Using blender, in File->user prefernces->Add-ons->Add Mesh->Add Mesh: Extra Objects
While Drawing Add->Mesh->Math Fuction->XYZ Math Surface
Flat circular rack for hyperboloid assuming a cone with base radius and upper radius same. Hyperboloid lower radius 3, height 4, max radius 5
Cone with base 5 and height 10 happens
Circular rack inner radius 3xroot5, outer radius 5*root5 side length of cone.
hyperboloid touches both inner and outer radius the 4*root2 generating edge
The cone of flat circular rack has
x=a*cos((6.28*u)-f)
y=a*sin((6.28*u)-f)
z=h*v or 0
u=0..1 24
v=0..1 24
a=sqrt( ((6.7082+b)*(6.7082+b))+(c*c))
b=3.577*v
c=4.38178*v
f=atan(c/(6.7082+b))
g=0
h=4
Hyperboloid as usual inner cylinder 3, height 4 and generater line 45 degree tilt Assuming the cone of base radius 5, middle radius 3 the height is 10 The Outer edge length of cone R=5*root5 the outer radius of flat circular rack The outer edge length for radius 3 and above is r=3*root5 The generator line of hyperboloid with height 4 and angle 45 is l=4*root2 R=r+a+b where a and b are result of perpendicular to radius
c*c =( R*R )- ((r+a)*(r+a))
solving it (R*R) - (l*l) -(r*r) = 2*r*a
a=3.5777
b=0.8951
c=4.38178
theta=50.77403 tilt of hyperboloid on flat circular rack
alpha = 23.0740
assuiming this as a cone type of fugure....