A Double Figure 8

Recently, a local artist had an intriguing question. Suppose you have a hook in the ceiling (who hasn’t?), and  two spot lights in front of the hook, slightly to the left and to the right. Suppose also that you have drawn two curves on the back wall. Can you bend a wire and suspend it from the hook so that the two projections match the drawings?

 

Sketch

I first thought: Yes, this means we just have to determine the intersection of two cones, so this is possible but maybe tricky.

Figure8001

After playing around with it a little I realized that this is simpler than I thought: Of both curves have the same height, this is essentially always possible, and even completely explicit. In fact, this is almost as simple as using two perpendicular parallel projections.

For instance, below you see a single red wire that has two figure 8 curves as projections.

Figure8010

Then of course one wants to play with it and rotate the wire.

Figure8032

Clearly, there are two more rotational positions where one of the projections is again a figure 8, the one above and the one below.

Figure8329

Now we need to find somebody who can accurately bend wires for us.

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