In my research, I came to a transformation problem. The simple version is an initial circle (or sphere) region is advected by some deformational flow. After some time the circle will be deformed into other shapes.
At the beginning, I used linear transformation (rotation, shearing, translating), but I found this is not enough when the flow is extremely deformative. The circle is stretched into long ellipse due the linear nature of the transformation, which should already be bent.
So I decide to try high-order polynomial transformation as shown in the figure. I am not very familiar with polynomial transformation, could it solve this problem? In addition, I also need an inverse transformation, but the high-order polynomial will add some difficulties.
Any input is appreciated!
Update:
I decided to use conformal transformation which should be easier to solve, especially when inverse the transformation, as suggested by @Shuchang. The new diagram is shown as:
If we know the coordinates of the control points and the rotation matrixes on them, how to define the transformation function $f(\mathbf{x}) = \mathbf{y}(\mathbf{x})$?