Some pictures illustrating parametric transformations

A polar region

Resizing a polar region

From Guichard's Calculus Section 15.7

The $\theta r$ coordinate system and the $xy$ coordinate system are related by $x=r\cos\theta$ and $y=r\sin\theta$. The small rectangle in the $\theta r$ plane corresponds under the mapping to the four-sided figure in the $xy$ plane. The large rectangle in the $\theta r$ plane is $[0,\pi]\times[0,1]$ and corresponds to the half-circle in the $xy$ plane.

Drag the blue dot in the $\theta r$ plane to move the rectangle; drag the green dot to resize the rectangle.

A more general transormation

The image of a the unit square under $T(u,v) = \langle u+v^2, u^2-v \rangle$ created with math3d.org.

Extending that to 3D

The previous image can be extended to 3D using a transformation like $T(u,v) = \langle u+v^2, u^2-v, (u+v^2)^2 + (u^2-v)^2 \rangle$ - again, created with math3d.org.