## Interrupted sinusoidal projection in model making

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Interrupted sinusoidal projection could be used to create interrupted world maps. We follow a more practical approach and ask: How could we use this in model making?

To unwrap a sphere and project it's surface to a 2-dimensional plane in form of an interrupted sinusoidal projection, we take a look at Figure 1. We assume that the whole surface of the sphere is partitioned by meridians into

What we're after is the geodesic horizontal width

with

Figure 2 shows the plot of the inverse function

Since the values of

As stated above, Figure 2 shows the

### Practical use

A recently implemented requirement from the field of model making called for a cylindrical body with a hemispherical hood. This is a concrete application of the theory discussed here. This means that we can halve the value range of the angle:

In an ideal world, after modeling the cut material, the perfect result should look like Figure 5. However, in the real world, due to compression and stretching of the material during bending, the result is more likely to be as shown in Figure 6. A perfect result can only be achieved by exactly replicating the corresponding bending radii. In general, this is hardly feasible, so that more or less large gaps or overlaps arise.

### References

- E. D. Demaine, M. L. Demaine, J. Iacono, and S. Langerman.
*Wrapping spheres with flat paper*. Computational Geometry, 42(8):748–757, 2009.