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Trochoid

A trochoid is the locus of a point located at a distance b from the center of a circle of radius a as the circle rolls without slipping along a straight line, forming a type of roulette curve that combines uniform translation with circular rotation. The specific shape depends on the ratio of b to a: when b < a, it produces a curtate cycloid with a flattened, wavy profile; when b = a, it generates the classic cycloid, known for its cusps and arches; and when b > a, it yields a prolate cycloid featuring loops and self-intersections. These curves are described parametrically by the equations
x = a\theta - b \sin \theta,
y = a - b \cos \theta,
where \theta is the parameter representing the rolling angle. The study of trochoids traces back to the , with early explorations in geometric constructions, and the variant was named and analyzed by around 1599 for its applications in motion. In the , trochoids gained prominence in for optimizing gear designs, as their smooth profiles reduce friction and wear in tooth fillets generated by rack cutters. Related generalizations of curves occur when the rolling takes place around a fixed circle rather than a line: an results from external contact, producing star-like patterns such as cardioids or nephroids, while a arises from internal contact, yielding roses or ellipses. In practical applications, trochoids appear in mechanical systems like pumps, where their profiles define rotor shapes for efficient fluid displacement, and in mechanisms to ensure uniform coverage. They also inspire educational tools, such as the toy invented in the 1960s, which uses geared disks to draw intricate trochoidal patterns, making abstract accessible. Key properties include variable speed along the curve—maximum |a + b| and minimum |a - b|—and arc lengths often involving elliptic integrals, underscoring their mathematical richness.

Definition

General Concept

A trochoid is the locus of a point fixed at a distance b from the center of a of a that rolls without slipping along a fixed straight line. This curve arises from the combined translational and rotational motion of the as it moves along the line, with the \theta determining the position of the tracing point at each instant. The key parameters governing the trochoid are the rolling circle's radius a, the radial b of the fixed point from the center, and the parameter \theta, which represents the angle through which the circle has rotated. As the circle rolls, the center translates horizontally by a a\theta, while the point's position relative to the center varies with the rotation. Depending on the ratio of b to a, the trochoid exhibits distinct visual forms, such as smooth arches when b < a or loops when b > a. Trochoids form a specific case within the broader class of curves, which are generated by a point attached to one curve rolling without slipping along another fixed curve.

Relation to Roulettes

A is defined as the traced by a fixed point attached to a moving as it rolls without slipping along a fixed . This construction generalizes various plane curves generated through rolling motion, encompassing a broad class of loci in classical . The generation of a relies on the prerequisite of no slipping during the rolling contact, which ensures that the motion decomposes into pure translation along the fixed curve and pure of the moving curve about its instantaneous . This condition maintains tangency and preservation between the curves, fundamental to the geometric integrity of the resulting path. Within this framework, a trochoid arises as a particular roulette where the moving curve is a circle and the fixed curve is a straight line, while epitrochoids and hypotrochoids emerge when the fixed curve is a circle. The term "roulette" itself originates from the French word roulette, meaning "small wheel," introduced by Blaise Pascal in his 1658 treatise Histoire de la Roulette, which explored the cycloid as a roulette curve.

Types

Cycloidal Trochoids

Cycloidal trochoids, also known as trochoids generated by a rolling along a straight line, are a class of curves where the path of the tracing point depends on its radial distance from the of the rolling relative to the 's . These curves arise in the context of a fixed straight line acting as the , resulting in purely translational motion of the rolling 's without rotational variation around a curved fixed . The common trochoid, specifically the , occurs when the tracing point lies on the of the rolling , producing a characterized by a series of cusps where the point touches the fixed line and smooth arches of equal height between them. This configuration yields a periodic with the point momentarily stationary at each cusp, creating the distinctive pointed arches. A curtate trochoid forms when the tracing point is located inside the rolling , resulting in a smoother, undulating without cusps, often described as a contracted or shortened version of the with reduced amplitude in its waves. In contrast, a prolate trochoid arises when the point is outside the , leading to a more extended with transverse loops and self-intersections, where the path crosses itself to form double points. These qualitative differences highlight how the position of the tracing point alters the 's : the balances arches and cusps, the curtate variant appears wavy and contained, and the prolate includes looping extensions that increase in complexity with greater radial distance. A special degenerate case occurs when the tracing point coincides with the center of the rolling circle, producing a straight line parallel to the fixed path, as the point simply translates without oscillation.

