## 5. Computing with ellipses and eggs

The main reason to prefer approximations of the egg curve with semi-ellipses of which the mathematical formulas are in canonical form is that in this case you easily read off the semiminor and semimajor axes of the ellipses and that you can apply mathematical formulas for the area and volume of the surface of revolution obtained by rotating an ellipse about its major axis. The following formulas hold for the surface of revolution obtained from an ellipse with a = the length of the semimajor axis and with b = the length of the semiminor axis:

and
with eccentricity e given by
.

The major perimeter of the surface of revolution is given by:

,

where E is the complete elliptic integral of the second kind. These formulas can be found on Internet (e.g. http://mathworld.wolfram.com/Perimeter.html) or can be computed via integration:

,

,

and
,
where
.

With the exception of volume you may need a computer algebra system to find the correct mathematical expressions. By the way, useful approximations of the perimeter of an ellipse exist (e.g., see Final Answers. The following approximation originates from the great mathematician Ramanujan (1913-1914):

The following combination of arithmetic and geometric mean (Bronshtein & Semendyaev, 1985) is also very attractive:

When the above formulas are applied to the approximation of the egg curve by two semi-ellipses with parameters as = 2.75 (small ellipse), ab = 3.43 (big ellipse) and b = 2.30 (both ellipses), then one gets a volume of 68.5 ml, an area of 82.2 cm2, a minor perimeter of 14.5 cm and a major perimeter of 17.0 cm. The computed volume and surface area are in agreement with the measured quantities. The computed area is in agreement with the estimated value according to the following allometric power-law between weight (in gram) and area of avian eggs (in cm2) (Paganelli et al., 1974):

According to this formula and the measured weight of the hen’s egg the surface area would be 83.5 cm2. Another beautiful allometric law for hen’s eggs is (Besch et al., 1968):

This gives the following value of surface area: 81.7 cm2. Our computer values lies between these two estimates. By the way, there also exist experimentally found relationships between the volume and area of an egg. For example (Hoyt, 1976):

If you fill out the measured volume of 68 ml, then the estimated surface area is also 81.7 cm2. In summary, the above surface area computation and the allometric laws found in the literature are in good agreement.

The equation for the volume V of an egg can also be written as

,

where λ = as + ab and β = 2b represent the length and width of the egg, respectively. This formula for the volume can be used to estimate the thickness of the eggshell as a root of a polynomial. I outline the method after (Narushin, 1998):

To my purpose, an egg consists of two components: the shell and the contents of the egg. The mass m of the egg is the sum of the mass ms of the eggshell and the mass mc of the egg’s contents. Let V, Vs and Vc be the volumes of the complete egg, of the shell only, and of the contents only, respectively. Let ρs and ρc be the density of the shell and the contents of the egg, respectively. It holds:

m = ρsVs + ρcVc
and
V = Vs + Vc.

Rewriting leads to the following expression for the volume of the contents of the egg:

.

Assume that the complete egg can be approximated as a surface of revolution of two semi-ellipses with canonical parameters as, ab and b. The same assumption can be made for the contents of the egg, but in this case with the canonical parameters

As = as - δ, Ab = ab - δ and B = b - δ.

So, the existence of an air cell in the egg and the variation in shell thickness, which depends on the particular spot on the egg, are neglected. The formula for the volume of the contents of the egg is:

.

Working out leads to:

.

Rewriting in terms of length λ and width β of the egg leads to:

.

Thus the eggshell thickness δ must satisfy the following third degree polynomial equation:

.

In the standard work (Romanoff & Romanoff, 1947) you can find the required densities for shell and contents of avian eggs: ρs = 2.3 g/cm3 and ρc = 1.037 g/cm3. Other data of the hen’s egg under consideration are: m = 74 g, V = 68 cm3, λ = 6.21 cm, and β = 4.62 cm. The shell thickness δ of the egg undr consideration must satisfy the following polynomial equation:

δ3 − 7.725δ2 + 19.6812δ − 0.6585 = 0.

The only real solution is δ = 0.034 cm. This is 2.5 μm less than the thickness according to the following allometric law (Ar et al., 1974):

δ = 5.126·10-3 · weight0.456.