A transformation is how you move a shape from one place to another. For example rotations, translations and reflections are all ways of moving a shape.
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Dilation, rotation, reflection and translation (Go to www.mathwarehouse.com/transformations/) For more information
They are points whose positions do not change under transformations.
Laplace' is known for transformations in math; as in a Laplace Transformation. Transformations are used extensively in matrix models in general equilibrium theory and econometrics such as Dominate Diagonal transforms. That is where I reached my level of incompetency; fond memories. See: Lionel McKinsey, Economic Theory and Matrices with Dominate Diagonals
The properties depend on what the transformations are.
Transformations can translate, reflect, rotate and enlarge shapes on the Cartesian plane.