[math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
As per David Cantrell's suggestion, I've been trying do define, using the simplest possible equations, an "oval" that has one curvature maximum and one minimum, such as k(s) = 2 + sin(s) where s is arclength, and k must of course be periodic. But without more critical points, such a k is impossible for a simple closed planar curve, thanks to the Four-Vertex Theorem. (Cf. < http://en.wikipedia.org/wiki/Four-vertex_theorem >.) So, can anyone think of a real-analytic simple closed convex planar curve as a candidate for the Simplest Oval ? (Perhaps there is a polynomial P(z) such that alpha(t) := P(exp(it)), 0 <= t <= 2pi works? In fact any solution must be of the form alpha(t) = f(exp(it)) for some analytic f.) Here's a possibility (that I haven't checked, since I'm about to leave on a 3-day trip back to Philly from Vancouver Island): Take an ellipse with semiaxes = 1 and 2 in the plane, with the end of a major semiaxis at the origin, and then apply the polar-coordinate transformation (r,theta) |--> (r^2, theta) to the ellipse. (I.e., (x,y) |--> (x sqrt(x^2+y^2), y sqrt(x^2+y^2).) --Dan
On 2/16/07, Greg Fee <gfee@cecm.sfu.ca> wrote:
So, can anyone think of a real-analytic simple closed convex planar curve as a candidate for the Simplest Oval ?
Try:
x^2=y*(1-y)*(1-y/2);
for 0<=y<=1 .
Or tinkering around with the parameters in a*x^2 = (b-y)*(c-y)*(d-y) improves this slightly, say a = 3, b = -1/2, c = 1, d = 2 ... Is this the earliest known example of a cubic egg? Fred Lunnon
On 2/17/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
Or tinkering around with the parameters in a*x^2 = (b-y)*(c-y)*(d-y) improves this slightly, say a = 3, b = -1/2, c = 1, d = 2 ...
Oops --- should have read a*x^2 = (y-b)*(y-c)*(y-d) say a = 3, b = -1/2, c = 1, d = 2, for -1/2 <= y <= 1 ... WFL
7-11 has eggs. Recall the 7.11 problem: Professor O'Blivet blunders into a 7-11 and selects four purchases, dutifully tallying their prices on his pocket calculator, but absent-mindedly using the Times button instead of the Add button, and is amused to note a total of $7.11, but not nearly as surprised as he should have been when the clerk reaches the same figure using proper addition. What are the four prices? The solution is unique (modulo permutation) when restricted to positive whole cents, but has infinitely many (a continuum?) of negative and rational solutions, e.g., $9, -5, -.05, 3.16 and $49/25, 4099683/2972866, 158623231/171679340, 33012030/11595311 . Plus a double-continuum of irrational solutions. Eliminating variables produced the "elliptic" curve d^2 + (790/(100 * b + 197)) = (((50 * b - 197)^2)/7500), which has sixfold symmetry and resembles a rounded triangle surrounded by three "hyperbolas". Solutions to the 7.11 problem correspond to points on the curve, and the straight line connecting any two will always meet the curve exactly once more, generating a third solution. Magically, such a "sum" of two rational solutions is rational. Permuting the prices generates multiple points, so you only need one solution to get started. (But does this generate them all?) The rounded triangle makes a decent egg when vertically squished. As noted in previous mail, when you plot y = +- sqrt(f(x)), with f a rational function, say, you get ovals on the x-axis with roots of f as their poles (animal and vegetal!). Question: how do we know the eggs are infinitely smooth at those poles? I.e., are they infinitely differentiable as functions of arc-length, say? One sure way to get a smooth egg is Ptolemaic (epicyclic) synthesis. Sum a_n cis(n t) gives a nicely Zwollen egg for a_1,...,a_6 = 1,.16,.125,0,.01,-.007 . These were just eyeballed. The exact van Zwolle figure has infinitely many such epicyclic syntheses, depending on the speeds you choose while tracing it. --rwg PS, intrument collector Jim Forderer showed me an article claiming that the best violins and guitars are also piecewise cicular, like the lute. PPS, I don't see how highlights reflected in a smooth surface with curvature discontinuites will have anything worse than curvature discontinuites. Is there some optical way to differentiate? segregate easteregg eggeaters isolative ovalities apivorous oviparous
On 2/19/07, R. William Gosper <rwg@osots.com> wrote:
PPS, I don't see how highlights reflected in a smooth surface with curvature discontinuites will have anything worse than curvature discontinuites. Is there some optical way to differentiate?
The key factor is that motion past a body has the effect of differentiating the surface. Imagine a polished vertical wall, comprising sections of different constant curvatures, meeting at pillars where only the tangent plane is common. Viewed from a stationary location, the wall appears smooth. Walking past a pillar however, the reflection of a circular source alters discontinuously in length, producing a potentially disturbing visual interruption similar to that of a lower-order discontinuity. The effect may be less noticeable in a musical instrument than in the coachwork of an automobile --- I can't claim to have researched the matter! Fred Lunnon
participants (4)
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Daniel Asimov -
Fred lunnon -
Greg Fee -
R. William Gosper