I really like Dan’s question about playing billiards with a hyperbola rather than an ellipse as the curve that the balls bounce off of. My offhand guess would be that when a ball goes off to infinity along a ray, you should draw the line that contains that ray and imagine the ball coming in from infinity along that line from the other side, until it hits the curve, at which point the usual angle-of-reflection rule applies. If we do this, does a billiard shot from one focus of a hyperbola hit the other focus (after passing through infinity)? Jim Propp On Friday, November 9, 2018, Henry Baker <hbaker1@pipeline.com> wrote:
You could come back to your daughter's class in a couple of years and present Tristan Needham's very cool demonstration that the focus-centered elliptical Newtonian orbit is merely the (complex) "square" of an origin-centered simple harmonic motion ellipse.
z = p*exp(i*t)+q*exp(-i*t), p>q>0 real, foci at +-2*sqrt(p*q).
(z = a*cos(t)+i*b*sin(t), a=p+q, b=p-q.)
z -> z^2 = (p*exp(i*t)+q*exp(-i*t))^2 = p^2*exp(i*2*t)+q^2*exp(-i*2*t)+2*p*q, foci at 0, 4*p*q.
"Bohlin's Theorem".
Newton and the Transmutation of Force, Tristan Needham The American Mathematical Monthly, Vol. 100, No. 2. (Feb., 1993), pp. 119-137. 484KB.
http://users.uoa.gr/~pjioannou/mech1/READING/newton20force.pdf
This work is a portion of a larger book:
Visual Complex Analysis, 612 pages. 9.2MB.
http://umv.science.upjs.sk/hutnik/NeedhamVCA.pdf
The laws of planetary motion, derived from those of a harmonic oscillator (following Arnold) 472KB.
https://arxiv.org/pdf/1404.2265.pdf
At 10:35 AM 11/8/2018, Henry Baker wrote:
You should definitely show them the "Dandelion" (actually, Dandelin!) Theorem, which I call the *ice cream cone* theorem. It proves the sum-of-the-distances theorem about ellipses w/o requiring analytic geometry.
When my plane geometry teacher showed it to us, it completely blew my mind!
In particular, I couldn't get over the fact that the ellipse was symmetrical, even though the spheres were of different sizes!
Even if the kids haven't yet had 3D geometry, they can still follow this easy theorem.
https://en.wikipedia.org/wiki/Dandelin_spheres
At 05:43 AM 11/8/2018, Cris Moore wrote:
I'm going to talk to my daughter's 8th grade class about conic sections.
I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry.
But I found it surprisingly tricky to work out examples.
Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4.
Using Pythagoras' theorem produces an equation with a bunch of square roots.
Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms.
Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2.
Am I doing something wrong?
Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.?
Of course, I then want to talk about light waves bouncing from one focus to another…
I'm not sure how to justify this without a little calculus.
- Cris
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