The following is a solution to Puzzle #13 – Fibre Choptic. Take a look if you haven’t already.
All that’s needed is to consider a single fibre a transverse distance from the point at which the fibres are cut.
Suppose the point is a distance above the base of the fibres. Then the length to which the fibre at is cut is given by
from Pythagoras’ theorem.
After the snipped fibre is straightened out, the height of the fibre above its base, , is equal to its length . Hence
This is the Cartesian equation for a hyperbola.
This gives an interesting geometric way of interpreting a hyperbola. Every point on a hyperbola can be rotated about its ‘shadow’ on the axis, and be made to pass through a common point at the hyperbola’s vertex.
Conversely, you can construct a hyperbola by rotating the vertex point about a series of points on the axis.