The class of Schoenberg transformations, embedding Euclidean distances into higher
dimensional Euclidean spaces, is presented, and derived from theorems on positive
definite and conditionally negative definite matrices. Original results on the arc
lengths, angles and curvature of the transformations are proposed, and visualized
on artificial data sets by classical multidimensional scaling. A simple distance-based
discriminant algorithm illustrates the theory, intimately connected to the Gaussian
kernels of Machine Learning.