Cycloid

Equations

The cycloid through the origin, generated by a circle of radius r, in the upper half-plane, rolling over the x-axis, consists of the points (x, y), with $${\displaystyle \begin{array}{cc}x& =r(t-\sin t)\\ y& =r(1-\cos t),\end{array}}$$ where t is a real parameter corresponding to the angle through which the rolling circle has rotated. For given t, the circle's centre lies at (x, y) = (rt, r).

The Cartesian equation is obtained by solving the y-equation for t and substituting into the x-equation:$${\displaystyle x=r{\cos }^{-1}\left(1-\frac{y}{r}\right)-\sqrt{y(2r-y)},}$$or, eliminating the multiple-valued inverse cosine:

${\displaystyle r\cos \left(\frac{x+\sqrt{y(2r-y)}}{r}\right)+y=r.}$

When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps on the x-axis, with the derivative tending toward ${\displaystyle \mathrm{\infty }}$ or ${\displaystyle -\mathrm{\infty }}$ near a cusp (where y=0). The map from t to (x, y) is differentiable, in fact of class C^∞, with derivative 0 at the cusps.

The slope of the tangent to the cycloid at the point ${\displaystyle (x,y)}$ is given by $\frac{dy}{dx}=\cot (\frac{t}{2})$.

A cycloid segment from one cusp to the next is called an arch of the cycloid, for example the points with ${\displaystyle 0\le t\le 2\pi }$ and ${\displaystyle 0\le x\le 2\pi }$.

Considering the cycloid as the graph of a function ${\displaystyle y=f(x)}$, it satisfies the differential equation:

${\displaystyle {\left(\frac{dy}{dx}\right)}^2=\frac{2r}{y}-1.}$

If we define ${\displaystyle h=2r-y=r(1+\cos t)}$ as the height difference from the cycloid's vertex (the point with a horizontal tangent and ${\displaystyle \cos t=-1}$), then we have:

${\displaystyle {\left(\frac{dx}{dh}\right)}^2=\frac{2r}{h}-1.}$

Involute

The involute of the cycloid has exactly the same shape as the cycloid it originates from. This can be visualized as the path traced by the tip of a wire initially lying on a half arch of the cycloid: as it unrolls while remaining tangent to the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length).

Demonstration

This demonstration uses the rolling-wheel definition of cycloid, as well as the instantaneous velocity vector of a moving point, tangent to its trajectory. In the adjacent picture, ${\displaystyle P_1}$ and ${\displaystyle P_2}$ are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, ${\displaystyle P_1}$ and ${\displaystyle P_2}$ coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, ${\displaystyle P_1}$ and ${\displaystyle P_2}$ traverse two cycloid curves. Considering the red line connecting ${\displaystyle P_1}$ and ${\displaystyle P_2}$ at a given time, one proves the line is always tangent to the lower arc at ${\displaystyle P_2}$ and orthogonal to the upper arc at ${\displaystyle P_1}$. Let ${\displaystyle Q}$ be the point in common between the upper and lower circles at the given time. Then:

• ${\displaystyle P_1,Q,P_2}$ are colinear: indeed the equal rolling speed gives equal angles ${\displaystyle \widehat{P_1{O}_{1}Q}=\widehat{P_2{O}_{2}Q}}$, and thus ${\displaystyle \widehat{O_{1}Q{P}_1}=\widehat{O_{2}Q{P}_2}}$ . The point ${\displaystyle Q}$ lies on the line ${\displaystyle O_1{O}_2}$ therefore ${\displaystyle \widehat{P_{1}Q{O}_1}+\widehat{P_{1}Q{O}_2}=\pi }$ and analogously ${\displaystyle \widehat{P_{2}Q{O}_2}+\widehat{P_{2}Q{O}_1}=\pi }$. From the equality of ${\displaystyle \widehat{O_{1}Q{P}_1}}$ and ${\displaystyle \widehat{O_{2}Q{P}_2}}$ one has that also ${\displaystyle \widehat{P_{1}Q{O}_2}=\widehat{P_{2}Q{O}_1}}$. It follows ${\displaystyle \widehat{P_{1}Q{O}_1}+\widehat{P_{2}Q{O}_1}=\pi }$ .
• If ${\displaystyle A}$ is the meeting point between the perpendicular from ${\displaystyle P_1}$ to the line segment ${\displaystyle O_1{O}_2}$ and the tangent to the circle at ${\displaystyle P_2}$ , then the triangle ${\displaystyle P_{1}A{P}_2}$ is isosceles, as is easily seen from the construction: ${\displaystyle \widehat{Q{P}_{2}A}=\frac{1}{2}\widehat{P_2{O}_{2}Q}}$ and ${\displaystyle \widehat{Q{P}_{1}A}=\frac{1}{2}\widehat{Q{O}_{1}R}=}$${\displaystyle \frac{1}{2}\widehat{Q{O}_1{P}_1}}$ . For the previous noted equality between ${\displaystyle \widehat{P_1{O}_{1}Q}}$ and ${\displaystyle \widehat{Q{O}_2{P}_2}}$ then ${\displaystyle \widehat{Q{P}_{1}A}=\widehat{Q{P}_{2}A}}$ and ${\displaystyle P_{1}A{P}_2}$ is isosceles.
• Drawing from ${\displaystyle P_2}$ the orthogonal segment to ${\displaystyle O_1{O}_2}$, from ${\displaystyle P_1}$ the straight line tangent to the upper circle, and calling ${\displaystyle B}$ the meeting point, one sees that ${\displaystyle P_{1}A{P}_{2}B}$ is a rhombus using the theorems on angles between parallel lines
• Now consider the velocity ${\displaystyle V_2}$ of ${\displaystyle P_2}$ . It can be seen as the sum of two components, the rolling velocity ${\displaystyle V_a}$ and the drifting velocity ${\displaystyle V_d}$, which are equal in modulus because the circles roll without skidding. ${\displaystyle V_d}$ is parallel to ${\displaystyle P_{1}A}$, while ${\displaystyle V_a}$ is tangent to the lower circle at ${\displaystyle P_2}$ and therefore is parallel to ${\displaystyle P_{2}A}$. The rhombus constituted from the components ${\displaystyle V_d}$ and ${\displaystyle V_a}$ is therefore similar (same angles) to the rhombus ${\displaystyle B{P}_{1}A{P}_2}$ because they have parallel sides. Then ${\displaystyle V_2}$, the total velocity of ${\displaystyle P_2}$, is parallel to ${\displaystyle P_2{P}_1}$ because both are diagonals of two rhombuses with parallel sides and has in common with ${\displaystyle P_1{P}_2}$ the contact point ${\displaystyle P_2}$. Thus the velocity vector ${\displaystyle V_2}$ lies on the prolongation of ${\displaystyle P_1{P}_2}$ . Because ${\displaystyle V_2}$ is tangent to the cycloid at ${\displaystyle P_2}$, it follows that also ${\displaystyle P_1{P}_2}$ coincides with the tangent to the lower cycloid at ${\displaystyle P_2}$.
• Analogously, it can be easily demonstrated that ${\displaystyle P_1{P}_2}$ is orthogonal to ${\displaystyle V_1}$ (the other diagonal of the rhombus).
• This proves that the tip of a wire initially stretched on a half arch of the lower cycloid and fixed to the upper circle at ${\displaystyle P_1}$ will follow the point along its path without changing its length because the speed of the tip is at each moment orthogonal to the wire (no stretching or compression). The wire will be at the same time tangent at ${\displaystyle P_2}$ to the lower arc because of the tension and the facts demonstrated above. (If it were not tangent there would be a discontinuity at ${\displaystyle P_2}$ and consequently unbalanced tension forces.)

