Area preserving continuous deformation of an arbitrary ellipse centered at origin into a circle of the same area centered at origin (graph included)

Question

I want to know if my derivation is correct

Model the ellipse by

$$f\left(t\right)=\left(A\cos t,B\sin t\right)$$

Model the resultant circle by

$$g\left(t\right)=\left(\sqrt{AB}\cos t,\sqrt{AB}\sin t\right)$$

Then in order to ensure that the area is constant throughout the transformation, model the transformation by

$$H\left(t\right)=\frac{\sqrt{AB}}{\sqrt{\left(A\left(1-T\right)+T\sqrt{AB}\right)\left(B\left(1-T\right)+T\sqrt{AB}\right)}}h\left(t\right)$$ such that $$h\left(t\right)=\left(\left(A\left(1-T\right)+T\sqrt{AB}\right)\cos t,\left(B\left(1-T\right)+T\sqrt{AB}\right)\sin t\right)$$

where $T$ goes from zero to 1, corresponding to the starting point of the transformation (arbitrary ellipse) and the ending point (circle of same area)

Here I have a graph modeling the transformation

Would this transformation be an example of Ricci flow?