Diffeomorphism $\varphi:(0,1)^2\rightarrow U$. $U$ is a parallelogram

Question

Let $U\subseteq\mathbb R^2$ be a open parallelogram with vertices $(2,3), (6,4), (8,6), (4,5)$. Find a diffeomorphism $\varphi:(0,1)\times (0,1)\rightarrow U$.

How can I find such an Diffeomorphism? I tried to do it with polar coordinates but I think it is wrong.

What did you try, exactly? Please share in an [edit]. – Shaun Jun 01 '22 at 19:18 — Shaun, Jun 01 '22 at 19:18

score 1 · Accepted Answer · answered Jun 01 '22 at 18:52

1

Hint: look at the coefficients in a linear combination of the vectors spanning the parallelogram.

answered Jun 01 '22 at 18:52

PinkyWay

4,565

1

Thanks. I got $\varphi(x,y)=(4,1)^Tx+(2,2)^Ty+(2,3)$. Is this correct? – Moritz Jun 02 '22 at 09:07
Moritz, yep. (: – PinkyWay Jun 02 '22 at 13:10

Diffeomorphism $\varphi:(0,1)^2\rightarrow U$. $U$ is a parallelogram

1 Answers1