[color=#0000ff][i][color=#0000ff][i][color=#999999]This activity belongs to the GeoGebra book [url=https://www.geogebra.org/m/mes4bgft]The Domain of the Time[/url].[/color][/i][/color][/i][/color][br][br]Imagine a wheel rotating on a horizontal plane without slipping. A point on its circumference will perform a [b]combination of two movements[/b]: the [b]uniform rectilinear motion[/b] of the wheel's center (the white point) and the [b]uniform circular motion[/b] (UCM) of the point rotating around that center (the green point). This combination results in the curved path traced by the point on the wheel (the orange point), which is known as a [i]cycloid[/i].[br][br]Press the button [img]https://www.geogebra.org/resource/yxbcmb2f/CZJZaLQBirTUHVXU/material-yxbcmb2f.png[/img] to see this curve.[br][br]As you can see from its definition, there are two clear consequences. First, the cycloid is a periodic curve because, with every turn of the wheel, the point starts the same path again. The second consequence is that its period is the length of the circumference (2π[i]r[/i], where [i]r[/i] is the wheel's radius), since with each rotation, the wheel travels its perimeter. Vary the value of [i]r[/i] to check this.[br][br]In the construction, we have limited the curve to angle values of [i]β[/i] between -2π and 2π. For each value of [i]β[/i], the green point’s angle is -[i]β[/i] - π/2. Thus, its position is (remember what we've seen about polar coordinates): ([i]r[/i] ; -[i][i][i]β[/i][/i][/i] - π/2). For that same [i]β[/i] value, the white point moves horizontally at a height of r, covering the corresponding arc length: [i]β[/i] [i]r[/i]. So its position is ([i]β[/i] [i]r[/i], [i]r[/i]). Therefore, the orange point’s position is:[br][br] ([i]β[/i] [i]r[/i], [i]r[/i]) + ([i]r[/i] ; -[i][i][i]β[/i][/i][/i] - π/2)[br][br]This is the equation of the cycloid (nicknamed "The Helen of Geometers", according to some because, like Helen of Troy, it was the source of numerous disputes among 17th-century mathematicians, and according to others, for the beauty of its properties). With GeoGebra, we can represent the two arcs of the curve shown as:[br][br] c([i]β[/i]) = ([i]β[/i] r, r) + (r ; -[i]β[/i] - π/2), -2π ≤ [i]β[/i] ≤ 2π[br][br]or, using the Curve command:[br][br] Curve(([i]β[/i] r, r) + (r ; -[i]β[/i] - π/2), [i]β[/i], -2π, 2π)[br][br]In the following activities, we will use this curve, but inverted. Activate the "Invert" checkbox to see it. In the inverted cycloid, the orange point's position for angle [i]β[/i] is given by:[br][br] ([i]β[/i] [i]r[/i], [i]r[/i]) + ([i]r[/i] ; [i][i][i]β[/i][/i][/i] + π/2)

[color=#0000ff][color=#999999][color=#999999][color=#0000ff][color=#0000ff][color=#999999][color=#999999]Author of the activity and GeoGebra construction: [/color][/color][/color][color=#0000ff][color=#999999][color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color][/color][/color][/color][/color][/color][/color]