# Automate re-creating this shape that looks like a paper sphere template

This is the 1977 "Charles Ross: Light Placed" poster by Jacqueline Casey and it's insane.

I was wondering how would one re-create a similar shape in Illustrator relatively easily. Just to clarify: I'm talking specifically about the shape which appears like a template for making a paper sphere, and not the contents.

Here's what I tried:

• creating

1. a 100×100mm circle with the inner stroke of 20mm and
2. a 20×314mm rectangle (h=2×PI×r)

and then blending the two - didn't work

• image-tracing an existing template - too limited

• drawing the shape manually using pen tool - too time consuming and not precise enough; also not flexible
• creating a small and a big circles then blending the two - this may work only when knowing which circle sizes to use, I couldn't figure out how to calculate

I feel like there's an easy way which I'm missing. Please share any ideas you may have

A simple way is to minimize calculations and let Illustrator find how long parts are needed. The slices are parts of cone surfaces. You should understand it if you watch the following revolved polyline. It's a half of approximated sphere.

The orange stripe can be cut as an edge slice of a circle sector which forms the surface of a cone. Some calculations are needed. You must find the sector angle of the circle.

In the leftmost drawing you can find AC, BC and CD. AC can be got exactly by inserting a node to the crossing A and removing the extras with the direct selection tool.

AC was drawn originally by extending the polyline segment BC holding the shift key and dragging the corner towards A.

Document Info panel's subpanel "Objects" gives the following lengths:

AC=107,715mm BC=11,14mm and DC=41,22mm

The revolution path of point C is a circle with radius DC. That gives cone bottom perimeter = 258,99mm. (=multiplied by 2*pi) On the other hand that must be equal with the length of an arc which has radius = AC. The angle of the arc in radians =(258,99mm)/AC= 2,4044 radians = 137,76 degrees.

Direct formula for the sector angle in degrees is (DC/AC)*360. So here we have the orange stripe:

We have 2 circles. Radiuses: the red 107,715mm and the blue 11,14mm less (=BC). The sector angle is made by rotating a horizontal line to 137,76 degrees.

The rest of the needed stripes can be constructed in the same way. There's no reason why the stripes should have equal widths. It can be useful to make equator stripes and pole stripes 50% narrower and simplify them. The pole caps could be circle caps and the equator stripe could be one vertical cylinder instead of 2 opposite cones around the equator. Actually one gets a polygon just for it from Illustrator when he doubleclicks the polygon tool and inputs the values instead of dragging.

The example from 1970's seems to need a 36 gon as the starting point:

Actually you need the colored sides of it and the dashed square as a marker of the center. Red sides become curved stripes as the orange one was above. The green one is a round pole cap and the blue becomes a vertical cylinder (=a rectangle in a plane) symmetrically around the equator.

Automatization: Programming is needed. You may want to input the size of the sphere and and how many stripes it must have. I do not believe blending does the trick, altough I cannot prove that nobody will ever find good blending settings.

What to program: Trigonometric formulas for the stripe dimensions, drawing actual curves and the nesting (=placements for effective cutting). Programming and deriving the needed formulas are beyond the scope of this answer. But you can try this old-style Excel worksheet which contains your example case: https://www.dropbox.com/s/8dlssrkrdh7ebd9/spherestripes.xls?dl=0

It gives the dimensions of the needed stripes. Only the northern hemisphere and the equator are calculated, copy the stripes for the southern hemisphere.

Input the three starting parameters to yellow cells. Blue cells contain intermediate results.

If you happen to have a high end CAD program such as Inventor, Solidworks, SpaceClaim, Catia or other high cost stuff in the same league, you can revolve a N-gon to get a surface:

Only the northern hemisphere + the equator are revolved to keep the image simpler.

Because the surface stripes can be unfolded to plane, you at first split the surface to stripes. Assign some thickness to them and unfold them with the sheet metal functionality. Then you will have the stripes. Unfortunately I cannot show it in practice because I haven't such (\$5000,-) software, the preceding images were only screenshots from entry level freeware CAD.

But nothing prevents to test the paperwork unfolder Pepakura that user Rafael suggested. Upvote him, if you see this useful:

I saved the revolved shape as STL file and imported it to Pepakura demo:

The circle in the left is the bottm cap. Not bad, I would say. Pepakura did the conversion in few seconds. It accepts also OBJ files.

In illustrator one can change the stokes, remove the unwanted parts, fill areas with the shape builder, duplicate those stripes which are needed for the southern hemisphere and move things to artistically acceptable places and positions. Here's what I got in a minute:

Filling with Shape Builder is actually a must, because the SVG from Pepakura contains a high number of unconnected line segments. There cannot be curves because STL file itself contains only straight lines.

My approach would be modeling a 3D sphere using Blender and converting it into a 3D craftable paper model using this: https://tamasoft.co.jp/pepakura-en/order/index.html (No affiliation here)

A slower, but a free method, would be taking the initial 3D model, and extracting a base polygon of each row aligning it with the camera viewport. https://blender.stackexchange.com/questions/15045/how-do-i-align-the-viewport-to-a-face-normal

Another option would be manually marking the seams of the 3D model and unwrapping it. The process is a bit specific, so look at some tutorials online pls.

https://en.wikibooks.org/wiki/Blender_3D:_Noob_to_Pro/UV_Map_Basics

This is the result of partial work... Just a test.

But unwrapping can produce not exact shapes, that potentially do not assemble well on a real paper model, so use the previous method, face per face one.

But a more exact way would be taking the mathematical approach @user287001 uses.

It's definitely a Blend, maybe made by parts:

• From the blue circle to the yellow and from the yellow to the pink

• Reflect/duplicate the blend taking as an axis the black path

• Make a rectangle on top to use it as a mask

• With the Direct Selection Tool select the mask top/left point and the bottom/right and use the Live Corner Widget to round these corners

End