3. The egg as a surface of revolution
It is for students more clear what is expected from them when the research question
“What is the volume and surface area of a given hen’s egg?”
is reformulated and split into the following tasks:
- Design and carry out an experiment to determine the volume and surface area of a given hen’s egg;
- Develop a mathematical model to compute the volume and surface area of a given egg;
- Compare the experimental results with the results obtained via the mathematical model.
This reformulation takes nothing away from the open character of the task, but it forces students to develop
and carry out at least one mathematical method and it invites them to compare various methods.
Assuming that students will indeed develop a variety of methods, an evaluation of their
techniques and a classroom discussion about the descriptive qualities of the mathematical models is appropriate.
I present in this article the following mathematical model: the egg as a surface of revolution of some
function, with the rotation axis in the longitudinal direction. The exact mathematical function that describes
the egg curve is still free to choose, but a preference for functions that keep the computations
as simple as possible holds. Circles, ellipses and parabolas are for this reason good candidates. In the process
of finding a suitable mathematical function I use a picture of the egg which was created with a digital camera and
I embed it in a worksheet made with the dynamic mathematics software package GeoGebra.
In GeoGebra you can place an image as background on the drawing pad:
select the Insert image button in the
toolbar, left-click on the drawing pad, and set the image to be in the background so that it is behind
the coordinate axes. If you connect it with the absolute position on the screen, it will remain the same
whatever you do with the scaling of the coordinate system. Hereafter an egg curve can be determined by
geometric constructions, via an algebraic approach, via regression, or a combination of these techniques.
I will discuss all these mathematical approaches to the problem, but not before I point at a possible false start
of the investigative work.
Figure 3 is a dynamic Geogebra figure in which the
background image is a digital photograph made with a web cam of an egg that was positioned on top of
a piece of graph paper. The coordinate system has been scaled so that it matches the scaling of the graph paper
and has its origin placed such that the horizontal axis matches the longitudinal axis of the egg and the
vertical axis intersects the egg at its widest points. This positioning and scaling of the
coordinate system is done with the tool button 
. To translate the drawing pad, drag with the left mouse a non-specific point of the
drawing pad. Scaling of the horizontal and/or vertical axis can be achieved
be dragging a point on the axis. The vertical scaling can be linked with the horizontal scaling by
right-clicking on a point of an axis, followed by selecting in the roll menu the wanted xAxis:yAxis
proportionality. On top of the background image two ellipses have been constructed. Each ellipse
describes well a particular part of the egg curve and the two ellipses are smoothly connected at the widest part of the egg.
Each ellipse is a conic section defined by five points. The formulas for the construction points and the
ellipses are displayed in the algebra window at the bottom of the figure.
You can replay or step through the geometric construction of Figure 3. If you
want to give it a try yourself, rewind the construction to its first step and choose the button
to construct an ellipse defined by five points (clicking five points of which no four points lie on a straight line
will produce a conic). You can always reset the construction to its original state by pressing the reset icon
in the upper right corner of the drawing pad.
So far so good, but when you look closer you will notice that the photograph was taken from too near distance.
By counting the squares on the graph paper or by looking at the coordinate axes you quickly find out that perspective
distortion (the egg seems to be 5.6 cm wide and 7.5 cm long) and lens distortion (the graph paper does not look like a nice rectangular grid)
cause problems. The best thing to do is to shoot a picture of the egg from a sufficiently large distance
with a digital camera with good zooming facilities and with no or negligible lens distortion.
Also cut a hole in the graph paper so that the egg fits into it and the paper can be positioned
halfway up the egg, as shown in
Figure 4.
An alternative for a digital camera or web cam could be a good-quality scanner.