Circular Trochoids

Circular trochoids are generated when a circle rolls around the exterior or interior of a fixed circle, with a point attached to the rolling circle tracing the path. This setup contrasts with the straight-line rolling case by introducing curvature to the fixed path, which typically results in closed curves or star-shaped patterns when the ratio of the fixed circle's radius R to the rolling circle's radius a is rational. The key parameters are the radius R of the fixed , the radius a of the rolling , and the distance b from the center of the rolling to the tracing point. When b = a, the tracing point lies on the circumference of the rolling , reducing the curve to an or . These generalize the straight-line trochoids, where the fixed path's infinite radius limit produces open, wavelike paths. Epitrochoids arise when the rolling of radius a moves externally around the fixed of radius R. This configuration produces loops or petal-like extensions outward from the fixed , with the curve closing after a finite number of rotations if R/a is rational. For instance, when R = a and b = a, the epitrochoid forms a cardioid, a heart-shaped with a single cusp. Another notable case occurs when R = 2a and b = a, yielding a , a kidney-shaped featuring two cusps. Hypotrochoids, in contrast, form when the rolling circle moves internally within the fixed circle. Here, the tracing point orbits inside the fixed circle, often creating star-shaped or rosette patterns with inward-pointing cusps for rational R/a. The hypocycloid subclass (b = a) includes the deltoid, obtained when R = 3a, which exhibits three cusps and resembles a three-pointed star. Similarly, the astroid emerges when R = 4a, producing a four-cusped, diamond-like known for its envelope properties in classical .

Parametric Equations

Rolling on a Straight Line

The equations for a trochoid generated by a point at distance b from the center of a of a rolling without slipping along a straight line are derived in a where the fixed line coincides with the horizontal x-axis at y = 0, and the y-axis is vertical, with the circle typically positioned above the line for the standard orientation. The position of the tracing point is the vector sum of the center's location and the point's position relative to the center. The center translates horizontally along the x-axis by a\theta, maintaining a constant y-coordinate of a, so its position is (a\theta, a). The relative position, assuming the initial position of the tracing point is along the toward the contact point (vertical downward) and the circle rotates as it rolls to the right, is (-b \sin \theta, -b \cos \theta). Adding these yields the parametric equations: x(\theta) = a\theta - b \sin \theta, y(\theta) = a - b \cos \theta. When b = a, these simplify to the equations of a standard : x(\theta) = a(\theta - \sin \theta), \quad y(\theta) = a(1 - \cos \theta). For the general case, as \theta ranges from 0 to $2\pi, the curve completes one full arch, with the shape scaling based on the ratio b/a: the arch is contracted for b < a (curtate trochoid) and extended for b > a (prolate trochoid), while the case (b = a) produces the characteristic cusp-to-cusp form. Eliminating the parameter \theta to obtain a Cartesian equation relating x and y directly leads to transcendental equations involving , resulting in no simple in terms of elementary functions.

