0302 Displaying reflection axioms in the P-model
1.
The reflection of a point across the same line is the original point;
We call geometric transformations involutory if, when applied to any geometric figure and then to its resulting image, we obtain the original figure. In Euclidean geometry, examples of this include axial reflection, central reflection, and inversion.
In the present case, the question is whether the HTükrözés[] interpreted on the P-model is involutory. In principle, it is, since this transformation is—in the Euclidean sense—an axial reflection or an inversion. Yet it is justified to ask: exactly how does this work in GeoGebra, specifically on the P-model?
Let us note in advance that a computer algebraic system that renders geometric figures (but obviously works with numbers of finite precision in the background) can never be perfectly accurate, so this cannot be expected from GeoGebra either.
GeoGebra considers two points to be identical if the difference between their coordinates is no greater than 10⁻⁸ times the unit of the GeoGebra drawing sheet for both coordinates. At the same time, zooming in on the drawing sheet cannot be continued indefinitely. You can zoom in on the drawing sheet as long as the unit of the displayed grid is greater than 10-13.
The program itself calculates to 15 decimal places, so detectable discrepancies may already occur in such precisely calculated results, which become visible on a drawing sheet subjected to sufficiently strong magnification.
030201 Is the HT Reflection involutory?
We leave it to our readers to try out the experiment of determining how far the t-axis had to be “extended” the t-axis to the edge of the unit circle in order to produce an image in which the mirror image of the mirror image no longer coincides with the original, as shown in the figure, even though GeoGebra still considers points C and C2 to coincide.
2.
o The reflections of points lying on a line also lie on a line; (Reflection preserves lines.)
We examine on the P-model whether, if point P lies on the line e = (C, D), then the reflection Pt of P with respect to the line t = (A, B) lies on the reflection et = (At, Bt) of the line e. In other words, is the HReflect() procedure truly "H-Line"-preserving?
There is one way to investigate this. Let’s define point P using the command P=Point[e]. Then P will lie on line e. Using the GeoGebra Relation[P_t,e_t] command, we can examine the relationship between the bjects Pt and et. We can now activate this command using a button.
Given this magnification of the GeoGebra drawing sheet, how large might the
base circle of the P-model be—say, relative to the size of the Earth—if its radius is 10 units in the drawing sheet’s coordinate system?
Note that the zoom function on the GeoGebra drawing sheet operates between 10-13 and 5 × 10⁸. Examining the link above has presumably convinced our readers that we can be satisfied with this level of precision.
030202 Is the HReflect isa line reflection?
030203
We must be content with an answer that is generally reassuring, but in quite extreme cases can be unsettling as it contradicts appearances:
3.
- The mirror images of an axially symmetric pair of points with respect to a single axis are also axially symmetric. (Reflection preserves symmetry.)
Having seen examples and counterexamples—in both directions—between the GeoGebra diagram of the P-model and the result obtained for a given relation, we can agree that it is not advisable to obsessively search for these contradictions that arise in extreme situations.
We can rest assured: the P-model illustrated with GeoGebra presents the fundamental phenomena of absolute and hyperbolic geometry with more than sufficient accuracy.