The Cycloid Arch Length Calculation

Introduction

A cycloid is the curve traced by a point on the rim of a circle as the circle rolls along a straight line without slipping. Its arches have a distinctive shape: the curve starts at a cusp, rises to a rounded top, and returns to a cusp. Although its parametric equations involve trigonometric functions, the length of one complete arch can be found exactly.

The key is to compute the speed of the parametrized curve before integrating. A trigonometric identity simplifies the speed considerably, but taking the square root introduces an absolute value. Keeping track of its sign on the arch’s parameter interval is essential.

Parametrizing One Arch

Let \(a>0\) be the radius of the rolling circle. A standard parametrization of the cycloid is

Here \(t\) is the parameter, measured in radians. The values \(t=0\) and \(t=2\pi\) describe consecutive cusps, so \(0\le t\le 2\pi\) traces one complete arch. At the endpoints, the curve is at height zero; at \(t=\pi\), its height is \(2a\). Its horizontal coordinate runs from \(0\) to \(2\pi a\).

For the cycloid, the derivatives exist and are continuous throughout the closed parameter interval. The velocity is zero at the two endpoint cusps, but the speed remains continuous, so the length integral over the complete arch is well-defined.

Deriving the Cycloid’s Speed

Differentiate each coordinate with respect to \(t\):

Substitute these derivatives into the speed formula. The identity \(\sin^2 t+\cos^2 t=1\) reduces the expression under the square root:

Now use the half-angle identity \(1-\cos t=2\sin^2(t/2)\). This gives \(2-2\cos t=4\sin^2(t/2)\), and therefore

The absolute value is required because \(\sqrt{u^2}=|u|\), not \(u\) for every real \(u\). On the interval \(0\le t\le2\pi\), the half-angle \(t/2\) lies in \([0,\pi]\), where the sine is nonnegative. Thus, specifically on this arch, the speed becomes \(v(t)=2a\sin(t/2)\).

The Length of a Complete Arch

Using the speed just derived, the length from the first cusp to the next is

To check the integration, differentiate the antiderivative: \(\frac{d}{dt}\bigl[-4a\cos(t/2)\bigr]=2a\sin(t/2)\), which is the speed on the stated interval. The result is positive, as a length must be.

Why the Calculation Works

The central simplification follows from combining the horizontal and vertical velocities before integrating. The horizontal component contains \(1-\cos t\), while the vertical component contains \(\sin t\). Squaring both and adding cancels the separate trigonometric terms into a single expression:

The half-angle identity then turns \(1-\cos t\) into a square, so the speed is proportional to the absolute value of a sine. On a complete arch, that sine has one sign throughout the interior of the interval. The absolute value can therefore be removed after the interval is specified, leaving a straightforward integral. This order matters: simplify the square root correctly first, then use the interval to determine the sign.

Worked Examples

Common Mistake: Dropping the Absolute Value Too Soon

A tempting but incorrect simplification is \(\sqrt{4\sin^2(t/2)}=2\sin(t/2)\) for every \(t\). In the preceding example, this would give the supposed speed \(6\sin(t/2)\) on \([-2\pi,0]\). Integrating that expression produces

A negative length signals that the expression used was not the speed. The correction is to retain \(6|\sin(t/2)|\), or, on this particular interval, replace it with \(-6\sin(t/2)\). The corrected integral is \(24\), as calculated above. Always use the specified parameter interval to determine the sign before removing an absolute value.

Why the Exact Length Is Useful

The formula \(L=8a\) gives a direct geometric measurement from the generating circle’s radius. It also provides a check on a longer calculation: after deriving a speed and integrating, the result should scale linearly with \(a\). Doubling the radius doubles every velocity magnitude and doubles the arch length. A result with a different dependence on \(a\), a negative value, or an incorrect endpoint interval should prompt a check of the setup.

The parameter interval is part of the calculation, not a detail to add afterward. The standard arch uses \(0\le t\le2\pi\); on that interval, the half-angle sine is nonnegative. Other intervals may describe a complete arch with the opposite sign of the sine, so the general speed \(2a|\sin(t/2)|\) is the safe starting point.