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Epitrochoid

An epitrochoid is a plane curve traced by a fixed point on a circle of radius b as it rolls without slipping around the exterior of a fixed circle of radius a, with the tracing point located at a distance h from the center of the rolling circle.^[1]^[2] The parametric equations for an epitrochoid are given by

x = (a + b) \cos t - h \cos \left( \frac{a + b}{b} t \right), \quad y = (a + b) \sin t - h \sin \left( \frac{a + b}{b} t \right),

where t is the parameter representing the angle of rotation.^[1] These equations describe the locus of the point as the rolling circle completes its motion, producing a variety of looped or cardioid-like shapes depending on the ratios of a, b, and h.^[2] Special cases include the epicycloid, which occurs when h = b (the tracing point lies on the circumference of the rolling circle), and the limaçon when a = b.^[1] Epitrochoids are closely related to hypotrochoids (where the rolling occurs inside the fixed circle) and form the basis for patterns generated by toys like the Spirograph, which simulate these curves using geared disks.^[1]^[3] Historically, epitrochoids were first illustrated by Albrecht Dürer in his 1525 treatise Underweysung der Messung mit dem Zirckel und Richtscheyt, where he referred to certain instances as "spider lines" due to their web-like appearance.^[1]^[2] The curves were further analyzed by mathematicians such as Philippe de La Hire, Girard Desargues, Gottfried Wilhelm Leibniz, Isaac Newton, and Leonhard Euler in the 17th and 18th centuries, contributing to the study of roulettes and cycloidal motions.^[2] In modern contexts, epitrochoids appear in engineering applications, such as the design of gerotor pumps and gear mechanisms, where their smooth, periodic paths facilitate efficient fluid displacement.^[4]

Definition and Geometry

Basic Definition

An epitrochoid is a roulette curve generated by the path traced by a fixed point attached to a circle of radius b that rolls without slipping around the exterior of a fixed circle of radius a, where the tracing point is located at a distance h from the center of the rolling circle.^[1] The key parameters defining the epitrochoid are the radius a of the fixed circle, the radius b of the rolling circle, and the distance h from the center of the rolling circle to the tracing point, typically satisfying $0 < h \leq b in standard configurations.^[1]^[5] Unlike the hypotrochoid, which involves a circle rolling internally around the fixed circle, the epitrochoid specifically describes external rolling motion.^[1]^[6] The resulting curve's appearance varies with the parameters: when h < b, it forms a smooth, loop-free path; when h = b, it produces cusps characteristic of an epicycloid; and when h > b, loops emerge.^[5] Additionally, the overall shape, including the number of loops or cusps, depends on the ratio k = a / b; for rational values of k, the curve closes after a finite number of rotations, often exhibiting rotational symmetry.^[7]^[2]

Geometric Construction

The geometric construction of an epitrochoid begins with a fixed circle of radius a centered at the origin. A second circle of radius b is positioned externally tangent to the fixed circle, with its center initially at (a + b, 0). As the rolling circle moves around the exterior of the fixed circle without slipping, its center traces a circular path of radius a + b centered at the origin. A point located at a fixed distance h from the center of the rolling circle—either on the circumference (h = b), inside (h < b), or outside (h > b)—traces the epitrochoid curve during this motion.^[1]^[2] The rolling motion involves two coupled rotations to maintain contact without slipping. The center of the rolling circle revolves around the fixed center by an angle \theta, measured from the positive x-axis. Simultaneously, the rolling circle rotates about its own center by an additional angle \phi = (a/b) \theta in the same direction, ensuring that the arc length traversed on the fixed circle (a \theta) matches the arc length unrolled on the rolling circle (b \phi). The total orientation of the vector from the rolling circle's center to the tracing point is thus \theta + \phi = ((a + b)/b) \theta, which determines the curve's oscillatory pattern.^[8]^[5] The shape of the resulting epitrochoid is strongly influenced by the parameters a, b, and h. When the ratio k = a/b is rational, the curve closes upon itself after a finite number of rotations, forming a periodic pattern with a number of lobes equal to the numerator p in the reduced fraction k = p/q (for the case h = b, yielding an epicycloid).^[9]^[10] For instance, with k = 3 (or a = 3b), the curve exhibits three lobes. Conversely, if k is irrational, the tracing point densely fills an annular region between concentric circles of radii |a + b - h| and a + b + h, without ever closing.^[1]^[7]^[5] In diagrams illustrating this construction, the fixed circle appears as a static outline, the path of the rolling circle's center as a dashed larger circle of radius a + b, and the rolling circle itself shown at successive positions with the tracing point marked, highlighting the external tangency and the evolving curve.^[2]

