We want to manufacture a cut-off slanted cone from a flat sheet of metal. If the cone were a normal right cone we know that we would simply cut out a sector of a circle and roll it up. However the cone is slanted. We want to know what the flattened shape looks like so that we can cut it out and roll it up to closely approximate correct final shape. We also want to minimize the amount of wasted metal after the shape is cut out.
The problem, and it generalizations may be solved analytically but the analytical solution is given in terms of indefinite integrals which rarely can be evaluated in closed form. The solutions may be found numerically which are good enough the create a picture of the flattened-out cone.
Broughton, Sean A., "Flattening a Cone" (2009). Mathematical Sciences Technical Reports (MSTR). 16.