## …what a metric is?

A metric is a function $d: X \times X \rightarrow \mathbf R$ such that

- $d(x,x) = 0$ (reflexivity)
- $d(x,y) \ge 0$ (non-negativity)
- $d(x,y) = d(y,x)$ (symmetry)
- $d(x,z) \le d(x,y) + d(y,z)$ (triangle inequality)

Prominent examples are the Euclidean $d(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2}$ and the Manhattan $d(x,y) = \sum_{i=1}^n |x_i-y_i|$ distance.

For completeness: A *norm* is a metric that also fulfills the parallelogram identity. This is always the case when the metric/norm is induced by a dot product, as for example with the Eucledian metric: $d_{Euclid}(x,y)=\sqrt{<x-y,x-y>}$. The Manhattan metric is also a norm, but it is not induced by a dot product.

## …Bezier simplices?

It's an unified theory that can be found in Prautzsch 10.1 & 10.2, basically describing d-variate Bezier constructs. The good thing is, if you know how it works, Bezier Curves and Patches both are quite simple since they are only special cases of this theory. This way, there are a lot of things you only need to learn once instead of twice.