Geometric Transformations

Transformations are very fundamental in computer graphics. They are used for Object Modelling, Viewing, and Projection.

Objects in a scene are a collection on points which have some location, orientation and size. These properties can be changed by Translations(\(\mathbf{T}\)), Rotations(\(\mathbf{R}\)), and Scaling (\(\mathbf{S}\)).

Homogeneous Coordinates

We can represent Rotation and Scaling as 2x2 matrices, but we cannot do so for Translation. There is non-linearity in Translation. To make Translation linear, we use Homogeneous Coordinates. This allows us to represent all 3 2D transformations as 3x3 matrices.

Proof that translation is non-linear

Let \(\mathbf{T}(p) = p + t\) be the translation of point \(p\) by vector \(t\). Then,

\[\mathbf{T}(p + q) = p + q + t \neq \mathbf{T}(p) + \mathbf{T}(q)\]

\[\mathbf{T}(ap) = ap + t \neq a\mathbf{T}(p)\]

Hence, Translation is non-linear.

To convert a point \((x, y)\) to Homogeneous Coordinates, we do the following:

\[\begin{bmatrix}x\\y\end{bmatrix} \rightarrow \begin{bmatrix}wx\\wy\\w\end{bmatrix}, w \neq 0\]

where \(w\) is a scaling factor. We can convert back to Cartesian Coordinates by dividing by \(w\). For linear transformations, embed the 2x2 matrix in a 3x3 matrix by adding a row and a column of 0s and a 1 in the bottom right corner.

\[\begin{bmatrix}a & b\\c & d\end{bmatrix} \rightarrow \begin{bmatrix}a & b & 0\\c & d & 0\\0 & 0 & 1\end{bmatrix}\]

2D Transformations in Homogeneous Coordinates

Transformation	Matrix
Translation	\(\begin{bmatrix}1 & 0 & t_x\\0 & 1 & t_y\\0 & 0 & 1\end{bmatrix}\)
Rotation	\(\begin{bmatrix}\cos\theta & -\sin\theta & 0\\\sin\theta & \cos\theta & 0\\0 & 0 & 1\end{bmatrix}\)
Scaling	\(\begin{bmatrix}s_x & 0 & 0\\0 & s_y & 0\\0 & 0 & 1\end{bmatrix}\)
\(\text{Shear}_x\) (\(\phi\) is angle along axis)	\(\begin{bmatrix}1 & \tan{\phi} & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}\)
\(\text{Shear}_y\)	\(\begin{bmatrix}1 & 0 & 0\\ \tan{\phi} & 1 & 0\\0 & 0 & 1\end{bmatrix}\)

Properties of Rotation Matrix

Rows are orthogonal to each other
Columns are orthogonal to each other
\(\mathbf{R}^\top = \mathbf{R}^{-1}\)
\(\det(\mathbf{R}) = 1\)

Make sure these are always satisfied.

Composition of Transformations

To apply multiple transformations to a point, we multiply the matrices of the transformations. The order of multiplication is important. For example, to rotate and then translate a point, we would do \(\mathbf{T}\mathbf{R}p\). Operations are evaluated from right to left.

Order of Multiplication

In general, Matrix Multiplication is not commutative. \(\mathbf{AB} \neq \mathbf{BA}\).

Transformation about Arbitrary Point

To rotate a point about an arbitrary point \((p_x, p_y)\), we first translate the point to the origin, rotate it, and then translate it back.

\[ \mathbf{R}_p(\phi) = \mathbf{T}(p_x, p_y) \hspace{10px} \mathbf{R}(\phi) \hspace{10px} \mathbf{T}(-p_x, -p_y) \]

Similarly for scaling, we can perform the following:

\[ \mathbf{S}_p(s_x, s_y) = \mathbf{T}(p_x, p_y) \hspace{10px} \mathbf{S}(s_x, s_y) \hspace{10px} \mathbf{T}(-p_x, -p_y) \]

Decomposition of Transformations

Any 2D matrix can be decomposed into a product of rotation, scale, rotation.

Eigenvalue Decomposition: \(\mathbf{A} = \mathbf{R} \hspace{10px} \mathbf{S} \hspace{10px} \mathbf{R}^{\top}\)

Note

Eigenvalue Decomposition only works for symmetric matrices.

Info

\(\mathbf{R}^{\top}\): Rotate the object to X and Y axes.
\(\mathbf{S}\): Scale the object.
\(\mathbf{R}\): Rotate the object back to its original orientation.