Mathematical Formulation

Parametric Equations

The parametric equations for an epitrochoid describe the path traced by a point at a fixed distance from the center of a circle of radius r that rolls externally around a fixed circle of radius R, without slipping.^[1] These equations are given by

\begin{align} x(\theta) &= (R + r) \cos \theta - d \cos \left( \frac{R + r}{r} \theta \right), \\ y(\theta) &= (R + r) \sin \theta - d \sin \left( \frac{R + r}{r} \theta \right), \end{align}

where \theta is the parameter representing the rotation angle of the center of the rolling circle, R > 0 is the radius of the fixed circle, r > 0 is the radius of the rolling circle, and $0 \leq d \leq r is the distance from the center of the rolling circle to the tracing point.^[1] These equations arise from vector addition of the position of the rolling circle's center and the position of the tracing point relative to that center. The center traces a circle of radius R + r as it moves with angular parameter \theta, contributing the term (R + r) (\cos \theta, \sin \theta). The tracing point, offset by distance d, undergoes an additional rotation relative to the center by angle -\frac{R + r}{r} \theta due to the combined orbital motion and the rolling without slipping, yielding the offset terms -d \left( \cos \left( \frac{R + r}{r} \theta \right), \sin \left( \frac{R + r}{r} \theta \right) \right).^[1] The curve closes after a finite interval when the ratio k = R/r is rational; specifically, if k = p/q in lowest terms with integers p and q, then \theta ranges from 0 to $2\pi q to complete the closed path.^[1] Restricting $0 \leq d \leq r ensures the tracing point remains within or on the rolling circle.^[1] For example, with R = 1, r = 1, and d = 1, the equations simplify to those of a cardioid: x(\theta) = 2 \cos \theta - \cos(2\theta) and y(\theta) = 2 \sin \theta - \sin(2\theta).^[1]

Key Properties

Epitrochoids possess notable symmetry and periodic properties derived from their parametric formulation. When the ratio k = R/r is an integer, the curve exhibits rotational symmetry of order k, meaning it maps onto itself under rotations by $2\pi / k. Additionally, the curve is periodic when R/r is rational, say k = R/r = p/q in lowest terms, with fundamental period $2\pi q for the parameter \theta, matching the closure condition described in the parametric equations. Cusps appear specifically when d = r and k is an integer, marking points where the curve touches the fixed circle tangentially. The curvature \kappa(\theta) of an epitrochoid is computed using the parametric derivatives, given by

\kappa(\theta) = \frac{ \dot{x}(\theta) \ddot{y}(\theta) - \dot{y}(\theta) \ddot{x}(\theta) }{ \left( \dot{x}(\theta)^2 + \dot{y}(\theta)^2 \right)^{3/2} },

where \dot{x} = dx/d\theta, \dot{y} = dy/d\theta, \ddot{x} = d^2x/d\theta^2, and \ddot{y} = d^2y/d\theta^2 are derived from the parametric equations x(\theta) = (R + r) \cos \theta - d \cos \left( \frac{R + r}{r} \theta \right) and y(\theta) = (R + r) \sin \theta - d \sin \left( \frac{R + r}{r} \theta \right). The arc length L over one full period is obtained via the standard parametric integral

L = \int_0^{T} \sqrt{ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 } \, d\theta,

where T is the period, though closed-form expressions are generally unavailable and require numerical evaluation. Regarding singularities and envelopes, epitrochoids develop self-intersections at points where the parametric equations satisfy f(t) = f(s) for distinct t, s, particularly when (R + r)/r is rational, leading to isolated crossing points. These occur more prominently in prolate cases (d > r), where the curve forms loops that may intersect. The envelope of a family of epitrochoids, parameterized by varying the tracing point distance, is itself another epitrochoid, determined by solving the system of the curve equations and their partial derivatives with respect to the parameter.^[11] Epitrochoids serve as a generalization of other classical curves: when d = r, the curve reduces to an epicycloid with cusps along the path; when d = 0, it simplifies to a circle of radius R + r. These relations highlight the epitrochoid's role in unifying roulette curve families.

