Cones, Planes, and Automobiles…err…Spheres

Ohh, something new? Well, seeing as I was out of town with essentially nothing to do, I do what I always do: math.

I was looking at an image posted earlier by KC LC with shadows cast by a sphere sitting on a plane. Something stuck out at me which I guess would only stick out if you’ve been staring at conics as long as I have (keep in mind, I’m in aerospace and conics show up big time in orbital mechanics). I noticed that the point of contact between the sphere and plane was pretty damn close to the focus of each of the conic section shadows (I explained the reason for them being conics in a previous comment). While it was easy to demonstrate that the shadows themselves were indeed conics, it’s not nearly as obvious that the point where the sphere touches the plane is a focus of the resulting conic.

For those of you that don’t know, the great thing about stuff that isn’t obvious is that it’s often remarkable. My limited human reasoning can’t see any particular reason why the point of contact should be a focus.

Now, the way I ended up proving this was probably not the most efficient way, but it was the most general (and probably the most prone to error; surprisingly I didn’t make any, though). I took an algebraic approach to solving this, but on further reflection I can see a geometric proof which works for everything except parabolas.

However, in explaining both proofs, I’m not sure which would take less effort on my part. The algebraic method is definitely more tedious for the readers (i.e. you) and it’s almost as bad for me since I have to render all the equations. The geometric method will be a lot less tedious for you but it will require me to render several images in order to describe what’s going on. I’m not sure which would be quicker. The geometric proof would take quite a few images because it’s very difficult to describe in words. However, I think it gives a lot more insight into the reason that all of this happens. The problem with algebraic approaches is that algebra seems to do a good job of making things look like mathematical accidents (although this can make results seem more remarkable). I feel I’d much rather give a deeper mathematical insight with a simpler explanation than an accident that looks like Sir Isaac Newton ate a physics book and threw up. However, the advantage of an algebraic proof is that it’s a little more concrete and provides some interesting intermediate results (one of which is a potentially useful formula). The geometric approach cannot provide formulae like the algebraic approach can.

So I’ll put this up to a poll to the handful of readers I have which method they want to see. I’m already pretty sure I know what the answer is. To help you with your decision, I will provide you with a little example of what each proof will involve.

Sphere, cone, plane, and a conic section.

Sphere, cone, plane, and a conic section.

{\frac{\left(\cot^2\phi - \tan^2\theta\right)^2\sin^2\phi}{d^2\tan^2\theta \cot^2\phi}\left[x - \frac{d\tan^2\theta}{\sin\phi\left(\cot^2\phi - \tan^2\theta\right)}\right]^2 + \frac{\cot^2\phi - \tan^2\theta}{d^2\tan^2\theta \cot^2\phi}y^2=1}

6 Responses to “Cones, Planes, and Automobiles…err…Spheres”

  1. xot says:

    I was reading this, and thought, “Hmmmmm, this seems very familiar”. Then I remembered A.K.Dewdney’s “The Tinkertoy Computer”. In chapter 13 he tells several stories originating from Ross Honsberger’s University or Waterloo mathematics lectures. I’ll recount one of them here:

    “Ross Honsberger has spent more than two decades collecting mathematical morsels for general computation. Attending one of his rare public lectures, I had the pleasure of sampling one of his mathematical feasts. I found it not just palatable but downright delicious.

    “I listened to Honsberger’s lecture near my hometown in Ontario. He served to the audience such delights as spheres in a cone, checkers on a board, dots on a dish and beans in a Greek urn.

    “Honsberger began by describing the marvelous spheres of Germinal Dandelin, a 19th-century Belgian mathematician. Dandelin discovered an amazing connection between the classical and modern concepts of the ellipse. The Greeks conceived of an ellipse as the figure that results when a plane cuts obliquely through a cone. Since the time of Descartes, however, the ellipse has been described analytically in terms of two special points called foci. The sum of the distances from the two foci to any point on the ellipse is constant.

    “Honsberger introduced Dandelin’s spheres by drawing our attention to a projected transparency of a plane cutting a cone. (Readers can follow Dandelin’s argument with occasional glances at Figure 13.1.) I cannot swear that what follows are Honsberger’s exact words, but he readily admits to a certain broad similarity:

    “‘It takes no genius to see that the plane divides the cone into two pieces. But it was Dandelin’s idea to insert a sphere into each piece. Like an over-inflated balloon. each sphere contacts the wall of the cone and touches the elliptical plane at a certain point. But where? One can imagine Dandelin’s heart leaping at the thought that the spheres might touch the plane at the two focal points of the ellipse.’

    “Honsberger places his marking pen on the transparency. He labels the two points of contact by the symbols F and G. Are these the foci of the ellipse?

    “‘Let’s take a look at what clever old Dandelin did. First, through any point P that we care to select on the ellipse, we may draw a straight line that runs up the side of the code to its tip. Second, the line will touch the two spheres at two points, say, A and B. No matter where we pick P to lie on the ellipse, the length of AB will be the same.

    “‘Ah, but that gives it away! The distance from the point F to P equals the distance from P to A. After all, both PF and PA are tangents to the same sphere from the same point. By the same reasoning, the distance from the second point, G, to P equals the distance from B to P. Are we not finished? PF PG = PA PB, and the latter sum is just the (constant) length of AB.

    “‘Now isn’t that the darndest thing?’ Honsberger sounds rural.

    “As I look around the lecture hall, students appear stunned. Professors alternately smile and frown. One of them behind me murmurs, ‘Well, I’ll be.’”

    A.K.Dewdney ran the Computer Recreations and Mathematical Recreations columns in Scientific American for many years. His columns (and more) appear reprinted in his books The Tinkertoy Computer, The Magic Machine, The Armchair Universe, and The Turing Omnibus. All are fascinating reads.

  2. xot says:

    Figure 13.1

    That should have appeared after paragraph 7, but it looks like it was filtered out.

  3. Yourself says:

    I’m not sure what it takes to get an image to show up in a comment. I don’t have problems doing it using the HTML image tag. I imagine that may be because I’m the admin, though.

    Figure 13.1

    I’ll also mention that this is exactly the geometric proof I referred to. The one for hyperbolas is similar. However, I can’t think of one that works for parabolas.

  4. xot says:

    Hmmm, I see it also filtered out my plus signs in “PF PG = PA PB” which should read:  PF + PG=PA + PB

  5. Yourself says:

    Hmm, it seems this theme’s AJAX sucks. It’s stripping benign characters. If I go into the edit area in the admin panel and edit the post, + signs work. I’ll have to look into this and fix this shit.

  6. Yourself says:

    Okay, I’m pretty sure I fixed it. I don’t know anything about AJAX or Javascript, but apparently there’s a serialize function which serializes form data (the comment form). An optional parameter can be provided to this method which creates some kind of associative array rather than a URL style string (i.e. foo1=bar1&x=5, which would make + into a space unless it was escaped). Changing the syntax to something what was suggested by Google results fixed everything.