Joy of Geometry and Conic Sections

DiveIntoMark is a continual source of fun anecdotes to show Sarah. He wins points for a good, sometimes self-deprecating, sense of humor, quotes from Winnie ther Pooh, and recent talk about babies. I like the added mix of zork, math and general geekiness. Today Mark has posted a story about the Cantor set, Sierpinski carpet, and Menger sponge. Sarah will not be amused, but I am. The ensuing discussion about various shapes of infinity has prompted me to publish the Vanishing Point animation I have been working on for the past several months.

When I was in fifth grade (age 10ish) there was a wooden cone at school sliced into three pieces that revealed three conic sections: circle, ellipse, and parabola. At the time I was curious to know what the relationship was between the slope of the intersecting plane relative to the axis of the cone and the resulting conic section. I didn't have the vocabulary to articulate it that way at the time, but that's what I wanted to know.

I have always loved geometry (fun and interesting) and tolerated algebra (boring). On the high-school geometry final exam (age 15ish), the instructor informed us that the final proof on the test required at least fourteen steps -- if you had less you had it wrong. I solved the proof in seven steps. I had finished the test about twenty minutes early and had plenty of time to double-check my work. I finally asked the instructor to go over my solution and he agreed that my solution was correct.

Fast forward to college. I enrolled in Calculus III as an elective even though it wasn't part of the architectural requirements. I was specificall looking forward to formally answering that question about cones and intersecting planes (finally). They were using the same text book as Calc II and the next couple chapters covered three-dimensional geometry.

One of the first things out of the instructor's mouth was "We're going to skip the chapters on three-dimensional geometry because it really isn't very interesting. Instead we'll jump straight into infinite series."

Bastard.

I tried to study those chapters on my own and couldn't quickly figure out how to apply the knowledge to answer my question. While I was trying, I managed to answer the question by inspection and gave up on the calculations.

There are two lines of interest: the axis and the edge of the cone. A circle is easy. The intersecting plane must be perpendicular to the axis of the cone. An ellipse is a generalization of a circle. The plane must intersect both edges of the cone. To create a parabola, the intersecting plane must be parallel to one of the edges. And for a hyperbola the plane must be somewhere in the range between the edges of the cone. I'm ignoring some edge cases (pun is intentional). Planes which intersect at the point of the cone generate a point. And Planes which intersect at the tangency generate lines. Perhaps that means that points and lines should be included in the family of conic sections. :-)

(yet another addition to my growing list of things to animate :)