Static Offset
This activity belongs to the GeoGebra book GeoGebra Principia.
The mathematical concept I will revolve around is a fundamental one: distance.
). If P is a point on curve c, the two parallel curves at distance k will be given by the locus of the points:
P ± k 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:
Sequence(Circle(O, 4 + k), k, 0, 20, 0.2)
Sequence(Circle(O, 4 – k), k, 0, 20, 0.2)
Furthermore, if we consider a point O as a circle with radius 0, we obtain a unique sequence of offsets centered on it:
Sequence(Circle(O, k), k, 0, 20, 0.2)
When placing a point in a space, the concept of distance to it behaves like what physicists call a "field": it doesn't manifest until we introduce another object into it. We will employ two simple procedures to visualize geometric places related to distance: the creation of implicit curves and the use of dynamic offset with activated trace.Classic Method: Sequences of Parallel Curves (Static Offset) Using the UnitPerpendicularVector command (and its opposite vector), it's simple to create sequences of parallels to a line, at progressive distances. For each line r, we find a pair of sequences: Sequence(Translate(r, k UnitPerpendicularVector(r)), k, 0, 20, 0.2) Sequence(Translate(r, -k UnitPerpendicularVector(r)), k, 0, 20, 0.2) 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 k will be given by the locus of the points:
P ± k 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:
Sequence(Circle(O, 4 + k), k, 0, 20, 0.2)
Sequence(Circle(O, 4 – k), k, 0, 20, 0.2)
Furthermore, if we consider a point O as a circle with radius 0, we obtain a unique sequence of offsets centered on it:
Sequence(Circle(O, k), k, 0, 20, 0.2)
In summary, we can easily create sequences of parallels to lines, circles and points
Author of the construction of GeoGebra: Rafael Losada.