Example 3: Cycloid

A point on the rim of a wheel rolling along a line traces out a curve called a cycloid. Find the vector equation for the cycloid if the wheel has radius one, rolls at one radian per second along the x-axis, and starts at t = 0 with the point at the origin.

As we can see from the close-up in Figure 10.6.3, the parametric equations are

x = t - sin t, y = 1 - cos t.

The vector equation is

X = (t - sin t)i + (1 - cos t)j.

A vector valued function in three dimensions can be written in the form

F(t) = f₁(t)i + f₂(t)j + f₃(t)k

and has the three components f₁, f₂, and f₃. The equation X = F(t) can be written as three parametric equations

x = f₁(t), y = f₂(t), z = f₃(t).

and as t varies over the reals we get a parametric curve in space.

Figure 10.6.3