[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][br]The mathematical concept I will revolve around is a fundamental one: distance. [br][quote]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.[br] [br]We will employ two simple procedures to visualize geometric places related to distance: the creation of [b]implicit curves[/b][b][/b] and the use of [b]dynamic offset with activated trace[/b].[/quote][br][color=#CC3300][b]Classic Method: Sequences of Parallel Curves (Static Offset)[/b][/color][br][br]Using the [b]UnitPerpendicularVector [/b]command (and its opposite vector), it's simple to create sequences of parallels to a [b]line[/b], at progressive distances. For each line [b]r[/b], we find a pair of sequences:[br][br] [color=#CC3300]Sequence(Translate(r, k UnitPerpendicularVector(r)), k, 0, 20, 0.2)[/color][br] [color=#CC3300]Sequence(Translate(r, -k UnitPerpendicularVector(r)), k, 0, 20, 0.2)[/color][br][br]Thanks to the [i]CurvatureVector[/i] command and the [i]Locus[/i] tool, we can generalize parallelism to many curves ([i]offset [/i][url=https://en.wikipedia.org/wiki/Parallel_curve][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]). If [b]P[/b] is a point on curve [b]c[/b], the two parallel curves at distance [b]k[/b] will be given by the locus of the points:[br][br] P ± k UnitVector(CurvatureVector(P, c))[br][br]Note that, in general, [b]offset curves are not congruent with the original curve[/b]. In other words, parallel curves are not simple translations, except in the case of lines.[br][br]However, in the case of the [b]circle[/b] (let's assume with center [b]O[/b] and radius [b]4[/b]), whose offset is also a circle, we don't need the [i]CurvatureVector[/i] command or the [i]Locus [/i]tool, as it's sufficient to vary the radius of the original circle appropriately:[br][br] [color=#CC3300]Sequence(Circle([b]O[/b], 4 + k), k, 0, 20, 0.2)[/color][br] [color=#CC3300][color=#CC3300]Sequence(Circle([b]O[/b], 4 – k), k, 0, 20, 0.2)[/color][/color][br]  [br]Furthermore, if we consider a [b]point O[/b] as a circle with radius [b]0[/b], we obtain a unique sequence of offsets centered on it:[br][br] [color=#CC3300]Sequence(Circle([b]O[/b], k), k, 0, 20, 0.2)[/color][br][quote]In summary, we can easily create sequences of parallels to lines, circles and points[/quote]

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]