_{0}\mathbb{T}(p_{0}) + c_{1}\mathbb{T}(p_{1}) + \cdots + c_{n}\mathbb{T}(p_{n})$

 

Homogeneous Coordinates

• Any Affine Transformation between 2D spaces can be represented by a 3×3 matrix.
  • $\mathbb{T}(p) = \begin{pmatrix} M_{2×2} & T_{2×1} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} p_{2×1} \\ 1 \end{pmatrix}$

• Any Affine Transformation between 3D spaces can be represented by a 4×4 matrix.
  • $\mathbb{T}(p) = \begin{pmatrix} M_{3×3} & T_{3×1} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} p_{3×1} \\ 1 \end{pmatrix}$

 

Affine Transformations (Homogenous Coordinates)

2D Spaces

3D Spaces

 

Examples of Affine Transformations

2D Rotation

2D Scaling

2D Shear

2D Reflection

2D Translation

 

• 2D Transformation for vectors
  • Translation is simply ignored.

 

Properties of Linear Transformations

• Any Affine Transformation between 3D spaces can be represented as a combination of a Linear Combination followed by Translation.
• An Affine Transformation maps lines to lines.
• An Affine Transformation maps parallel lines to parallel lines.
• An Affine Transformation preserves ratios of distance along a line.
• An Affine Transformation does not preserve absolute distances and angles.

 

 

③ Rigid Transformation(강체 변환)

2D Spaces

3D Spaces

 

• A Rigid Transformation $\mathbb{T}$ is a mapping between affine spaces.
  • $\mathbb{T}$ is a function that maps vectors to vectors, and points to points.
    • $\mathbb{T}: v → v', $        $v, v' \in \mathbb{R}^{n}$
    • $\mathbb{T}:p → p', $        $p, p' \in \mathbb{R}^{n}$
  • $\mathbb{T}$ preserves distances between all points.
  • $\mathbb{T}$ cross product for all vectors (to avoid reflection)

 

Rigid Body Rotation

• Rigid body transformations allow only rotation and translation.
• Rotation matrics from $SO(3)$

 

Taxonomy(분류) of Transformations

 

 

Composite Transformation(합성 변환)

Functionalization

Translation

Scaling

Rotation

 

Composite Transformations

Composite 2D Translation

Composite 2D Scaling

Composite 2D Rotation

 

• Suppose we want,

• We have to compose two transformations.

 

• Matrix Multiplication is not commutative(가환성의, 제시된 수의 순서에 상관 없이 결과가 동일한)
  • $T(x, 3) × R(-90˚) ≠ R(-90˚) × T(x, 3)$

 

Pivot-Point Rotation

• Rotation with respect to a pivot point (x, y)

 

Fixed-Point Scaling

• Scaling with respect to a fixed point (x, y)

 

Scaling Direction

• Scaling along an arbitrary axis