Verb for transforming a sphere into a plane

I'm designing tools for a 3D modeling software. I've programmed two tools – one which turns a cylinder into a straight plane (and back), and another one which turns a sphere into a plane and vice versa. Now I need to know how to call those tools.

The act of transforming a cylinder into a plane can be called unwrapping or unrolling.

» Click to unwrap the cylinder «

But is there any verb to describe the transformation from a sphere into a plane?

» Click to (?) the sphere «

• Does the sphere turn into a circle or does it cut it kinda like what you would do to turn a globe into a map? – Hank Feb 14 '17 at 20:44
• @Hank – I actually convert the Cartesian coords to spherical and then map them back to X,Y,Z – so it becomes a sphere. – m93a Feb 14 '17 at 21:36
• Whatever – I mean it becomes a square. – m93a Feb 14 '17 at 21:43
• Sorry, but as far as I know, you are either dealing with Euclidean geometry or topology, but not both at the same time. – Lambie Feb 14 '17 at 22:19
• math.rice.edu/~pcmi/sphere/sphere.html Can't be done. See the diagram. A plane and a circle, can miss each other OR: First they can meet in a single point. In this case the plane is tangent to the sphere at the point of intersection. In the other case the sphere and the plane meet in a circle. – Lambie Feb 14 '17 at 22:26

In math, a surface that can be transformed into a plane is called a developable surface. Without stretching it, a sphere cannot be directly developed. Same for a saddle curve or a torus (donut shape) .

So, » Click to develop the sphere «

(Math.) a surface described by a moving right line, and such that consecutive positions of the generator intersect each other. Hence, the surface can be developed into a plane.

Image of a saddle curve Reversing the transformation? How about back-developing, globalizing (deliberate humor), rounding out, and orbating or sphering (neologisms)?

• Cool! Any ideas how the opposite could be called? (Transforming the plane back to the sphere.) But both 'undevelop' and 'round off' seems fine... – m93a Feb 14 '17 at 21:41
• Right. You can't make a silk purse out of a sow's ear. How can a three-dimensional sphere (topological figure) be transformed into a two-dimensional surface (Euclidean plane)? Or am I missing something? – Lambie Feb 14 '17 at 22:21
• @Lambie You are missing something. The surface of the sphere can also considered to be a manifold with only 2 dimensions – Cascabel Feb 14 '17 at 22:32
• Hmm. Well, then, the verb is continuously deform, is it not? You just pull it into a sphere since the two are topologically equivalent. Or am I wrong, based on what you said. – Lambie Feb 14 '17 at 22:50

Cartographers and mathematicians have a word for mapping a sphere onto a plane: projection.

Wikipedia's entry on Mercator projection has some useful diagrams and equations, including: 