Legendre Transformation explained (with Animation)

A legendre transformation is a way of transforming a function of some variable into another function of another variable without losing any information. It is used for thermodynamic potentials or in theoretical mechanics when deriving the Hamilton function from the Lagrange function. In this video I show a geometric interpretation for the legendre transformation via a helpful little animation before deriving the legendre transform of a differentiable function from the definition.
My explanation is mainly intended for people who have at least some basic knowledge of single and multi variable calculus and need a quick explanation and derivation of the legendre transformation. For that reason I am not overly exact and thorough in the mathematical details. Consider my explanation a Physicist's approach to a mathematical topic.