How to use polynomial or conformal transformation

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 $f(\mathbf{x}) = \mathbf{y}(\mathbf{x})$?

• There are many nonlinear transformation to make a deformation and polynomial transformation, in my opinion, may not be a good choice. Could you specify what's the intention behind this? – Shuchang Dec 28 '13 at 8:39
• @Shuchang I am developing a Lagrangian numerical transport scheme, which uses many particles to discretise the continuous tracer. In deformative flow, the shape that the particle presents will be changed. This shape is used when interpolating the tracer mass carried by particles onto other spots (like a mesh). – Li Dong Dec 28 '13 at 8:43
• I'm not sure but suggest conformal transformation. Essential you are aligning two curves. – Shuchang Dec 28 '13 at 8:49
• Is it mandatory to use polynomial transformations? If not, you could represent the initial shape using spline curves and transform it be moving the parameters or control points. -- Second thought: You would gain some flexibility at moderate degrees by considering rational functions instead of polynomials. – Lutz Lehmann Dec 28 '13 at 15:43
• @LutzL I used to represent the shape by polygon, but it was a nightmare when the flow is extremely deformative. I would like to use transformation because it provide a convenient way to calculate the transformed coordinate. – Li Dong Dec 29 '13 at 5:48