Rolling on a Fixed Circle

When a circle of radius a rolls around the outside of a fixed circle of radius R, the path traced by a point on the rolling circle at a distance b from its center is an . The parametric equations for this curve are \begin{align*} x(\theta) &= (R + a) \cos \theta - b \cos \left( \left( \frac{R}{a} + 1 \right) \theta \right), \\ y(\theta) &= (R + a) \sin \theta - b \sin \left( \left( \frac{R}{a} + 1 \right) \theta \right), \end{align*} where \theta is the parameter representing the angle of rotation of the center of the rolling circle around the fixed circle. If the rolling circle instead rolls around the inside of the fixed circle, the resulting path is a , with equations \begin{align*} x(\theta) &= (R - a) \cos \theta + b \cos \left( \left( \frac{R}{a} - 1 \right) \theta \right), \\ y(\theta) &= (R - a) \sin \theta - b \sin \left( \left( \frac{R}{a} - 1 \right) \theta \right). \end{align*} These equations assume a < R for the hypotrochoid to avoid intersection issues. The derivation of these equations follows from the position of the tracing point as the sum of the position of the rolling circle's center and the position of the point relative to that center. For the epitrochoid, the center traces a circle of radius R + a with angular parameter \theta, so its position is ((R + a) \cos \theta, (R + a) \sin \theta). The rolling circle rotates by an additional angle \phi = -(R/a) \theta due to the gear ratio k = R/a, leading to the relative position of the point being -b (\cos( (k + 1) \theta ), \sin( (k + 1) \theta )) to account for the opposite rotation direction and initial phase; combining these yields the full parametric form. A similar decomposition applies to the hypotrochoid, where the center traces a circle of radius R - a and the relative rotation is \phi = (R/a) \theta in the same direction, resulting in the adjusted angular multiple k - 1. Special cases arise when b = a, the distance to the tracing point equals the rolling radius. For the epitrochoid, this produces an epicycloid, the path of a point on the circumference of the rolling circle. Similarly, b = a in the hypotrochoid yields a hypocycloid. The curves exhibit periodicity when the gear ratio k = R/a is rational, say k = p/q in lowest terms, closing after \theta advances by $2\pi q; otherwise, the path is dense on an annular region. In the limit as R \to \infty, both epitrochoid and hypotrochoid equations reduce to the parametric form of a trochoid generated by rolling on a straight line, as the fixed circle approximates a line.

Properties

Geometric Characteristics

Trochoids exhibit distinctive geometric features arising from the rolling motion of a circle, including cusps, arches, and patterns of symmetry. In the case of the common cycloid, a specific type of trochoid generated by a point on the circumference of the rolling circle, cusps occur at parameter values θ = 2πn (where n is an integer), corresponding to points where the curve intersects the rolling line at y = 0 and the tangent becomes vertical. These cusps mark the instants when the tracing point contacts the base line, creating sharp points that define the curve's arch-like structure. The arches of vary by type, reflecting the position of the tracing point relative to the rolling circle of radius a. For the common cycloid, each arch rises to a height of 2a, forming smooth, rounded vaults between cusps. In curtate cycloids, where the tracing point lies inside the circle at a distance b < a from the center, the arches are reduced in height to 2b, resulting in a flatter, more elongated profile without cusps. Prolate cycloids, with the point outside the circle at b > a, produce arches extended to height 2b, often featuring self-intersecting loops that add complexity to the curve's outline. Straight-line trochoids, including cycloids, display and periodicity along the direction of rolling. These curves repeat every interval of 2πa in the x-coordinate, generating an infinite sequence of identical arches. Circular trochoids, formed by a circle rolling around a fixed of radius R, exhibit when the ratio R/a is an , resulting in star-like or patterns with the corresponding number of-fold . Closure in these curves occurs after a finite number of rotations when R/a is a p/q in lowest terms, producing a bounded, non-repeating that tiles the plane periodically; otherwise, the curve is dense and non-closing. A notable envelope property distinguishes the among trochoids: its , the locus of centers, is another congruent to the original but translated, typically shifted vertically by -2a. This self-similar underscores the cycloid's unique geometric harmony in classical curve theory.

