T-Equidistances
This activity belongs to the GeoGebra book GeoGebra Principia.
When contracting T-circles with the trace activated, at each point in the plane the color corresponding to the nearest center survives.
With multiple points, we can visualize the Voronoi diagram and compare it with the one corresponding to the Euclidean distance.
To analyze the equidistance point-line, we need to determine the distance from a point (x, y) to a line r: a x + b y + c = 0. This distance is (this formula is provided to students and can be directly introduced in the algebraic view):
Xr(x,y) = |a x + b y + c| / Max(|a|, |b|)
From the point-line equidistance the T-parabola arises, while from the point-circle equidistance the T-ellipse and the T-hyperbola emerge.
If we consider equidistance to the sides of a polygon, its skeleton and median axis arise. We can traverse it with a bitangent disk to verify this.
Finally, we can also find the T-equidistant path between two curves, either through offset (as shown here) or by generating a heat map.
- Note: For a better view of the construction, it is recommended to download the ggb file here.
Author of the construction of GeoGebra: Rafael Losada.