Some find it counterintuitive that slicing a cone gives an ellipse and not an oval.
What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse.
An oval is egg-shaped. The two ends have different curvatures. There's one axis of symmetry, while an ellipse has two.
Is there a technical definition of "egg-shaped"? A way to measure the ovality? Ellipses have eccentricity, do ovals then have eggcentricity?
More seriously, what constraints make sense to give a well defined "egg shape"? You have one one axis of symmetry and two curves that must fit together smoothly, very smoothly. I don't see how you can glue halves of two different ellipses together to get the smoothness constraint. (Can you?) There should be lots of other "almost ellipse" definitions that will glue together. Any takers on a simple definition or process to make an egg? Err... oval?
Here's the simplest egg-shape known to me:
IIRC, it is the shape used by Arnault of Zwolle (ca. 1450) in describing his lute. [...]
The construction is very simple, requiring only arcs of circles of three different radii. [...] Anyway, the figure at
should make the construction abundantly clear. I suspect that that beautiful egg shape was used a good bit during the Middle Ages (perhaps in architecture?) but have no supporting evidence.
Having been a maker of historical instruments, I analyzed the outlines of many lutes (and some other instruments) from the Renaissance. The outlines were always done, it seemed, by ruler-and-compass constructions. But the outlines of lutes in the later Renaissance involved somewhat more complicated constructions than did Arnault's lute.
A highly polished surface reflecting a bright light would show up the discontinuous curvature of such an elementary spline; as technology advanced, so perfectionist makers might well have become dissatisfied with this situation --- as have the designers of modern cars.
[quoting somewhat out of context:]
So, can anyone think of a real-analytic simple closed convex planar curve as a candidate for the Simplest Oval ?
x2=y*(1-y)*(1-y/2);for 0<=y<=1 .
Or tinkering around with the parameters in
a*x2 = (b-y)*(c-y)*(d-y)improves this slightly, say a = 3, b = -1/2, c = 1, d = 2 ...
What about the locus of points such that d(A)+kd(B) is constant, where k is a real number and A,B are the focii. Wouldn't that make an oval?
This gives a rather nice pointy oval, if k is chosen just less than (string length)/(focal distance), being the critical value where the curve becomes limacon-like.
The equation is
((x2 + (y-a)2) + c2*(x2 + (y+a)2) - 4*b2)2 -where
4*(x2 + (y-a)2)*c2*(x2 + (y+a)2) = 0,
2a = focal distance,with say
2b = string length,
c = Gray weighting k;
a = 1, b = 1.5, c = 1.35 inside a 3x3 box.
Here's the example I mentioned in my posting, together with the Maple code producing it.
# Steve Gray's weighted ellipse egg: 1 < c < b/a (limacon) # 2a = focal distance, 2b = string length, c = Gray weighting (k) a := 1; b := 1.5; c := 1.35; s := 1.5; stegg := expand(((x^2 + (y-a)^2) + c^2*(x^2 + (y+a)^2) - 4*b^2)^2 - 4*(x^2 + (y-a)^2)*c^2*(x^2 + (y+a)^2));
Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 .
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