CTK Insights

24 Jan

Finding a Parallelogram in 3D

Imagine a pyramid with no symmetries or regularities whatsoever. To construct a pyramid like that, pick a plane, four arbitrary points in the plane and one point outside. The lines (or rays) joining the latter to the four points in the plane serve as the edges of a slanted and likely irregular pyramid. However, the following exercise shows some regularity is still present regardless of how irregular the pyramid may appear.

There are a couple of ways to formulate the problem I have in mind.

  1. For a random pyramid built as described above, with the four coplanar points forming a convex quadrilateral, there is always a plane that cuts the pyramid in a parallelogram.

  2. Consider a class of pyramids constructed by choosing four coplanar points and joining an extra fifth point not in the same plane to the chosen four. The second class is constructed similarly with a restriction that the four coplanar points form a parallelogram. The fact is that the two classes of pyramids coincide.

Parallelogram is a convex quadrilateral with the opposite side lines parallel. Among several characterizations of parallelogram there is a couple that will help solve the problem:

  1. Parallelogram is a convex quadrilateral with a pair of equal and parallel sides.

  2. Parallelogram is a quadrilateral in which the diagonals bisect each other.

The three solutions below are each based either on the definition or one of the characterizations of parallelogram.

First lets look into how to get a plane cut parallel lines on two opposite faces of a pyramid. Start with just two faces, i.e., two planes that meet in a line:

A plane that cuts parallel lines on the two faces is necessarily parallel to their line of intersection (otherwise, this is where the two lines would meet). Chose one such plane and rotate it around one of the parallel segments on one of the faces of the pyramid; the segment on the opposite face - while remaining parallel to itself - will change in length from zero (near the apex of the pyramid) to infinity. The change is continuous, so that at some intermediate position the plane will cut to equal segments.

Note that there is a simple planar construction of the segment inscribed into an angle equal and parallel to the given one.

Now it is possible to have a solution based on the definition of parallelogram as a convex quadrilateral with two pairs of parallel sides. The opposite faces of a pyramid meet at two lines through the apex of the pyramid:

This two lines determine a plane such that a plane parallel to the latter cuts parallel lines on both pairs of the opposite faces of the pyramid.

The third solution to the problem is based on the fact that the diagonals in a parallelogram bisect each other. The opposite edges of the pyramid define two planes that meet in a line through the apex of the pyramid:

Focus now on each of the two planes. In each, two edges of the pyramid define an angle with a third ray in-between. The task is to construct a line segment with the end points on the sides of the angle which is divided in half by the middle ray. This is a nice planar problem that admits a simple and instructive solution. Solving two planar problems - one for each of the planes - produces to segments with the end points on the opposite edges of the pyramid that bisect each other and, therefore, define a parallelogram.

In addition to the line-wise properties, parallelogram can be characterized as a convex quadrilateral with the pairs of equal opposite angles. The three forgoing constructions show then how to find a plane that cuts equal angles on two dihedral angles - 3D angles formed by two intersecting planes - that share an apex. I have a difficulty finding a construction of the plane based on this property of parallelogram

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