Special Cases

Epicycloid

An epicycloid is a special case of the epitrochoid where the tracing point lies on the circumference of the rolling circle, corresponding to the parameter d = r. In this configuration, the parametric equations simplify from the general epitrochoid form. The position of the tracing point is given by:

\begin{align} x(\theta) &= (R + r) \cos \theta - r \cos \left( \frac{R + r}{r} \theta \right), \\ y(\theta) &= (R + r) \sin \theta - r \sin \left( \frac{R + r}{r} \theta \right), \end{align}

where R is the radius of the fixed circle, r is the radius of the rolling circle, and \theta is the rolling angle.^[12] Geometrically, the epicycloid exhibits distinct traits compared to the general epitrochoid. Unlike epitrochoids with d > r, which can form loops, the epicycloid produces no such loops due to the tracing point's position on the rolling circle's boundary. The curve always touches the fixed circle at its cusps, where the tracing point momentarily aligns with the point of contact between the circles. The number of cusps equals the number of divisions of the fixed circle by the rolling circle's path; specifically, for a rational ratio k = R/r = n/m in lowest terms, the epicycloid closes after m full rotations of the rolling circle and features n cusps. These cusps occur at points where the radius vector from the fixed center reaches length R.^[12]^[13] This subclass of epitrochoid is notable for its cusp-driven structure, which has been applied in designing gear teeth profiles to achieve smooth meshing.^[12]

Cardioid

The cardioid arises as a special instance of the epitrochoid when the radii of the fixed circle and the rolling circle are equal, R = r, and the tracing point lies on the circumference of the rolling circle, so d = r. This setup produces a distinctive heart-shaped curve characterized by its smooth, dimpled profile and a prominent cusp.^[14] The generation process involves the rolling circle rotating externally around the fixed circle without slipping, with the point on the rim tracing the path as the centers separate by $2r.^[15] The parametric equations describing this curve are

x(\theta) = 2r (1 - \cos \theta) \cos \theta, \quad y(\theta) = 2r (1 - \cos \theta) \sin \theta,

where \theta ranges from 0 to $2\pi.^[14] In polar coordinates, it takes the form \rho = 2r (1 - \cos \phi).^[15] These equations yield a curve symmetric about the x-axis, with the cusp located at the origin corresponding to \theta = 0. The cardioid belongs to the broader limaçon family, serving as the transitional case where the parameter ratio causes the inner loop to contract into a single point of tangency at the cusp. Key properties include the presence of exactly one cusp at the origin, marking the point of initial contact between the circles. The enclosed area of the cardioid is $6 \pi r^2.^[14] As a limiting case of the epicycloid with one cusp, it exemplifies the simplest non-circular epitrochoid.^[12]

Nephroid

The nephroid is a special case of the epitrochoid generated when the fixed circle has radius R = 2r and the rolling circle has radius r, with the tracing point located at a distance d = r from the center of the rolling circle, placing it on the rim.^[16] This configuration produces a kidney-shaped curve with two cusps, distinguishing it from other epitrochoids like the cardioid.^[17] The parametric equations for the nephroid are given by

x(\theta) = 3r \cos \theta - r \cos 3\theta, \quad y(\theta) = 3r \sin \theta - r \sin 3\theta,

