Complex Elliptical Map

z plus inverse z 001
map: z ⟶ f(z) = z + 1/z, domain: a ≺ |z| ≺ 4, morph: 1 ≺ a ≺ 2.
The radial parameter lines r ⟶ r*exp(i*t) , t = constant, are mapped to Hyperbolas r ⟶ r*exp(i*t) + 1/r*exp(-i*t) = (r+1/r)*2cos(t) + i*(r-1/r)*2sin(t)
The angular parameter lines t ⟶ r*exp(i*t), r = constant, are mapped to Ellipses t ⟶ r*exp(i*t) + 1/r*exp(-i*t) = 2(r+1/r)*cos(t) + i*2(r-1/r)*sin(t)

All these ellipses and hyperbolas have the same focal points, therefore they are called “confocal”.

[see Hyperbola]

[see Ellipse]

ellip rs
Stereographic Projection onto a sphere.

z_to_z_plus_1_over_z.pdf