Surface area of an egg

The first post in this series looked at a possible formula for the shape of an egg, how to fit the parameters of the formula, and the curvature of the shape at each end of the egg.

The second post looked at the volume. This post looks at the surface area.

If you rotate the graph of a function f(x) around the x-axis between c and d, the area of the resulting surface is

2 \pi \int_c^d f(x)\sqrt{1 + (f'(x))^2} \, dx

This integral cannot be computed in closed form for the function f describing an egg. (At least I can’t find a closed form, and neither can Mathematica.) But we can do it with numerical integration.

Here’s the Mathematica code.

f[x_, a_, b_, k_] := b Sqrt[(1 - x^2/a^2) / (1 + x k)]
area[a_, b_, k_] := 
    2 Pi* NIntegrate[ 
    f[x, a, b, k] Sqrt[1 + D[f[x, a, b, k], x]^2], 
    {x, -a, a}

As a sanity check, let’s verify that if our egg were spherical we would get back the area of that sphere.

area[3, 3, 0] returns 113.097 and N[36 Pi] also returns 113.097, so that’s a good sign.

Now let’s plot the surface area as a function of the parameter k.

Plot[area[4, 2, k], {k, -0.2, 0.2}]

The y-axis starts at 85.9, so the plot above exaggerates the effect of k. Here’s another plot with the y-axis starting at zero.

Plot[g[4, 2, k], {k, -0.2, 0.2}, PlotRange -> {0, 100}]

As with volume, the difference between an egg and an ellipsoid is approximately a quadratic function of the parameter k.