Measurement

Area

Using the above parameterization $x=r(t-\sin t)$ and $y=r(1-\cos t)$, the area under one arch, ${\displaystyle 0\le t\le 2\pi ,}$ is given by: $${\displaystyle A={\int }_{x=0}^{2\pi r}ydx={\int }_{t=0}^{2\pi }{r}^2(1-\cos t{)}^{2}dt=3\pi r^2.}$$ This is three times the area of the rolling circle.

Arc length

The arc length S of one arch is given by $${\displaystyle \begin{array}{cc}S& ={\int }_0^{2\pi }\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt\\ & ={\int }_0^{2\pi }r\sqrt{2-2\cos t}dt\\ & =2r{\int }_0^{2\pi }\sin \frac{t}{2}dt\\ & =8r.\end{array}}$$

Another geometric way to calculate the length of the cycloid is to notice that when a wire describing an involute has been completely unwrapped from half an arch, it extends itself along two diameters, a length of 4r. This is thus equal to half the length of arch, and that of a complete arch is 8r.

From the cycloid's vertex (the point with a horizontal tangent and ${\displaystyle \cos t=-1}$) to any point within the same arch, the arc length squared is ${\displaystyle 8r^2(1+\cos t)}$, which is proportional to the height difference ${\displaystyle r(1+\cos t)}$; this property is the basis for the cycloid's isochronism. In fact, the arc length squared is equal to the height difference multiplied by the full arch length 8r.

Cycloidal pendulum

If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the string is constrained to be tangent to one of its arches, and the pendulum's length L is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle, L = 4r), the bob of the pendulum also traces a cycloid path. Such a pendulum is isochronous, with equal-time swings regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by: $${\displaystyle \begin{array}{cc}x& =r[2\theta (t)+\sin 2\theta (t)]\\ y& =r[-3-\cos 2\theta (t)],\end{array}}$$ where ${\displaystyle \theta }$ is the angle that the straight part of the string makes with the vertical axis, and is given by $${\displaystyle \sin \theta (t)=A\cos (\omega t),{\omega }^2=\frac{g}{L}=\frac{g}{4r},}$$ where A < 1 is the "amplitude", ${\displaystyle \omega }$ is the radian frequency of the pendulum and g the gravitational acceleration.

The 17th-century Dutch mathematician Christiaan Huygens discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be used in navigation.

Related curves

Several curves are related to the cycloid.

• Trochoid: generalization of a cycloid in which the point tracing the curve may be inside the rolling circle (curtate) or outside (prolate).
• Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line.
• Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line.
• Hypotrochoid: generalization of a hypocycloid where the generating point may not be on the edge of the rolling circle.
• Epitrochoid: generalization of an epicycloid where the generating point may not be on the edge of the rolling circle.

All these curves are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the similitude ratio of curve to evolute is 1 + 2q.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Other uses

The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used by Wallace K. Harrison in the design of the Hopkins Center at Dartmouth College in Hanover, New Hampshire.

Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves. Later work indicates that curtate cycloids do not serve as general models for these curves, which vary considerably.