Confocal Ellipse Map (original) (raw)

map: z ⟶ f(z) = z + 1/z, domain: a ≺ |z| ≺ 4, morph: 1 ≺ a ≺ 2.
The radial parameter linesr ⟶ r*exp(i*t), t = constant, are mapped to Hyperbolasr ⟶ r*exp(i*t) + 1/r*exp(-i*t) = (r+1/r)*2cos(t) + i*(r-1/r)*2sin(t)
The angular parameter linest ⟶ r*exp(i*t), r = constant, are mapped to Ellipsest ⟶ 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”.