Generate 2D Math Objects by Collapsing 3D Math Objects

I had some excellent high school mathematics instructors. They both loved their field and took an interest in their students. Since high school, I have had a deep interest in collapsing 3D mathematical equations to derive equations for 2D mathematical objects or modifying 2D objects into other 2D, or 2D objects into 1D.

A 3D sphere becomes a 2D circle. A 2D parabola becomes a 1D line. The 2D hyperbola shown, if collapsed along the x-axis, becomes two 1D line segments stretching at one end to infinity. The same hyperbola collapsed along the y-axis becomes a complete line. A 2D circle becomes a single 1D line segment of a length equal to the diameter.

An Example of Collapsing 3D into 2D

What can be obtained by collapsing 3D math objects of a more complex sort? Let’s consider collapsing a simple cone aligned along the y-axis and centered at the origin, described by the equation, x² + z² = ky². Such a cone can be collapsed because all the terms containing one variable can be isolated from all terms that contain the other variables.

If x = 0, for instance, the equation becomes z² = ky². This graphs as two intersecting lines. That makes sense, since if you could cast a shadow of a solid 3D cone, you would basically see two open-ended triangles. However those triangles are actually hollow, so you could really only expect to see the outline.

If, instead, z = 0, then x² = ky². The same situation applies here as in the first case, except the lines are facing in directions perpendicular to the others.

If y = 0 then z² = ‒x². This is only one point, since the square root of a negative number in the real world is imaginary (see however, this article on imaginary numbers. That point coincides with the origin.

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