thales.swf
DrGeoII is designed for a smooth integration within the Squeak interactive system. Particularly it is compatible with the Squeak Morph user interface system and its Etoys components. In this demonstration we will show you how a DrGeoII interactive geometry content can be designed and enriched with various multimedia features. At the end the teacher will have designed a complete interactive content the student can manipulate. Of course such content could be designed by the students themselves.
It is about the calculus of a pyramid height with the help of the pyramid shadow and a stick. The solution is reported to be from Thales[1]. Here for the clarity of the example, we are proposing a simplified model, independent of the sun position.
The example will involve:
The pyramid base, a parallelogram, is designed using the center property with central symmetry. The pyramid height is designed to have a perpendicular direction to the front side of the base. The stick is parallel to the pyramid height. Dashed segments are used to represent the hidden sides of the pyramid. An arc is used to represent the sun position.
The sun is modelized as a point on the arc. The pyramid shadow is constructed semi-arbitrary with the ray going from the sun position to the pyramid summit. An arbitrary point in this ray is chosen as the shadow of the pyramid summit. The shadow of the pyramid height is constructed from this point and the pyramid base center.
The sun beams are considered all parallel to each other and a set of sun beam are represented as parallel segments. The stick shadow is constructed with this property. Various styles are used to enrich the visual aspect of the interactive figure. Intermediate constructions are hidden.
The triangles formed by the stick and its shadow, the pyramid and its shadow are homothetic. Therefore we can calculate the pyramid height thanks to the Thales theorem. To do so we first ask for the length of the known values: the stick and its shadow length, the pyramid shadow length.
Next we calculate the pyramid height with Etoys script. Within DrGeoII, mathematic value items do have one read-only slot for Etoys: the value slot. We are using it to construct graphically a Etoys script to compute the pyramid height. Then we display the result of this calculus with a detailed Etoys watcher.
We also want the sun to move in the sky, so we can realize our calculus does not depend on the position of the sun in the sky. To do so we are using another feature of DrGeoII, free point on curve do have a specific slot: the curveAbscissa slot. It is a read and write slot, it represents the position of the point on its curve. The value range is [0 ; 1]. We use this slot to program the motion of the sun in a portion of the arc it belongs to.
It is about enriching the resulting interactive figure we have now. First we draw a sun sketch, we next attach it to the geometric point representing the sun. We also add a picture representing the Sneferu's Pyramid and a text annotation. That's all folks !
In this simple example, I show you step by step a construction to modelize a mathematic situation. Next, you have seen how Etoys graphic scripting can be used within DrGeoII to do simple calculus and animation of mathematic item in an interactive geometric figure.
Moreover, I show you the benefit we have from a smooth integration of the DrGeoII geometric display engine within the Squeak Morph system: enriching the geometric figure with external multimedia resources, changing the graphic aspect of mathematic items, etc. It brings interesting and new opportunities of enriched pedagogical contents for geometry.
If you want to know more about DrGeoII and related research work, you can contact me at <hilaire AT ofset.org>
