Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
I plotted this with equal scale using the Mac utility "Grapher", and from the plot it's clear that this closed curve is strictly convex everywhere. It does have arcs (near the maximum of curvature) where the curvature comes very close to 0. NOTE: There is a converse to the four-vertex theorem (which says that the curvature function k(s) (s=arclength) of a C^2 simple closed planar curve must have a least two local maxima and two local minima). The converse says that for any continuous real-valued function on the circle with at least two local maxima and two local minima is the curvature function of some simple closed curve in the plane. (Cf. < http://www.ams.org/notices/200702/fea-gluck.pdf >.) We want our egg to be strictly convex, bilaterally symmetric, and as simple as possible. Hence we'd like the curvature function to be everywhere positive, satisfy k(t) = k(-t) (0 <= t <= 2pi), and have the fewest possible critical points: two local maxes and two local mins. One simple way to obtain such a function is via k(t) = (cos(t) + 1/2)^2 + 1/2 (or more generally, k(t) = (cos(t) + a)^2 + b^2 where 0 < a < 1 and b > 0). I'd like to see simple closed curves with curvature functions like these. --Dan -------------------------------- RWG wrote: <<
Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 .
WFL>Looks more like a squash or pear to me Whoa, nonconvex even? It looks quite Mossy when I plot it with Macsyma, equalscale:true, but I don't know how to get equalscale in Mma, which gives me various avocados. (Are there any birds with nonconvex eggs?)
--- must be some funny-shaped birds in your part of the world, Bill! Amen. I recently rounded a corner on which calmly browsed an East African crested crane, perhaps in town to see "The Last King of Scotland." Or maybe it just wanted some action with an American whoopee crane.
P.S. Fred, they're called *whooping* cranes, not whoopee cranes.
I don't suppose the biology of egg formation might give us hints as to nature's oval formula?
On 2/22/07, David Wilson <davidwwilson@comcast.net> wrote:
I don't suppose the biology of egg formation might give us hints as to nature's oval formula?
The point being, I presume, that the topography and musculature of the avian reproductive tract should have a major influence on the shape of an ovum, prior to its shell hardening? Hang on a minute --- are you suggesting somebody sneaks up behind an unsuspecting chicken and, er ... Well, it's not exactly "whoopee"; still, I think this might be one for RWG! WFL
On 2/22/07, Daniel Asimov <dasimov@earthlink.net> wrote:
I plotted this with equal scale using the Mac utility "Grapher", and from the plot it's clear that this closed curve is strictly convex everywhere.
It does have arcs (near the maximum of curvature) where the curvature comes very close to 0.
Correct --- I hadn't meant to imply that the curvature actually went negative --- just that it looked as if it was thinking about it! [Also I cheated, and tried varying Bill's constant coefficients ...]
NOTE: There is a converse to the four-vertex theorem (which says that the curvature function k(s) (s=arclength) of a C^2 simple closed planar curve must have a least two local maxima and two local minima).
The converse says that for any continuous real-valued function on the circle with at least two local maxima and two local minima is the curvature function of some simple closed curve in the plane. (Cf. < http://www.ams.org/notices/200702/fea-gluck.pdf >.)
What an interesting and well-presented account! I wish I claim to have understood it all, but it would easily bear re-reading.
We want our egg to be strictly convex, bilaterally symmetric, and as simple as possible. Hence we'd like the curvature function to be everywhere positive, satisfy k(t) = k(-t) (0 <= t <= 2pi), and have the fewest possible critical points: two local maxes and two local mins.
The important point here is that t is not arclength, gradient, or any other intrinsic feature of the curve. Quoting from loco. cit. p195 [S^1 = circle, |R^2 = plane] : "Full Converse to the Four Vertex Theorem. Let κ : S^1 → |R be a continuous function that is either a nonzero constant or else has at least two local maxima and two local minima. Then there is an embedding α : S^1 → |R^2 whose curvature at the point α(t) is κ(t) for all t ∈ S^1."
... I'd like to see simple closed curves with curvature functions like these.
This would surely make an interesting project, but a highly nontrivial one. Any plotting algorithm must recreate the embedding referred to above, the construction of which constitutes the major portion of the proof! Furthermore, it's not at all clear to me at this stage whether such an embedding, and hence the resulting curve, would necessarily be unique --- indeed, I strongly suspect it may not be.
RWG wrote: ...
Or maybe it just wanted some action with an American whoopee crane.
P.S. Fred, they're called *whooping* cranes, not whoopee cranes.
Um, not guilty, I'm afraid. And I'm quite sure that Bill's ornithological nomenclature is impeccable, at any rate when he can restrain himself from punning in dubious taste! Fred Lunnon
On 2/22/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
It's not at all clear to me at this stage whether such an embedding, and hence the resulting curve, would necessarily be unique --- indeed, I strongly suspect it may not be.
On reflection, the curve is obviously not unique! Any (diffeo-)automorphism of the circle could precede a chosen embedding, generating in general a distinct curve from the same "curvature function". All this complication is a result of the multiple connectivity of the closed curve: the end-points and their differentials have somehow to be "joined up". WFL
participants (3)
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Daniel Asimov -
David Wilson -
Fred lunnon