Is there a straightforward way to map all my vertices from a cubic space to a cylindrical space?

Question

I have an existing model I've built up which currently has cubic geometry and fits neatly inside a cube. This was built up in Blender manually by duplication, not by scripting, in case you're curious about that.

I'd now like to map the vertices to fit the thing inside a cylinder.

In 2D, this turns out to be called a Schwarz-Christoffel mapping.

Is there a straightforward way to deform my existing model in this fashion without going back to the start and writing a script to generate the object? I think I could probably knock out a script to generate an OBJ file if I had a day to do it, but I feel like Blender might have an even faster option that lets me reuse my existing geometry.

I've been looking through all the "Deform" modifiers but nothing seems to jump out as immediately appropriate except for Cast, but I can't seem to get that one to do what I want either.

Edit:

For the next person hunting this out, the transformation I'm actually going with:

https://gist.github.com/trejkaz/222f687ef394f7430517868b254dc1b6

Answer 1

Squircle and shapekey

Subdivided cube run thru script to give squircle generated cylinder shapekey.

In How can I morph a flat plane to be a flat cirlce? I have shown one of a number of square to circle mappings.

The mapping maps XY coords to a radius 1 circle, result shown above using default cube. For radius not equal to one would have to scale and rescale accordingly, or make radius an argument of mapping.

Note: This is the squircle mapping, as demonstrated I did go thru and provide an example of all the mappings, if I find the file will add the conformal mapping. As explained in link in linked answer.

import bpy
import bmesh
from math import sqrt
from bpy import context
collection = context.collection
ob = context.object
me = ob.data
def squircle(x, y):
    u = x * sqrt(1 - y * y / 2)
    v = y * sqrt(1 - x * x / 2)
    return u, v


sk = ob.shape_key_add(name="Basis")
ci = ob.shape_key_add(name="Cylinder")    
for v in me.vertices:
    ci.data[v.index].co.xy = squircle(*v.co.xy)

Note: Changing space would be more akin to changing Cartesian x, y, z to cylindrical r, theta, h where r is the radius, theta the angle around the axis and h the height of the cylinder. This wouldn't change the look.

Casting to Cylinder via modifier

Could instead add a cast modifier

Result on subdivided default cube, radius set to sqrt(2)

Not sure the top is the result desired, to which would possibly require using a weighted vertex group for interior verts with cylinder axis aligned normals.

Answer 2

You can add a modifier Cast. By default it will deform your model into a sphere shape, but you can set the shape to a cylinder or cuboid.

You might want to plya with the axes and factor to get precisely Hat you want.

Answer 3

If you have the formula of a Schwarz-Christoffel mapping, you could write a simple script to do it, mapping vertex by vertex. But frankly, there are easier ways to do it, like using map projections. It is trivial to map a cube to sphere. Subdivide a cube several times and then apply Mesh > Transform > To Sphere. And most map projections are cylindrical projections.