Similarity Transformation

The similarity transformation has one more degree of freedom than the Euclidean Transformation, which allows the object to be uniformly scaled, and its matrix is expressed as:

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

Note that that the rotation part has an extra scaling factor $s$, which means that we can evenly scale the three coordinates of $x$, $y$, and $z$ of a vector after it is rotated.

Due to the scaling, a similarity transformation no longer keeps the volume of the transformed boy unchanged. You can imagine a cube with a side length of 1 transforming into a side with a length of 10 (but still being a cube). The set of three-dimensional similarity transform is also called similarity transform group, which is denoted as $\text{Sim}(3)$.