Interpenetration of Prisms

Interpenetration of Prisms

When two prisms interpenetrate, the result is not a smooth curve but a series of straight lines meeting at corners — because prisms have flat faces, and two flat faces always meet in a straight line.

The method is simpler than for cylinders. A prism has a finite number of edges, so instead of twelve arbitrary generators you use the actual edges of each prism:

1. Number the edges of both prisms.
2. Find where each edge of the first prism pierces a face of the second — that is a corner of the intersection.
3. Find where each edge of the second pierces a face of the first.
4. Join the points in order, with straight lines.

The number of points is finite and known. A square prism through a square prism gives eight points, not an infinite curve. Every point matters — miss one and the outline is visibly wrong, since there is no smooth curve to hide the error.

Worked example. A square prism entering a larger square prism squarely gives a rectangular intersection. Rotate the smaller prism 45° about its axis and the intersection becomes a hexagon — same solids, different orientation, completely different result.

Core checkpoint: Use the real edges, not arbitrary generators. Join with straight lines, never a curve.