Find diffeomorphism transforming the following areas:

Question

Find diffeomorphism transforming the following: interior of the triangle T with vertices in $(0,0),(0,1),(1,0)$ onto the interior of the circle of radius 1 and centre in $(0,0)$. Obviously i am looking for the function but how to find this function? What is the easiest method to do that?

Answer 1

Let's do this for the triangle with vertices $(-1,1),(0,0),(1,1)$ instead. If we can do it for this triangle, we can easily do it for the given triangle.

The idea is to make $(0,y)$ in this triangle correspond to a point on the diameter from $(0,-1)$ to $ (0,1)$ in the unit disc. (Time to draw a picture.) A natural way to do this is send $(0,y)$ to $(0,-1+2y).$ Now let's try to make the horizontal slice through $(0,y)$ in the triangle correspond to the horizontal slice through $(0,-1+2y)$ of the disc. (Note that the slice in the triangle is the set $(-y,y)\times \{y\}.$) You have a simple equation to solve. I am getting the map

$$(x,y) \to (2x(y-y^2)^{1/2}/y, -1 + 2y).$$