Math CentralQuandaries & Queries


Question from Leo:

I want to build a sundial where the shanow falls on an elliptical cylinder. I can calculate the coordinates of the points on the cylinder that I want to mark. My problem is that I will have to make the surface as a flat sheet and bend it into an elliptical shape. However, I cannot work out a scheme to transfer the coordinates I know into a distance that will work when I bend the shape.

Hi Leo!

Interesting project, interesting problem!

Distances along ellipses cannot be computed in terms of elementary functions: when you integrate the distance element you get something called an "elliptic integral". These functions are not found on standard calculators, though they can be approximated by power series, etc.

Also, results for one ellipse cannot be adapted to another ellipse with differing eccentricity (otherwise the circular case, which is easy, could be transformed to give all ellipses!) As a result, tables of elliptical arc lengths would have to be two-parameter tables.

Now, software and formulae are available. See:

But, to be honest, you would probably get an answer accurate enough for construction purposes most easily by cutting out a cardboard ellipse quadrant and measuring the edge with a flexible ruler (or rolling it carefully on a rigid one.)

Good Hunting!

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