Representation of a Triangle

Question

In this document, a Triangle is represented as: $$ T(s,t) = B + sE_0 + tE_1\\for~all~(s,t)\in D=\{(s,t):s\in[0,1], t\in[0,1],s+t\le1\} $$ Can someone explain this representation of a Triangle?

Answer 1

In that representation, the triangle is the image of $T$ (i.e. $T(D)$). Try thinking what points would satisfy this equation.

$B$ here is a location parameter, which (assuming the constants belong to $\mathbb{R}^2$, correct me if I'm wrong) corresponds to the node of the triangle closer to $0$ (if $E_{0}$ and $E_{1}$ are positive).

Then $B + xE_0$ and $B + xE_1$ maps (as we vary $x$) to each of its connecting edges of the triangle, and finally $B + sE_0 + tE_1\ |\ s + t =1$ maps to the opposing edge. Making $s, t$ and $s + t < 1$ we get the inside of the triangle.