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 \(p\) equal pieces. This in turn also divides the angles around each pole into \(p\) equal parts of \(A = 2\pi/p\). The piece shown in Figure 1 is cut from pole to pole following the lines of two adjacent meridians.

What we're after is the geodesic horizontal width \(h\) of our piece in terms of the angle \(\varphi \in [0, \pi]\) which has its maximum at \(\varphi = \pi/2\) ("equator"). Without further proof, we could express this functional dependency as follows [1]:

\[h(\varphi, p) = \arctan\bigl(\sin\varphi \cdot \tan(\pi / p )\bigr)\]

with \(\pi / p = A/2\).

Figure 2 shows the plot of the inverse function \(h^{-1}\) which flips function \(h(\varphi, p)\) upright:

\begin{equation} h^{-1}(\varphi, p) = \arcsin \bigl(\cot(\pi/p) \cdot \tan(\varphi)\bigr) \label{eq:inverse} \end{equation}

Since the values of \(\varphi \) by definition cover \([0, \pi]\), equation \eqref{eq:inverse} plots it's values only for the lower right edge. The whole curve is covered by plotting \(\pm h^{-1}, \pi \pm h^{-1} \).

As stated above, Figure 2 shows the \(p\)-th part of the complete surface. So, to cover the entire surface, we need \(p\) pieces next to each other. Figure 3 shows an example with \(p = 8\) pieces and takes the spheres radius \(R\) into account.

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: \(\varphi \in [\pi/2, \pi]\); \(R\) also becomes the radius of the cylinder. Figure 4 shows the cutting pattern that we need to apply to our chosen material:

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

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