Parametric Equations
A wheel touches a flat surface at point \( P \), assumed to be a fixed point on the wheel. As the wheel, with radius \( r \), rotates around its center on the flat surface (without sliding), point \( P \) describes a curve called a cycloid (see video below).
A Rotating Wheel Describing a Cycloid
What the video shows is explained in Fig. 1 below: point \( P \), a fixed point on the wheel, rotates around the center of the rotating wheel and moves horizontally and vertically, taking positions \( P_1, P_2, \dots \)
In Fig. 2 below, point \( P \) is shown after a rotation of the wheel around its center \( C \), measured by angle \( t \) made by \( CP \) and \( CA \) such that \( CA \) is parallel to the \( y \)-axis. Hence the components of vector \( \vec{CP} \) are given by:
\[ \vec{CP} = \left\langle r \cos \theta \; , \; r \sin \theta \right\rangle \qquad (I) \]where \( \theta \) is the angle in standard position. We need to express the components of \( \vec{CP} \) in terms of angle \( t \), noting that:
\[ \theta = \dfrac{3\pi}{2} - t \]Substitute \( \theta \) into (I) and write:
\[ \vec{CP} = \left\langle r \cos \left(\dfrac{3\pi}{2} - t\right) \; , \; r \sin \left(\dfrac{3\pi}{2} - t\right) \right\rangle \]Use the trigonometric formulas \( \cos(A - B) = \cos A \cos B + \sin A \sin B \) and \( \sin(A - B) = \sin A \cos B - \cos A \sin B \), along with \( \cos\left(\dfrac{3\pi}{2}\right) = 0 \) and \( \sin\left(\dfrac{3\pi}{2}\right) = -1 \), to obtain:
\[ \vec{CP} = \left\langle -r \sin t \; , \; -r \cos t \right\rangle \]
To find the equation of the cycloid, we need the coordinates of point \( P \) with respect to the origin \( O \) of the coordinate system (Fig. 3).
After a rotation of angle \( t \), the \( x \)-coordinate \( OO' \) of the center \( C \) of the circle is equal to the length of the arc \( O'P \), which is given by \( rt \) (\( t \) in radians). The \( y \)-coordinate \( O'C \) of the center \( C \) is equal to \( r \). Hence, the components of vector \( \vec{OC} \) are:
\[ \vec{OC} = \left\langle rt \; , \; r \right\rangle \]Vector \( \vec{OP} \) is given as a sum of vectors:
\[ \vec{OP} = \vec{OC} + \vec{CP} \]Substituting the components of \( \vec{OC} \) and \( \vec{CP} \):
\[ \vec{OP} = \left\langle rt - r \sin t \; , \; r - r \cos t \right\rangle \]The coordinates of \( P \) are given by the parametric equations:
where \( t \) is the angle, in radians, through which the wheel has rotated around its center, measured starting from the negative y-axis in a clockwise direction.
Length of a Cycle of a Cycloid
The length \( L \) of a curve given by \( y = f(x) \) is:
\[ L = \int_{x_1}^{x_2} \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} dx \]The derivative \( \dfrac{dy}{dx} \) of parametric equations is given by:
\[ \dfrac{dy}{dx} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}} \]Using \( dx = \dfrac{dx}{dt} dt \), the arc length formula becomes:
\[ L = \int_{t_1}^{t_2} \sqrt{1 + \left(\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}\right)^2} \dfrac{dx}{dt} dt \qquad (II) \]Which can be rewritten as:
\[ L = \int_{t_1}^{t_2} \dfrac{1}{\left|\dfrac{dx}{dt}\right|} \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \dfrac{dx}{dt} dt \]One cycle of a cycloid corresponds to one full rotation of the wheel, so the integration limits are \( t_1 = 0 \) and \( t_2 = 2\pi \). Using the parametric equations, the derivatives with respect to \( t \) are:
\[ \dfrac{dx}{dt} = r(1 - \cos t) \quad \text{and} \quad \dfrac{dy}{dt} = r \sin t \]Since \( \dfrac{dx}{dt} = r(1 - \cos t) \) is non-negative on \( [0, 2\pi] \), we have \( \left|\dfrac{dx}{dt}\right| = \dfrac{dx}{dt} \). Substituting this into (II):
\[ L = \int_{0}^{2\pi} \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} dt \]Substitute the derivatives and simplify:
\[ L = \int_{0}^{2\pi} \sqrt{(r(1 - \cos t))^2 + (r \sin t)^2} dt \] \[ L = r \int_{0}^{2\pi} \sqrt{1 + \cos^2 t - 2\cos t + \sin^2 t} dt \]Using the trigonometric identity \( \sin^2 t + \cos^2 t = 1 \):
\[ L = r \int_{0}^{2\pi} \sqrt{2(1 - \cos t)} dt \]Using the identity \( 1 - \cos t = 2 \sin^2(t/2) \):
\[ L = r \int_{0}^{2\pi} \sqrt{4 \sin^2(t/2)} dt = 2r \int_{0}^{2\pi} |\sin(t/2)| dt \]Since \( \sin(t/2) \) is non-negative on \( [0, 2\pi] \):
\[ L = 2r \int_{0}^{2\pi} \sin(t/2) dt \]Evaluating the integral:
\[ L = 2r \left[-2 \cos\left(\frac{t}{2}\right)\right]_{0}^{2\pi} = 4r \left[-\cos(\pi) + \cos(0)\right] = 4r(1 + 1) = 8r \]The length of one cycle of a cycloid is:
\[ \boxed{L = 8r} \]