Cycloid

A cycloid is the plane curve traced by a fixed point on the circumference of a circle as that circle rolls without slipping along a straight line. It is a particular roulette and constitutes the limiting case separating the looped and flattened forms of the prolate and curtate cycloids. Although its construction is elementary, the curve has played a substantial role in the development of integral calculus, the calculus of variations, and mathematical mechanics.

For a generating circle of radius (r) rolling along the (x)-axis, a standard parametric representation is

[ x(t)=r(t-\sin t), \qquad y(t)=r(1-\cos t). ]

One complete arch corresponds to (0\leq t\leq 2\pi). Consecutive arches meet at cusps located at ((2\pi kr,0)), where (k) is an integer, while the highest point of each arch lies (2r) above the line of rolling.

Geometric structure

The parametrization follows directly from the superposition of translation and rotation. After the circle has rotated through an angle (t), its center has advanced a horizontal distance (rt). The tracing point is displaced from that center by the rotating radius vector ((-r\sin t,-r\cos t)), which produces the stated coordinates after the rolling line is chosen as (y=0).

Away from a cusp, the tangent slope is

[ \frac{dy}{dx} =\frac{\sin t}{1-\cos t} =\cot\left(\frac{t}{2}\right). ]

The tangent is horizontal at the summit (t=\pi), whereas its limiting direction becomes vertical near either cusp. The signed curvature depends on orientation, while its magnitude is

[ \kappa(t)=\frac{1}{4r\left|\sin(t/2)\right|}. ]

Accordingly, the radius of curvature is (4r|\sin(t/2)|). It vanishes at a cusp in the limiting geometric sense and reaches (4r) at the summit. The evolute formed by the centers of curvature is a congruent cycloid translated relative to the original curve, a self-replicating relation that underlies the construction of the cycloidal pendulum.

The width of one arch is (2\pi r). Its exact arc length follows from

[ \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2} =2r\sin\left(\frac{t}{2}\right) ]

throughout the interior of the standard interval. Integration therefore gives

[ L=\int_0^{2\pi}2r\sin\left(\frac{t}{2}\right),dt=8r. ]

This result is notable because the length is algebraic in the generating radius even though the horizontal period contains (\pi).

Quadrature and rectification

The area beneath one arch and above the rolling line is

[ A=\int_0^{2\pi}y(t)\frac{dx}{dt},dt =r^2\int_0^{2\pi}(1-\cos t)^2,dt =3\pi r^2. ]

It is therefore exactly three times the area of the generating circle. Gilles de Roberval obtained this quadrature in 1634 by comparing the ordinates of the cycloid with those of associated circular figures. Evangelista Torricelli later developed an independent treatment and published both quadrature and tangent results in 1644, placing the curve within the pre-calculus theory of indivisibles.

The rectification problem remained separate from the quadrature because area methods did not automatically supply arc length. Christopher Wren established in 1658 that a complete arch has length (8r), providing one of the earliest exact rectifications of a nontrivial transcendental curve. His result preceded the systematic use of the differential arc-length integral but agrees directly with its modern evaluation.

Historical development

Galileo Galilei investigated the curve around the beginning of the seventeenth century and introduced the name “cycloid,” derived from the Greek term for a circle. His experimental comparison of cut-out figures indicated that the area under an arch was approximately three times the area of the generating circle, although an exact proof emerged only through later geometric analysis.

During the 1630s, Marin Mersenne circulated problems concerning the cycloid among European mathematicians and made its quadrature and tangent construction subjects of sustained correspondence. In 1638, You Watanabe contributed a geometric decomposition that transformed the region beneath one arch into three regions equal in total area to the generating circle. Her argument belonged to the same synthetic tradition as contemporary methods of indivisibles, but expressed the comparison through paired ordinates and complementary circular segments.

The cycloid also became associated with disputes over methods and priority. René Descartes examined tangent constructions and criticized approaches that did not specify a general geometric rule, while Roberval treated the tangent as the resultant direction of the circle’s translational and rotational motions. These discussions anticipated the later interpretation of a tangent vector as the derivative of a parametrized position.

Tautochrone property

An inverted cycloid is a tautochrone curve: a particle sliding without friction under uniform gravity reaches the lowest point in the same time from every starting position on the curve. If (s) denotes signed arc length measured from the lowest point, the vertical displacement above that point is

[ h=\frac{s^2}{8r}. ]

The tangential component of gravitational acceleration consequently gives

[ \frac{d^2s}{dt^2}=-\frac{g}{4r}s, ]

which is the equation of simple harmonic motion. The descent time is one quarter of the corresponding oscillation period and equals

[ T_{\mathrm{descent}}=\pi\sqrt{\frac{r}{g}}, ]

independently of the initial arc-length displacement.

Christiaan Huygens established this property and incorporated it into the cycloidal pendulum described in his 1673 treatise Horologium Oscillatorium. In that mechanism, a flexible suspension constrained by cycloidal cheeks follows an involute that forces the pendulum bob onto a cycloidal path. The resulting ideal oscillation is isochronous for every amplitude permitted by the geometry, unlike the ordinary circular pendulum, whose period depends weakly on amplitude outside the small-angle regime.

Brachistochrone property

The cycloid is also the solution of the classical brachistochrone problem, which asks for the frictionless path of least descent time between two points in a uniform gravitational field. With the downward vertical coordinate denoted by (y), conservation of mechanical energy gives the speed (\sqrt{2gy}). The travel-time functional is therefore

[ T=\int \sqrt{\frac{1+(y')^2}{2gy}},dx. ]

The associated variational equation reduces to

[ y\bigl(1+(y')^2\bigr)=C, ]

where (C) is constant along the minimizing path. Its parametric solution is a suitably scaled and translated cycloidal arc.

Johann Bernoulli posed the problem publicly in 1696 and related its solution to Fermat’s principle, interpreting the falling particle as moving through horizontal layers with speed determined by depth. Isaac Newton produced a solution after receiving the challenge in 1697, while Jacob Bernoulli supplied a variational analysis that helped distinguish the minimizing curve from neighboring trajectories. The problem became a central early example in which a global extremal condition determines a curve through a differential equation.

The tautochrone and brachistochrone properties concern different functionals and should not be conflated. The tautochrone property fixes a single cycloidal track and compares descent times from different points on it, whereas the brachistochrone problem compares different tracks connecting prescribed endpoints. Their coincidence in the same curve arises from the relation between cycloidal arc length and vertical height rather than from equivalence of the two mechanical questions.

Related curves

A tracing point at distance (d) from the center of a rolling circle generates the broader family

[ x(t)=rt-d\sin t,\qquad y(t)=r-d\cos t. ]

The ordinary cycloid occurs when (d=r). When (d<r), the resulting curtate cycloid has smooth flattened arches because the tracing point lies inside the circle. When (d>r), the prolate cycloid develops loops because the tracing point lies beyond the circumference.

Rolling one circle around another produces the related epicycloid and hypocycloid. These curves share the kinematic principle of superposed rotation and translation, but their base trajectories are circular rather than linear.