# Polynomial Example

• function: `z + c*z^4/4`
• domain: conformal polar grid, `0.8 ≦ |z| ≦ 1.25`, `0 ≦ angle ≦ 2*pi`
• morph : `0.3 ≦ c ≦ 1.0`

Note that the circles `t → exp(i*t)` in the domain are mapped to `exp(i*t) + c*exp(4i*t)` in the range. We recognize them as cycloidal curves. (that is, curve traced by rolling a circle on another circle.)

Compare the inwards pointing cusps of the examples on this page with the outward pointing cusp of A Example of a Non-Conformal Mapping

In this case we have three zeros of the derivative. But the behavior of the map near these critical points is the same as in the earlier cases (near the zeros of their derivative). Observe an important fact of complex analysis: No interior point of the domain is mapped to the boundary of the image. In other words: The boundary of the image set consists only of images of boundary points. No fold lines as in the non-conformal example can occur.

• function: `z + c*z^5/5`
• domain: conformal polar grid, `0.8 ≦ |z| ≦ 1.25`, `0 ≦ angle ≦ 2*pi`
• morph : `0.3 ≦ c ≦ 1.0`

Observe for all conformal maps that the parameter quadrilaterals in the range near f(z) are much smaller than the parameter quadrilaterals in the domain near z where the derivative f'(z) vanishes.

z_z_plus_1_squared.pdf