Analytic Features

The analytic features of trochoids are derived from their parametric representations using techniques from calculus and differential geometry, yielding explicit expressions for quantities such as arc length, enclosed areas, and curvature. These properties highlight the trochoid's smooth variation except at singular points, with the cycloid serving as a canonical example where closed-form results are particularly accessible. For the cycloid, a special trochoid with generating point on the circumference (b = a), the arc length L of one complete arch (from \theta = 0 to \theta = 2\pi) is found via the parametric arc length integral L = \int_{0}^{2\pi} \sqrt{\left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2} \, d\theta. Substituting the derivatives \frac{dx}{d\theta} = a(1 - \cos \theta) and \frac{dy}{d\theta} = a \sin \theta simplifies the integrand to a \sqrt{2(1 - \cos \theta)} = 2a \left| \sin \frac{\theta}{2} \right|, yielding L = 8a. The area A under one arch of the , relative to the directrix, is computed as A = \int_{0}^{2\pi a} y \, dx = \int_{0}^{2\pi} y(\theta) \frac{dx}{d\theta} \, d\theta = a^2 \int_{0}^{2\pi} (1 - \cos \theta)^2 \, d\theta = 3\pi a^2. This result, three times the area of the generating circle, underscores the 's efficient spatial coverage. The \kappa(\theta) of the is given by \kappa(\theta) = \frac{\left| x' y'' - y' x'' \right|}{\left( (x')^2 + (y')^2 \right)^{3/2}} = \frac{1}{4a \sin(\theta/2)}, for $0 < \theta < 2\pi, where the second derivatives are x'' = a \sin \theta and y'' = a \cos \theta. The is thus \rho(\theta) = 4a \sin(\theta/2), which approaches zero (infinite ) at the cusps (\theta = 0, 2\pi), reflecting the curve's sharp turns there; analysis often begins at these cusps due to their geometric prominence. The pedal curve of the , taken with respect to a cusp, is a straight line to the directrix at a $2a$ from it. In terms of asymptotic behavior, trochoids generated by rolling along a straight line, like the , are unbounded and extend infinitely in the direction of motion. In contrast, trochoids from rolling around a fixed circle (epitrochoids or hypotrochoids) are typically bounded within an annular region, becoming closed curves when the ratio of the fixed circle radius to the rolling circle radius (R/a) is rational.

Applications

In Mechanics

In mechanics, trochoids describe the trajectories of points attached to rolling objects, providing fundamental insights into motion under constraints like no-slip conditions. The path traced by a point on the rim of a wheel rolling without slipping along a straight line is a cycloid, a special case of the trochoid family, while points inside or outside the rim generate curtate or prolate trochoids, respectively. This geometric property explains the characteristic cusps and arches in wheel paths, where the instantaneous velocity at the contact point is zero during pure rolling. In vehicle dynamics, the trochoid trajectory illustrates ideal rolling motion; however, during turns, skidding occurs when lateral friction forces exceed available tire grip, causing the wheel to slip and deviate from the expected trochoidal path, leading to understeer or oversteer. Trochoids also play a key role in kinematic analysis of mechanisms involving rolling contact. In bicycles during uniform forward motion, the path of a pedal relative to the ground forms a curtate cycloid, as the crank rotates around the moving bottom bracket, combining translational velocity of the frame with rotational motion of the crank. In theoretical mechanics, the inverted cycloid solves the brachistochrone problem, identified by Johann Bernoulli in 1696 as the curve of fastest descent for a particle sliding under gravity from one point to another, outperforming straight lines or other paths due to its balance of potential energy loss and velocity gain. Bernoulli's solution, derived via the calculus of variations, demonstrates that the cycloid minimizes travel time by enabling higher speeds in steeper sections. Practical applications in machinery leverage trochoidal profiles for efficient and reduced . Cycloidal employ trochoid-based profiles, such as epicycloids and hypocycloids, to achieve conjugate action and smooth meshing, minimizing backlash and compared to ; this is particularly valued in instruments like clocks, where uniform extends , and in pumps, where low-noise operation supports reliable fluid handling. In rotary engines, such as the Wankel , the approximates a envelope, enabling compact, high-speed operation with three combustion chambers formed by the rolling motion inside an epitrochoidal housing.