where \theta ranges from 0 to $2\pi, and r is the radius of the rolling circle.^[16] Geometrically, the curve arises from rolling a circle of radius r around the outside of a fixed circle of radius $2r, tracking the path of a point on the circumference of the rolling circle.^[18] The nephroid can also be expressed as an epicycloid formed by this rolling motion, and equivalently as the envelope of diameters of the rolling circle in the generation of a cardioid from two equal circles.^[19] Key properties of the nephroid include its two sharp cusps, located at the points corresponding to \theta = 0 and \theta = \pi, which mark the points where the tracing point touches the fixed circle. The total arc length of the nephroid is L = 24r.^[16] In optics, the nephroid serves as a caustic curve, specifically the envelope of rays reflected from a circular boundary under parallel incident light, such as those observed at the bottom of a cylindrical glass illuminated obliquely.^[20] This reflective property highlights its role in the concentration of light rays along the curve's boundary.^[21]

History and Development

Origins in Roulette Curves

The study of roulette curves, encompassing epitrochoids as a key variant, originated with early explorations of the cycloid, the path traced by a point on a circle rolling along a straight line. In 1599, Galileo Galilei named the cycloid and began a decades-long investigation into its properties, including an attempt to compute its area, which he estimated as three times that of the generating circle. Marin Mersenne advanced this work in 1628 by providing the first precise definition of the cycloid and coining the term "roulette" to describe curves generated by such rolling motions, distinguishing them from static geometric constructions. These foundational efforts laid the conceptual groundwork for more complex roulettes like epitrochoids, though the specific external rolling mechanism of epitrochoids emerged later. An early visual representation of an epitrochoid appeared in Albrecht Dürer's 1525 treatise Underweysung der Messung mit dem Zirckel und Richtscheyt (Instruction in Measurement with Compass and Straight Edge), where he depicted the curve using intersecting lines that evoked a spider's web, leading him to call such figures "spider lines." In the 17th century, Girard Desargues examined epitrochoids in 1640 as part of his broader geometric inquiries. Christiaan Huygens studied epitrochoids around 1679. Further contributions came from Philippe de La Hire (1694), Gottfried Wilhelm Leibniz, Isaac Newton (1686), and Leonhard Euler (1745, 1781), who analyzed properties of epitrochoids and related roulettes. Evangelista Torricelli, Galileo's student, advanced the study of roulette curves through his work on the cycloid, while John Wallis examined its properties and historical context, including epicycloids—a subset of epitrochoids where the tracing point lies on the rolling circle's rim—through their analyses of curve properties and areas.^[2]^[1] Epitrochoids differ from cycloids, which roll along a fixed straight line, and from hypocycloids, generated by internal rolling within a fixed circle, highlighting the roulette family's diversity based on the fixed path's geometry. The modern term "epitrochoid" was coined in the 19th century, derived from the Greek epi ("upon" or "over") and trochos ("wheel"), to precisely denote the external rolling configuration.

Modern Mathematical Analysis

In the 19th century, epitrochoids received formal mathematical treatment through their integration into cycloidal gear theory, as detailed by Robert Willis in his seminal work Principles of Mechanism, where he analyzed the geometric profiles generated by rolling circles to optimize tooth forms for smooth meshing.^[22] This inclusion highlighted the parametric representation of epitrochoids as essential for engineering applications, marking a shift toward rigorous algebraic descriptions of the curves.^[23] By mid-century, mathematicians had refined these parameterizations, establishing epitrochoids as a subclass of roulette curves with explicit equations derived from the rolling motion.^[1] The 20th century brought advancements in differential geometry, with Robert C. Yates's Curves and Their Properties (1952) providing detailed analyses of epitrochoid curvature, including formulas for radius of curvature and evolute properties to quantify local bending.^[24] This work emphasized invariants such as total curvature, facilitating deeper study of the curve's geometric behavior under transformations. Computer-assisted plotting emerged in the 1960s, enabling visualization of complex epitrochoid families through early digital plotters and simulation software, which accelerated exploration beyond manual construction.^[25] Key contributions underscore the close ties between epitrochoids and epicycloids, as epitrochoids generalize the latter when the tracing point lies on the rolling circle's circumference, a connection extensively documented in mathematical histories.^[2] Modern texts, such as Alfred Gray's Modern Differential Geometry of Curves and Surfaces with Mathematica (1997), extend this analysis by computing differential invariants like geodesic curvature using symbolic tools, bridging classical parameterization to computational geometry.^[26] Over time, the study evolved from descriptive geometry to symbolic computation, allowing precise evaluation of arc lengths via elliptic integrals and envelopes through envelope theorems in gear design contexts.^[27] These parametric equations emerged as a direct outcome of such analyses, enabling versatile modeling in both theoretical and applied settings.^[2]

