How to create quad-sphere with evenly spaced edges?

Question

On the left is a quad sphere which is the result of subdividing a cube, and using Alt+Shift+S to turn it into a sphere. On the right is my desired quad sphere, where I have painstakingly evened out the edges, so that all edges in the same edge loop are of the same length. What is the best/easiest/fastest way to create this kind of quad sphere?

My current method, which is rather tedious, looks like this:

I take one quadrant of one side of the original cube:

Then I alternate selecting all of the vertical and horizontal edge loops, and Looptools > Space them:

Finally, I duplicate and rotate this section of the sphere as necessary:

Is there a significantly more efficient way to achieve the same result?

This is a demonstration of the type of quadsphere I am attempting to achieve:

Answer 1

The Extra Objects addon can create a quadsphere with even edges
Just enable it in Preferences > Addons

Answer 2

A Geometry nodes modifier will produce an evenly spaced quad sphere and allow you to set up a template with your own options and input any quad count. Constructed with arcs, the spacing is perfect for the lines going one direction but not the other. The step labeled "Make Symmetrical" is only necessary if you want 90° symmetry using a compromise best fit for the spacing running both directions.

You can download this .blend file and use Append to add a Quadsphere or the Modifier itself into your file. Append the Modifier itself from the Node Tree folder, or an example quadsphere from the object folder. Change the number of steps in the geometry nodes modifier.

Answer 3

I ended up here after realizing a cylinder extruded from the cross-section of a quad sphere made with the Extra Objects addon does not have equal sides. I was dismayed. DISMAYED.

Here is a comparison of it to a normal circle (orange):

The ideal you are asking for appears to be possible. While I have no formal proof in favor or against, your samples have a lot going for them, and can be approximated with code.

Looking at add_mesh_round_cube.py of the Extra Objects addon, one can see that its idea of a quad sphere is a specific parameterization of a more general mathematical formalism. Apparently not one designed to get equal edge lengths.

To find the math which would do the job, I studied your sample quad spheres for a bit. One thing I thought is noteworthy is that the ring segments on each of the 6 "sides" each appear to have their vertices within a plane, just as other quad spheres do. (There is a bit of inaccuracy in the samples, which I am not sure whether it is safe to ignore.) From there, we can construct the three axis-aligned circles as well as the twelve 45°-angled circle segments, using some known math to determine the angle between (1, 1, -1) and (1, 1, 1) as well as rotate vertices around an axis.

To get the positions of the remaining vertices, it is important to notice that the circle segments between the axis-aligned circle segments and the 45°-angled circle segments have a smaller radius than 1. This is because the plane in which they lie demonstrably does not intersect the center of the quad sphere. While there is nothing unusual about this (given other quad spheres), we need to know the exact radii to figure out the angle spanned by each circle segment. To do this we can use two opposing points p0 and p1 of two 45°-angled circle segments of one "side" of the quad sphere as well as the center point halfway in between pCenter. This gives us a vector with the correct direction of the center of the circle segment's arc as w = pCenter - 0.5 * (p0 + p1). This can then be used to calculate the axis-aligned slant angle of the circle segment and determine a rotation axis rotAxis perpendicular to the circle segment's plane. Then math:

circleSegmentDist = pCenter.dot(rotAxis)
circleSegmentRadius = sqrt(1.0 - circleSegmentDist * circleSegmentDist) # equal to sin(acos(abs(circleSegmentDist)))
arcAngle = 2 * w.angle(p0 - (pCenter - ringSegmentRadius * w.normalized()))

The rest is a loop which finds the vertices by rotating p0 around rotAxis using angles between 0.0 and arcAngle.

This gives us this result (which matches your samples quite well):

However, by rotating a copy of the sphere around its center by 90°, we can show that there still is some error, revealing the math turns out slightly differently depending on whether we rotate vertices in x- or in y-direction within a "side":

I am not sure right now where the problem comes from. I suspect the assumption about the circle segment vertices lying within a plane might be wrong. I believe the error is too large to be caused by floating point arithmetic.

Either way, you can get it on Github: https://github.com/Zyl9393/even_quad_sphere