Equation to fit an egg
How would you fit an equation to the shape of an egg? This site suggests an equation of the form
Note that if k = 0 we get an ellipse. The larger the parameter k is, the more asymmetric the shape is about the y-axis.
Let's try that out in Mathematica:
ContourPlot[ x^2/16 + y^2 (1 + 0.1 x)/4 == 1, {x, -4, 4}, {y, -3, 3} ]
Here's another plot with k = 0.05. This one's a little closer to an ellipse.
If you measured an egg, how would you find a, b, and k?
Setting y = 0 shows that 2a is the length of the egg. Setting x = 0 shows that 2b is the width of the egg at the midpoint of the length. Note that it's not the maximum height of the egg because that occurs to the left of the midpoint. (To the left if k is positive. The parameter k could be negative, which flips the egg about the y-axis so that the flatter side is on the right.)
To find k we measure the point x where the maximum height occurs.
We have the equation
and implicit differentiation shows
At the maximum height the derivative of y is zero, and so the right side also equals zero. This lets us solve for k.
As k increases, the egg gets flatter on the left side and more pointed on the right side. We can quantify this by calculating the curvature at both ends.
For a curve given implicitly by F(x, y) = 0, the curvature is given by
The expression above simplifies greatly at the two points we're interest in, (a, 0).
And so the curvature reduces to
So in our first example above, with a = 4, b = 2, and k = 0.1, we have a curvature of 0.6 on the left and 1.4 on the right. In the second example with k = 0.05, we have a curvature of 0.8 on the left and 1.2 on the right.
See the next post for the volume of an egg, assuming the equation for the shape in this post.
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