Function a *z^b + b*z

a square z plus2z 001
map: z — > f(z) = a*z^2 + 2*z, domain: 0.7 < |z| < 1.3, morph: 0.5 < a < 1.5.
Observe that the slightly curved parameter squares in the domain are mapped to “curved squares” in the range. All angle preserving (=conformal) maps have this property. When a parameter circle of the domain hits the point z0 where f'(z0) = 0 then the image of this circle is a cardioid. The other parameter circles are mapped to limacons.
a cube z plus3z 001
map: z ⟶ f(z) = a*z^3 + 3*z
domain: 0.7 ≺ |z| ≺ 1.3
morph: 0.25 ≺ a ≺ 1.25
As for all conformal maps we see: The “curved parameter squares” in the domain are mapped to “curved squares” in the range. Near the zeros of f’ the image squares are very small. The parameter circles which hit the zeros of f' are mapped to Nephroids

square2 pre square2

function: z → a *z^b + b*z

Preimage: polar grid.
Domain: 0.7 ≤ |z| ≤ 1.3 and 0 ≤ arg(z) ≤ π

Stereographic Projection onto a sphere.

z_z_plus_1_squared.pdf