Loci (2008)

Mathematical Brooding over an Egg, André Heck

So far an egg curve was approximated via geometric constructions with a digital image at the background. The algebraic expressions for the constructed geometric objects were read off in order to use them in computations. By trial and error the geometric objects were adjusted to give formulas as simple as possible. A more convenient route for mathematical modeling of the egg curve is to start with algebra and move to geometry. Figure 5 is a dynamic GeoGebra figure in which the drawn ellipses have been defined by parameterized mathematical equations in which the parameter values are specified via vertical slider bars. The corresponding mathematical formulas are also displayed in a nicely formatted way in the drawing pad. You can replay or step through the geometric construction, or you can even give it a try yourself after rewinding the construction to its first step. You can always reset the construction to its original state by pressing the reset icon in the upper right corner of the drawing pad.

As a matter of fact, the digital image is not a background picture in the GeoGebra activity, but it is a picture that has been placed at a fixed position, linked with the coordinate system, and has been made transparent so that the coordinate axes can shine through the image. The reason for working in this way is that an image of any size can then be resized and matched with a convenient coordinate system: in this case it is easy to calibrate the coordinate axes and have their scaling match with the graph paper shown in the image. It is fair to say that this is a matter of personal taste because you can also resize the image before insertion in the GeoGebra activity to a convenient format. The calibration process can be done as follows. First insert the digital image and put it at some convenient place in the drawing pad, e.g., with its lower left corner at the origin of the current coordinate system. Next create a horizontal segment, measure its length, and resize it so that it has length 6 cm in the current horizontal scaling and its left point coincides with a corner point of the graph paper. Now rescale the horizontal axis such that the calibration segment matches with a real distance of 6 cm, i.e., with a distance of 6 cm revealed in the graph paper. Adjust the scaling of the horizontal and vertical coordinate axes so that they have 1:1 proportionality again. The digital image is properly transformed during this process and can then be translated to a more suitable position on the screen (and even be fixed to that position).

The following parameter values are found via the algebraic approach: `a`_{s} = 2.8 (small ellipse), `a`_{b} = 3.4 (big ellipse) and `b` = 2.3 (both ellipses). With these values one finds a volume of 68.7 ml, an area of 82.2 cm^{2}, a minor perimeter of 14.5 cm, a major perimeter of 17.1 cm, and a shell thickness of 0.027 cm. The computed values, with the exception perhaps of the eggshell thickness, are in good agreement with the quantities measured in an experimental way and with literature values.