One of my favourite Useful Coincidences in Mathematics is that a paraboloid: (a) has the reflector property; (b) is the shape assumed by the surface of rotating liquid; which together enable the construction of 'liquid mirror telescopes': https://cp4space.wordpress.com/2014/05/28/liquid-mirror-telescopes/ ^^ this also has a calculus-free proof of the reflector property. Best wishes, Adam P. Goucher
Sent: Thursday, November 08, 2018 at 5:06 PM From: "Mike Speciner" <ms@alum.mit.edu> To: "Cris Moore" <moore@santafe.edu>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] conic sections
Actually, using the focus-directrix definition of a parabola, and the fastest-path property of light, it's easy to see that parabolic dishes focus light at the focus. (Fastest path defaults to shortest path when there's only one medium; refraction happens with different media.)
And the focus to focus reflection of ellipses is easy to see when using the constant sum of distances to foci property.
And, by the way, the 3-D proof of the properties of conic sections (with tangent spheres to cone and plane cutting cone) is really quite beautiful if they don't know it.
--ms
On 08-Nov-18 11:39, Cris Moore wrote:
These comments are all very helpful… sort of :-)
My goal is to show students the unity between algebra and geometry. I don’t mind doing some algebra on the board, and it’s not _that_ hard. They know Pythagoras’ theorem, but they haven’t really used it to go back and forth between algebra and circles or ellipses. They know what a parabola is, and they’ve heard that parabolic dishes collect incoming rays, but they don’t know quite what this means, or why it might be true. And they haven’t seen the physical way of drawing an ellipse with a string connected to two thumbtacks...
And I have one class period to do this.
- Cris
On Nov 8, 2018, at 9:26 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Yes, it does have these operations, but they aren't "general", but have to do specifically with transforming the current coordinate system.
(Yes, one could save the coordinate system, utilize these operations, and then restore the coordinate system -- which might be a good way to implement a more general transform, but that more general transform isn't "built in", although it is also a one-liner.)
Notice also, that the 3x3 matrix ops in Postscript aren't general matrix ops, but are to implement *homogeneous* representations:
the triple (x,y,z) represents the 2D point (x/z,y/z).
So Cris also needs to teach his daughter homogeneous representations!
BTW, lots of kids in the early 1980's were already hacking Apple ]['s and assembly language, so 8th grade is NOT too early to start learning computer graphics.
At 08:01 AM 11/8/2018, Mike Speciner wrote:
PostScript has matrix operations, including concat (multiply) for 2D affine transformations.
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