Applications and Uses

In Mechanical Engineering

In mechanical engineering, epitrochoids find prominent application in the design of the Wankel rotary engine, where the engine housing bore follows an epitrochoidal profile generated with a fixed circle radius R and rolling circle radius r in the ratio R = 2r, along with a specific offset distance d tailored to the triangular rotor shape.^[28] This configuration enables a compact design by allowing the three-lobed rotor to perform continuous rotary motion without reciprocating parts, converting combustion pressure directly into shaft rotation for improved power density compared to piston engines.^[29] Epitrochoidal profiles also appear in early 20th-century positive displacement air compressor designs, such as trochoidal rotary compressors, where the rotor traces an epitrochoid within the housing to achieve sealed chambers for efficient air intake and compression.^[30] These machines utilize the curve's geometry to maintain constant volume displacement during operation, providing smooth, pulsation-free output suitable for industrial pneumatic systems. In epicyclic gear systems, epitrochoid-based tooth profiles, often as modified cycloidal curves, are employed in planetary transmissions to enhance meshing characteristics, particularly in high-torque applications like automotive differentials and robotic drives.^[31] By generating tooth shapes via epitrochoid envelopes, these profiles minimize interference and distribute contact stresses evenly, thereby reducing wear and extending service life under load.^[32] Overall, epitrochoids offer advantages in these engineering contexts through smooth rolling motion derived from their parametric geometry, enabling constant volume sealing and reliable operation when R/r ratios are optimized to prevent rotor-housing interference.^[23]

In Toys and Visualization

The Spirograph toy, invented by British engineer Denys Fisher in the 1960s, is a popular geometric drawing device that generates epitrochoids and hypotrochoids through the interaction of geared rings.^[33] It features a fixed outer ring of radius R and a smaller rolling disk of radius r with evenly spaced teeth for meshing; a pen inserted at a distance d from the disk's center traces the curve as the disk rolls around the fixed ring.^[34] This mechanical setup allows users, particularly children, to create intricate, symmetrical patterns by varying the gear ratios and pen positions, fostering creativity through hands-on exploration of roulette curves.^[35] Digital tools have expanded access to epitrochoid visualization beyond physical toys, enabling precise parametric plotting and dynamic simulations. Software such as GeoGebra provides interactive applets where users can adjust parameters like the fixed circle radius R, rolling circle radius r, and offset distance to generate and animate epitrochoids in real time.^[36] Similarly, Mathematica supports parametric equations for epitrochoids, with built-in demonstrations that animate the rolling motion of the circle, illustrating the curve's formation step by step.^[3] In mathematics education, epitrochoids serve as an engaging introduction to parametric curves and periodic motion, often integrated into STEAM (Science, Technology, Engineering, Arts, and Mathematics) curricula to demonstrate concepts like trigonometry and gear ratios.^[37] Educators use tools like GeoGebra animations to show how rational ratios of R/r produce closed loops, while irrational ratios yield non-closing, dense patterns that fill space over iterations.^[1] This approach helps students visualize abstract ideas, connecting geometric construction to real-world dynamics without requiring advanced computation.^[38] Epitrochoids find artistic applications in graphic design, where their symmetrical, looping forms inspire decorative patterns for logos, textiles, and digital illustrations.^[39] For instance, Spirograph-generated designs have influenced modern vector art tools that replicate epitrochoid curves for scalable motifs, echoing historical ornamental styles while enabling fractal-like variations through iterative scaling.^[40]

References

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The roulette traced by a point P attached to a circle of radius b rolling around the outside of a fixed circle of radius a.
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