Interpenetration: The Principle
≈ 42 minInterpenetration: The Principle
When two solids pass through one another, their surfaces meet along a curve of interpenetration. Finding that curve is a Grade 11 core skill and it underpins every development that follows.
The method is the generator method from Grade 10, applied twice.
- Divide one solid's circular or polygonal section into equal parts and number them — these are the generators.
- Project each generator into the other views.
- For each generator, find where it crosses the other solid's surface. That is one point on the curve.
- Join the points in numbered order with a smooth curve.
The curve is found point by point. There is no formula and no shortcut, and the number of points you take determines the accuracy — twelve is standard.
Worked example. Two equal cylinders meeting at 90° with axes intersecting: the curve of interpenetration is two straight lines crossing at 45° when viewed from the front. This is a special case — as soon as the diameters differ, the curve becomes genuinely curved. Learners who memorise the straight-line result apply it to unequal cylinders and get it badly wrong.
Core checkpoint: Number the generators, work one at a time, join in order. Never sketch the curve.
How is the curve of interpenetration between two solids found?
Under what conditions do two cylinders meeting at 90° give straight interpenetration lines?
How many generators are conventionally used when constructing a curve of interpenetration on a cylinder?
A learner applies the straight-line result to two cylinders of different diameters meeting at 90°. What is the outcome?