Loci (2008)

Mathematical Brooding over an Egg, André Heck

I continue the mathematical modeling of egg shapes as surfaces of revolution around the major axis on the basis of digital images. A much-used method that is not yet discussed in this paper is the following technique of changing the canonical equation of an ellipse

with a function `f` into

such that the curve looks more like an egg curve. The function

where `c` is some constant, is an example of a function that does such a job. But secretly I assume here that the longitudinal direction of the egg is the same as the horizontal axis. Thus, if you want to compare the egg curve with a digital image of a particular egg, then the major axis of the egg must be horizontal. For a picture that you take yourself this is no problem. But for a picture that you obtain from the Internet this does not have to be the case or in some cases the picture from internet may be too small to be used conveniently. With image processing software you can adjust the size and orientation of a picture, but using such a program is not really necessary: Geogebra allows you to apply affine transformations on digital images as long as they are not background pictures positioned with respect to the computer screen. The easiest way of doing this is by linking the picture to one, two, or three points, or by linking the picture to a geometric object like a line segment or a triangle. For a triangle, the digital image is transformed such that it becomes a parallelogram. In version 3 of GeoGebra you cannot link a digital image to a 4-gon and have it transformed such that it fits in the 4-gon. In other words, perspectivisms cannot be applied on the image in GeoGebra 3. The transformations that are possible on a digital image depend on the geometric object linked with the corner points. The possibilities are tabulated below:

geometric object | # degrees of freedom | transformation |
---|---|---|

1 corner points | 2 | translation |

2 corner points, segment, vector, line | 4 | similarity |

3 corner points, triangle | 6 | affine mapping |

This freedom is sufficient for bringing an image of an egg in any size and orientation that you want. In Figure 12 I have enlarged a picture of a partridge egg to twice the size of a real egg, which is 37 × 27 mm, and hereafter I have modeled it with the mathematical formula. Drag point A or B if you want to see how a dynamic image is made available in GeoGebra.

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As a matter of fact, this kind of model also works for the hen's egg that is used almost all the time in this paper.