In a way it's directional scaling.

SVD Decomposition: \(\mathbf{A} = \mathbf{U} \hspace{10px} \mathbf{S} \hspace{10px} \mathbf{V}^{\top}\)
Paeth Decomposition: Applies only to rotations. Uses shear to represent non-zero rotations. Very useful in raster image rotations.

\[ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} = \begin{bmatrix} 1 & \frac{\cos \phi - 1}{\sin \phi} \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ \sin \phi & 1 \end{bmatrix} \begin{bmatrix} 1 & \frac{ \cos \phi - 1}{\sin \phi} \\ 0 & 1 \end{bmatrix} \]

3D Transformations

Properties which were applicable in 2D transformations are also applicable in 3D transformations. The only difference is that we have an additional axis.

Transformation	Matrix
Translation	\(\begin{bmatrix}1 & 0 & 0 & t_x\\0 & 1 & 0 & t_y\\0 & 0 & 1 & t_z\\0 & 0 & 0 & 1\end{bmatrix}\)
Rotation about X-axis	\(\begin{bmatrix}1 & 0 & 0 & 0\\0 & \cos\theta & -\sin\theta & 0\\0 & \sin\theta & \cos\theta & 0\\0 & 0 & 0 & 1\end{bmatrix}\)
Rotation about Y-axis	\(\begin{bmatrix}\cos\theta & 0 & \sin\theta & 0\\0 & 1 & 0 & 0\\-\sin\theta & 0 & \cos\theta & 0\\0 & 0 & 0 & 1\end{bmatrix}\)
Rotation about Z-axis	\(\begin{bmatrix}\cos\theta & -\sin\theta & 0 & 0\\\sin\theta & \cos\theta & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}\)
Scaling	\(\begin{bmatrix}s_x & 0 & 0 & 0\\0 & s_y & 0 & 0\\0 & 0 & s_z & 0\\0 & 0 & 0 & 1\end{bmatrix}\)

Arbitrary Axis Rotation

Given 3 mutually orthogonal unit vectors \(\mathbf{u} = (x_u, y_u, z_u)\), \(\mathbf{v} = (x_v, y_v, z_v)\), \(\mathbf{w} = (x_w, y_w, z_w)\), \(R_{uvw}\) transforms the coordinate system from \(xyz\) to \(uvw\).

\[ R_{uvw} = \begin{bmatrix} x_u & y_u & z_u \\ x_v & y_v & z_v \\ x_w & y_w & z_w\end{bmatrix} = \begin{bmatrix} u \\ v \\ w \end{bmatrix} \]

To rotate around arbitrary vector \(\mathbf{a} = (x_a, y_a, z_a)\)

Form an orthonormal \(uvw\) coordinate system with \(w = \mathbf{a}\)
Rotate \(uvw\) basis to canonical basis \(xyz\)
Rotate about \(z\) axis by \(\phi\)
Rotate back to \(uvw\) basis

\[R_a(\phi) = R_{uvw}^\top \hspace{10px} R_z(\phi) \hspace{10px} R_{uvw}\]

Constructing a basis from a vector

make \(w\) a unit vector \(w = \frac{a}{||a||}\)
Choose any vector \(t\) not parallel to \(w\), and build \(u = \frac{t \times w}{||t \times w||}\)
Complete the basis by \(v = w \times u\)

Transforming Normals

Normals are very common in computer graphics, so we need a way to also transform surface normals.
They do not transform the way surfaces do
Let the surface points be transformed by the matrix \(\mathbf{M}\). Then the normal vectors are transformed by the the following

\[ \mathbf{N'} = \mathbf{M}^{-\top} \mathbf{N} \]

Proof of Transforming Normals

Say you have a point \(p\) and a normal \(n\) on a surface. The normal is perpendicular to the surface at that point, i.e.

\[n^\top p = 0\]

If we transform the point to \(p' = \mathbf{M}p\), then the new normal \(n'\) should be perpendicular to the new surface at \(p'\), i.e.

\[n'^\top p' = 0\]

Let the new normal be \(n' = \mathbf{N}n\). Then,

\[n'^\top p' = (\mathbf{N}n)^\top \mathbf{M}p = n^\top \mathbf{N}^\top \mathbf{M}p = 0 \]

Since we already know that \(n^\top p = 0\), we can say that \(\mathbf{N}^\top \mathbf{M} = I\). Hence, \(\mathbf{N} = \mathbf{M}^{-\top}\).