Generate 2D Math Objects by Collapsing 3D Math Objects

collapsing 3dI 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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