Now it's Apr 2023 and the question was posted in Dec 2015. But it's interesting, so here's what I can give.
The writer of the original question inserted this explanation as his own comment:
Imagine you have a large grapefruit in your hand along with a white
piece of paper. You want to wrap the grapefruit in one layer of white
paper without wrinkles. How would you do it? This is what I'm trying
to achieve with my artwork.
Later he revealed his frustration by writing this comment:
no luck on upwork, none seem to know how to do it :(
Seriously, nobody can know how to do it, because already in 1800's mathematicians proved it's impossible. No pieces cut from a plane can be bent exactly along a spherical surface so that
- there's no wrinkles
- there's no free space between the the paper and the spherical surface
- the paper is not stretched nor shrunken anywhere
Most of us probably see it without any proofs by intuition or at least after trying to do it in practice.
If we use some more flexible material than paper and make unlimited stretching and shrinking possible, it's no problem to cover even a whole sphere with a single piece of artwork which was originally a rectangular piece of plane. The next image shows a simulation of it in Illustrator:
The sphere is, of course, made with classic 3D effect Revolve. The effect offers a possibility to map art on the surface.
The amount of stretching or shrinking or both can be reduced substantially, if the artwork is divided onto several pieces of plane which are scaled differently already before placing them on the sphere. Even more helps if only a part of the sphere must be covered. Clever division and mapping the artwork already in a plane to several pieces can really make in practice possible to have the artwork on a sphere without too extreme material stretching nor shrinking.
Your pieces named "top, middle and bottom" maybe are selected in such way. But it's absolutely sure that they do never get bent exactly along a sphere with no stretching nor shrinking - no matter whatever new will be invented in the future. That's the major common property of math facts.
It's totally different thing if you allow some free space between the artwork and the sphere. Arcs which resemble your top, middle and bottom pieces can easily be bent to conical surfaces. By tiling few of them one can get quite close to the spherical form. An example:
This one needs 4 arcs, one rectangle in the middle and 2 circles as top and bottom caps. It's well possible to divide a rectangular image to slices and map them on the rectangle, 4 arcs and the pole circles so that the seams fit and the distortion is minimized. Of course, the distortion is fully zero only on the middle part, which is a bent rectangle. At the north and south pole the distortion is infinite - the top and bottom edges of the artwork are squeezed to singular points. But that's avoided if we leave the top and bottom circles uncovered.
There's no automatic way in Illustrator to find which pieces are needed and how to slice the artwork and map the slices on the parts of the surface. But it can be calculated with elementary geometry. No advanced math tricks are needed.
One thing should be clear: The edges of the surface pieces must fit against each other. Any two adjacent edges must be as long. Your arcs do not fulfill that requirement (I measured them in Illustrator), so your arcs together are not anything which resembles my previous grey 3D shape.
But that was not the original problem. In your question you have given a set of surface arcs. To join their edges some shrinking or stretching or both is needed. Maybe someone knows how and how much. You only asked how to split and map your beautiful artwork onto the arcs. You assumed Illustrator should have what's needed to do the job. We'll see later that it needs some work, but it's possible.
You can slice the artwork horizontally to separate rectangles and map each piece onto an arc. To do the job you can in theory use Illustrator's Envelope Distort > Make with Warp > Arc. But that's only a theory. It's not as easy as one may at first think, because arc warping makes the slices longer than they were. Much calculations are needed and they must be based on assumptions of how the arc warp work. Adobe has not published it. The amount of calculations is doubled by the fact that Illustrator's arc warp makes max. 180 degrees wide arcs, but you want arcs with wider opening angle.
So, forget the arc warp, do it in another way: Use a custom Art Brush. You bend with it an artwork slice along a circular arc. Not asked, but it works also in Inkscape, if the artwork is a single or combined vector path. In illustrator CC brushes can also be raster images or groups of objects.
I must admit that slicing complex vector artwork needs much work. Masking is useless, because the slices must not contain any invisible extras. Masking removes nothing, it makes only a part of an image invisible. Illustrator's Shape Builder is a valuable help in slicing. Inkscape doesn't have it, there you have only the Boolean Path operations such as cut and intersect.
Let's try a simple example. It's the grey approximated ball above made of rings and 2 circles. The next image shows its dimensions:
(A note for the Moderator: The images have no source links, but do not remove them. Each image in this answer and all content shown in them is my own.)
Each zone is 30 mm wide. It's actually a revolved 12-gon. The "artwork" is 90 mm wide and 70 mm high rectangle with some holes. That's done for fast drawing, easy slicing and to show that the result fits.
The 70 mm total height is divided to 30 mm + 30 mm + 10 mm. To keep the result easily verifiable only about 15% of the ball surface is covered. It doesn't affect the method.
The slices are placed on the ball in Illustrator to see the rendered result. Unfortunately there's no way to extract from Illustrator the surface arcs with the curved slices on them. Advanced 3D programs have that capability. The arcs with the curved slices must be calculated and drawn separately.
The flattened middle zone of the ball is a rectangle. Its length is the diameter multiplied by Pi. That's 351.736 mm. We simply draw a rectangle WxH = 351.736 x 30 millimetrs and align the bottom slice in the middle of it.
The 2nd artwork slice is trickier. It needs some geometric calculations to get the right curvature. The smallest part needs about the same amount of work, it only covers 10 mm of the full 30mm. All needed flat pieces which contain a part of the artwork are shown in the next image:
The straight parts in the right are used as "Art Brushes" to convert circular arcs to the surface arcs. How to get their outline dimensions (still without art) is explained in the next image and below
This shows the dimensions of one arc surface. A = the opening angle, Ra = the radius of the circle where the green spine arc was sliced from, Lav = the length of the spine arc and the length of the centerline of the ready surface arc. It's the average of the lengths of the inner and outer edges.
The rectangle (height = H, width = Lav) is applied as an art brush to the spine arc. In theory any width shorter than Lav could be used, because illustrator stretches the result to full length of the spine arc, but it's useful to make the rectangle exactly as long as needed. That's because you can place on the rectangle (+center and group) a slice of the artwork. We return to it soon.
In the left there's shown the dimensions of a bent surface ring and the elementary geometry formulas how to calculate Lav, A and Ra for the flat surface arc. The drawing is not in scale. The image of the circular arc surface is zoomed bigger to make the details more visible.
As said the art brush rectangle should be drawn to the full final width =Lav. That's because the artwork slice can be placed on it before dragging both together into the brush pane.
The slice must generally be scaled horizontally to narrower size to match the edge to the slice on the adjacent surface ring.
The rule: You know how many percents the image covers of the perimeter. It's in my grey-blue example 90 mm which means 25,587% of the perimeter in the biggest ring.
To make the seam to match the next curved slice must cover as many percents of the centerline length Lav of the surface arc. That's 77,942 mm, so the slice is squeezed horizontally to that width. The last slice is only 45 mm wide in the brush.
The calculations become reasonable fast if the formulas are held in a spreadsheet. But to be productive one really should use some professional grade 3D drawing or package design software.