A "cuenco" (bowl) in Cuenca
This activity belongs to the GeoGebra book The Domain of the Time.
Imagine a wheel rotating on a horizontal plane without slipping. A point on its circumference will perform a combination of two movements: the uniform rectilinear motion of the wheel's center (the white point) and the uniform circular motion (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 cycloid.
Press the button 
to see this curve.
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 (2πr, where r is the wheel's radius), since with each rotation, the wheel travels its perimeter. Vary the value of r to check this.
In the construction, we have limited the curve to angle values of β between -2π and 2π. For each value of β, the green point’s angle is -β - π/2. Thus, its position is (remember what we've seen about polar coordinates): (r ; -β - π/2). For that same β value, the white point moves horizontally at a height of r, covering the corresponding arc length: β r. So its position is (β r, r). Therefore, the orange point’s position is:
(β r, r) + (r ; -β - π/2)
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:
c(β) = (β r, r) + (r ; -β - π/2), -2π ≤ β ≤ 2π
or, using the Curve command:
Curve((β r, r) + (r ; -β - π/2), β, -2π, 2π)
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 β is given by:
(β r, r) + (r ; β + π/2)
to see this curve.
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 (2πr, where r is the wheel's radius), since with each rotation, the wheel travels its perimeter. Vary the value of r to check this.
In the construction, we have limited the curve to angle values of β between -2π and 2π. For each value of β, the green point’s angle is -β - π/2. Thus, its position is (remember what we've seen about polar coordinates): (r ; -β - π/2). For that same β value, the white point moves horizontally at a height of r, covering the corresponding arc length: β r. So its position is (β r, r). Therefore, the orange point’s position is:
(β r, r) + (r ; -β - π/2)
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:
c(β) = (β r, r) + (r ; -β - π/2), -2π ≤ β ≤ 2π
or, using the Curve command:
Curve((β r, r) + (r ; -β - π/2), β, -2π, 2π)
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 β is given by:
(β r, r) + (r ; β + π/2)Author of the activity and GeoGebra construction: Rafael Losada.