In Design and Graphics

In , trochoids are generated using equations to create intricate patterns for animations and visualizations, with the toy serving as a classic example that simulates hypotrochoids through geared rolling circles. The form allows for efficient plotting of these curves in software, enabling dynamic rendering of looping paths in educational simulations and artistic designs. This approach has influenced digital recreations of effects, where hypotrochoids emerge from fixed and rolling circle interactions, fostering interactive graphics in tools like or Java-based environments. The adoption of trochoidal curves extends to visual arts, where Spirograph-inspired hypotrochoids have shaped patterns in , , and movements since the 1960s, emphasizing repetitive, hypnotic geometries. In , algorithms based on trochoid generation produce (SVG) exports for patterns, as seen in visualizations that adapt hypotrochoid forms for abstract representations. Architectural design incorporates cycloidal trochoids for aesthetic and functional curves, such as in pendulum clocks where the tautochrone property ensures isochronous motion; Christiaan Huygens implemented cycloidal cheeks in his 1673 clock design to constrain the pendulum bob to a cycloid path, enhancing timekeeping accuracy. Cycloidal arches also appear in bridge structures for their efficient load distribution and elegant form. In (CAD) and , epitrochoid paths guide CNC machining for precise profiles, enabling smooth, constant-velocity motion in mechanical components like pumps and drives. These profiles are interpolated in during milling, as detailed in algorithms for trochoidal pockets that integrate boundaries directly into CNC cycles for high-precision parts. Such applications leverage parametric definitions to minimize and achieve complex geometries unattainable with linear paths. Analytic arc lengths of trochoids aid in precise rendering for these design contexts, ensuring accurate scaling in both simulations and physical outputs.

History

Early Developments

The specific curve now recognized as the cycloid—a fundamental trochoid formed by a point on a circle rolling along a straight line—was first systematically noted by Galileo Galilei in 1599, who named it from the Greek kyklos (circle) and eidos (form). Galileo studied the cycloid for over 40 years, attempting to determine its area by comparing weights of metal pieces cut to its shape against those of the generating circle, estimating the ratio as approximately 3:1 but unable to prove it irrational analytically; in a 1639 letter to Evangelista Torricelli, he described it as a path traced by a point on a wheel's rim. In the 1620s and 1630s, provided the first detailed mathematical treatment of the , defining it precisely as the locus of a point on a rolling circle and noting that the length of each arch's base equals the circle's circumference (2πa, where a is the radius). posed the challenge of finding the 's area to de Roberval in 1628, leading to Roberval's 1634 solution via Cavalieri's method, establishing the area of one arch as 3πa². Christiaan Huygens advanced the study significantly in the 1670s, proving in his 1673 publication Horologium Oscillatorium sive de motu pendulorum that the is the tautochrone—the curve along which a body slides under gravity in equal times regardless of starting position—building on its properties to design clocks with cycloidal cheeks for isochronous swings. This work integrated the into practical mechanics, influencing horology. In 1696, posed the brachistochrone problem—finding the curve of fastest descent between two points under gravity—which was solved independently by Bernoulli, , , and , all identifying the as the solution. This demonstration elevated the 's status in the emerging field of . The 17th-century rise of , particularly ' coordinate methods in (1637), enabled parametric equations for such curves, shifting focus from geometric construction to algebraic description and facilitating trochoid analysis.

Modern Contributions

In the 20th century, significant advancements in the mathematical representation of trochoids facilitated their integration into computational design and . A key contribution was the development of rational representations for segments of epitrochoids and hypotrochoids, enabling precise modeling in (CAD) systems. This method, which combines circular arcs using Bernstein-like trigonometric basis functions, produces even-degree rational Bézier curves whose control points and weights are computed via discrete convolutions, addressing a prior gap in trochoid parameterization for graphical software. Building on this, early 21st-century research explored alternative geometric formulations to simplify the study of trochoid properties. In , Sibel Pasali introduced map equations for trochoids, representing them as mappings of the unit circle in the . This approach proved necessary and sufficient conditions for trochoids to form —curves passing through the with radial symmetry—and analyzed rosette petals, while linking cusp formation to generalized discriminants from broader curve theory. More recent work has incorporated physical interpretations to derive novel parametric forms. In 2024, H. Arbab and A. Arbab proposed a physics-oriented redefining centered trochoids as paths traced by points on a circle undergoing controlled combinations of rolling and sliding motions along a fixed circle. Using techniques like the Virtual Rotating Circles Technique and Virtual Sliding Simulator, they derived parametric equations that unify trochoids with co-centered ellipses, generated by co-polarized rotations at commensurable angular frequencies, offering new insights into their construction beyond pure rolling. These contributions have extended trochoid applications in , such as modeling meshing in trochoidal gear profiles for pumps and reducers, where general mathematical frameworks account for clearances and contact dynamics. Overall, modern efforts emphasize computational efficiency, geometric generalizations, and interdisciplinary links, enhancing trochoids' utility in , , and .