Tigric coordinates are actually the same as the parametric equations for a tiger. There is a focal duocylinder with radii a and b.

x = (a + rho Cos[gamma]) Cos[theta];

y = (a + rho Cos[gamma]) Sin[theta];

z = (b + rho Sin[gamma]) Cos[phi];

w = (b + rho Sin[gamma]) Sin[phi]

The isosurfaces are tricky, I thought I knew them but I had them wrong. Isosurfaces for rho are obviously tigers:

(Sqrt[x^2 + y^2] - a)^2 + (Sqrt[z^2 + w^2] - b)^2 == rho^2

Isosurfaces for gamma are weird cone shapes. We have

(sqrt(x^2 + y^2) - a)/cos(gamma) = rho = (sqrt(z^2 + w^2) - b)/sin(gamma)

From this you can show that it's a surface of the form

tan(gamma) sqrt(x^2 + y^2) + sqrt(z^2 + w^2) = a tan(gamma) - b.

Do we have a name for this? It's clearly related to cones.

Surfaces of constant theta and phi are half-hyperplanes. Unlike the other coordinate systems, they don't start at the axis, they start at 'a' or 'b' units away from the axis respectively. For example, if we fix phi = 0, then x and y can be anything, but z > b and w = 0.