Let
- $\hat K\subseteq\mathbb R^2$ be the triangle spanned by $\hat v_1:=(0,0)$, $\hat v_2:=(1,0)$ and $\hat v_3:=(0,1)$
- $K\subseteq\mathbb R^2$ be a nondegenerate triangle spanned by $v_1,v_2$ and $v_3$
- $T:\hat K\to K$ with $T(\hat K)=K$
Please consider the ordering of vertices on $\partial\hat K$ depicted in the following figure:
Which assumptions on $T$ do I need to impose in order to ensure that the ordering is preserved in the transformation of $\hat K$ into $K$?
For example, in the figure above, $T_m$ preserves the ordering while $T_m'$ does not.
Clearly, we want that $T$ is of the form $$T(x)=Ax+b\;\;\;\text{for all }x\in\hat K\tag1$$ for some invertible $A\in\mathbb R^{2\times 2}$ and $b\in\mathbb R^2$. Since we need to have $$T(\hat v_i)=v_i\;\;\;\text{for all }i\in\left\{1,2,3\right\}\tag2\;,$$ we obtain $$b=v_1\tag3$$ and $$A=\underbrace{\left(v_2-v_1,v_3-v_1\right)}_{\text{columns}}\;.\tag4$$
So, $T$ is already uniquely determined by the vertices. So, I guess my question shouldn't be how we need to restrict $T$. The question should be how we need to restrict the numbering of further nodes.
If you want to keep straight lines straight, then your map is affine and preserves the order of points at the line.