An "equiangular surface"
as defined by K. Boyadzhiev
Reference: K. Boyadzhiev, Equiangular surfaces, self-similar surfaces, and the geometry of seashells, College Math. Journal (4) 38 (2007), 265 - 271.

Maple code:

restart; 1 

 

h := mu*(sqrt(a^2+1)*arctan(sqrt(a^2+1)*tan(psi)/sqrt(a^2-tan(psi)^2))-arctan(tan(psi)/sqrt(a^2-tan(psi)^2))); 1
h := mu*(sqrt(a^2+1)*arctan(sqrt(a^2+1)*tan(psi)/sqrt(a^2-tan(psi)^2))-arctan(tan(psi)/sqrt(a^2-tan(psi)^2))); 1 

mu*((a^2+1)^(1/2)*arctan((a^2+1)^(1/2)*tan(psi)/(a^2-tan(psi)^2)^(1/2))-arctan(tan(psi)/(a^2-tan(psi)^2)^(1/2))) 

> rho1 := R*exp(mu*theta+h); 1
 

(Typesetting:-mprintslash)([rho1 := R*exp(mu*theta+mu*((a^2+1)^(1/2)*arctan((a^2+1)^(1/2)*tan(psi)/(a^2-tan(psi)^2)^(1/2))-arctan(tan(psi)/(a^2-tan(psi)^2)^(1/2))))], [R*exp(mu*theta+mu*((a^2+1)^(1/2)... 

r := eval(rho1, {mu = .1, a = 2, R = 1, psi = 1/2*Pi-phi}); 1 

exp(.1*theta+.1*5^(1/2)*arctan(5^(1/2)*cot(phi)/(4-cot(phi)^2)^(1/2))-.1*arctan(cot(phi)/(4-cot(phi)^2)^(1/2))) 

> plot3d(r, theta = 0 .. 4*Pi, phi = 1/2*Pi-arctan(2) .. 1/2*Pi+arctan(2), coords = spherical, shading = zgrayscale, grid = [100, 25], scaling = constrained, orientation = [45, 67]); 1
plot3d(r, theta = 0 .. 4*Pi, phi = 1/2*Pi-arctan(2) .. 1/2*Pi+arctan(2), coords = spherical, shading = zgrayscale, grid = [100, 25], scaling = constrained, orientation = [45, 67]); 1
plot3d(r, theta = 0 .. 4*Pi, phi = 1/2*Pi-arctan(2) .. 1/2*Pi+arctan(2), coords = spherical, shading = zgrayscale, grid = [100, 25], scaling = constrained, orientation = [45, 67]); 1
 

Plot 

>
 

POV-Ray picture:

(exported from Maple)