Deformations can be interpreted as resulting from a force applied to the shape. Mathematically, a deformation is defined as a mapping from a shape x to a shape y by a transformation function , i.e., . Given a notion of size of deformations, the distance between two shapes can be defined as the size of the smallest deformation between these shapes.
Shape alterations can be seen as consequences of external forces acting upon them. Mathematically, a shape modification is described as a process that transforms one shape into another through a specific function , i.e., . The distance between two shapes can be understood by considering the minimum amount of change required to transform one into the other.