Dynamic Offset with Activated Trace
This activity belongs to the GeoGebra book GeoGebra Principia.
Dynamic Offset with Activated Trace
Now we will replace each sequence of parallels with a single dynamic parallel. As before, using the UnitPerpendicularVector command (and its opposite vector), it's straightforward to create parallels to a line, at a given distance d.
For each line r, we find a pair of parallels:
Translate(r, d UnitPerpendicularVector(r))
Translate(r, –d UnitPerpendicularVector(r))
Thanks to the CurvatureVector command and the Locus tool, we can generalize parallelism to many curves (offset). If P is a point on curve c, the two parallel curves at distance d will be given by the locus of the points:
P ± d UnitVector(CurvatureVector(P, c))
Note that, in general, offset curves are not congruent with the original curve. In other words, parallel curves are not simple translations, except in the case of lines.
However, in the case of the circle (let's assume with center O and radius 4), whose offset is also a circle, we don't need the CurvatureVector command or the Locus tool, as it's sufficient to vary the radius of the original circle appropriately:
Circle(O, 4 + d)
Circle(O, 4 – d)
Furthermore, if we consider a point O as a circle with radius 0, we obtain a unique sequence of offset centered on it:
Circle(O, d)
In summary, we can easily create sequences of offsets of lines, circles and points.Also, we can create the intersection points of two objects and the corresponding locus. The problem with using the Locus command or tool is that in many situations (more complex than the one shown here) it's not possible to use it properly. Since GeoGebra is a Dynamic Geometry program, we can not only move geometric objects at our will but also establish automatic animations [22]. To achieve this, we add a trace to the offset and choose a decreasing value of d (opposite to a slider "increasing once"). Note: alternatively, we can choose an increasing value for d (increasing once) and assign it a speed of -1 instead of 1. By doing so, simultaneously offsetting a point and a line, for instance, we can visualize the parabola through color contrast.
The advantage of using offset over implicit curves, which we will see next, is that it allows us to pause the procedure's playback at any time and observe how the traces of the lines intersect. This helps us understand why these intersection points are part of the sought-after locus.
Author of the construction of GeoGebra: Rafael Losada.