# Similarity Transformation / Sim(3)

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}{*{20}{c}} {s \mathbf{R}}& \mathbf{t}\\ {{ \mathbf{0}^T}}&1 \end{array}} \right] \end{equation}$$ Note 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 $\mathrm{Sim}(3)$. ### Related - [[notes/Special Euclidean Group|SE(3)]]