## 10. Dynamic use of digital images in GeoGebra

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 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 it is not really needed to use such a program: Geogebra allows you to apply affine transformations on digital images as long as they are not picture at the background 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. Which transformations are possible on a digital image depends 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 13 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

.

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

Figure 13. An aproximation of a partridge egg.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now

The dynamic use of digital images and the feature of adjustable transparency of images in Geogebra also offers the opportunity the compare two pictures via similarity transformations. An example of such an activity is shown in Figure 14, in which a facial mask, taken from www.beautyanalysis.com, that would model the ideal facial contours on the basis of the golden ratio has been placed on top of a photograph of a student. Such an investigation on facial dimensions is for many a pupil more interesting, more funny, and more contemporary (think of many current television programmes about beauty, cosmetics, and plastic surgery) than only modeling the facial profile with an egg shaped curve.

Figure 14. Facial investigation with photographs.
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You can translate, rotate, and rescale the photographs and masks of the above GeoGebra figure by dragging points or line segments. You can always reset the construction to its original state