Interpenetration of a Cone and a Cylinder

Interpenetration of a Cone and a Cylinder

A cylinder passing through a cone is the most demanding intersection in the syllabus, because the two surfaces curve in different ways: the cylinder's radius is constant, the cone's changes with height.

The cutting-plane method is the reliable approach. Instead of using either solid's generators, introduce a series of horizontal cutting planes. Each plane cuts the cone in a circle and the cylinder in a rectangle — two shapes that are easy to draw in the plan. Where that circle and rectangle cross in the plan are points on the intersection, which are then projected back up into the elevation.

Why this works better here. Both solids are cut by the same plane, so the two sections are directly comparable in the same view. Using the cone's generators alone would require finding where each meets a curved cylinder — a harder problem at every step.

Choose plane positions deliberately. Add planes where the curve turns fastest, and at the highest and lowest points of the intersection. Evenly spaced planes miss those.

Worked example. A horizontal plane partway up gives a circle of the cone's radius at that height, crossing the cylinder's rectangle at two points — one intersection point on each side.

Core checkpoint: Cut both solids with the same plane, find the crossings in the plan, project back. Repeat.