Homogeneous Coordinates

The formula ${\mathbf{a}}^{\prime }=\mathbf{Ra}+\mathbf{t}$ from Euclidean Transformation expresses the rotation and translation of Euclidean space, but this form gets messy if you’re doing multiple transforms. For example, two transforms ${\mathbf{R}}_1,{\mathbf{t}}_1$ and ${\mathbf{R}}_2,{\mathbf{t}}_2$ would yield:

$$\mathbf{b}={\mathbf{R}}_1\mathbf{a}+{\mathbf{t}}_1,\mathbf{c}={\mathbf{R}}_2\mathbf{b}+{\mathbf{t}}_2$$

So the transform from $\mathbf{a}$ to $\mathbf{c}$ would be:

$$\mathbf{c}={\mathbf{R}}_2({\mathbf{R}}_1\mathbf{a}+{\mathbf{t}}_1)+{\mathbf{t}}_2$$

which will get even messier if we use more transformations. Thus, we introduce homogeneous coordinates and the Transform Matrix, rewriting ${\mathbf{a}}^{\prime }=\mathbf{Ra}+\mathbf{t}$ to:

$$\left[\begin{array}{c}{\mathbf{a}}^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ 0^T& 1\end{array}\right]\left[\begin{array}{c}\mathbf{a}\\ 1\end{array}\right]=\mathbf{T}\left[\begin{array}{c}\mathbf{a}\\ 1\end{array}\right]$$

We add $1$ at the end of the 3D vector and turn it into a 4D vector (called homogeneous coordinates). This way, the rotation and translation can be written in one matrix, making the whole thing linear. Matrix $\mathbf{T}$ is called the transform matrix.

If we call $\tilde{\mathbf{a}}$ the homogeneous coordinates of $\mathbf{a}$ such that $\tilde{\mathbf{a}}=[\mathbf{a}1{]}^T$, then the combination of the transforms has a nice form:

$$\begin{array}{r}\tilde{b}={\mathbf{T}}_1\tilde{\mathbf{a}},\tilde{\mathbf{c}}={\mathbf{T}}_2\tilde{\mathbf{b}}\\ ⟹\tilde{\mathbf{c}}={\mathbf{T}}_1{\mathbf{T}}_2\tilde{\mathbf{a}}\end{array}$$