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Involute

In geometry, an involute is a plane curve that is generated as the locus of a point on a taut, inextensible string as the string is unwrapped from a fixed base curve, known as the evolute.^[1] The string remains tangent to the evolute at the point of contact, and the involute represents the path traced by the free end of the string.^[2] This construction ensures that the evolute is the envelope of the normals to the involute, and vice versa, linking the two curves through the geometry of curvature centers.^[1] The parametric equations for the involute of a circle of radius a, a particularly significant case, are given by x = a(\cos t + t \sin t) and y = a(\sin t - t \cos t), where t is the parameter representing the unwind angle.^[3] More generally, for any curve parameterized by arc length s with position vector \mathbf{r}(s) and unit tangent \mathbf{T}(s), the involute is \mathbf{r}_i(s) = \mathbf{r}(s) - (c - s) \mathbf{T}(s), where c is a constant related to the total string length.^[4] Key properties include the fact that the normal to the involute at any point passes through the corresponding point of tangency on the evolute, and the arc length along the involute corresponds to the angle of unwrap.^[3] The curvature \kappa of the circle involute decreases as \kappa = 1/(a t), reflecting its spiraling nature away from the base circle.^[3] Involutes were first systematically studied by Christiaan Huygens in 1673 as part of his work on pendulum clocks, where the involute of a cycloid was used to achieve isochronous oscillations.^[1] Today, the involute profile is the dominant tooth shape in modern gearing systems, employed in spur, helical, and other gear types due to its ability to maintain a constant velocity ratio between meshing gears regardless of minor variations in center distance. This property arises because the common normal at the point of contact is fixed along the line of action, ensuring smooth power transmission with minimal wear.^[2] The involute's ease of manufacture via hobbing or shaping tools, combined with its tolerance for manufacturing imperfections, has made it the standard in automotive, industrial, and precision machinery applications.^[5]

Definition and Construction

Parametric Definition

The parametric definition of an involute curve arises from the geometric process of unwinding a taut string from a base curve, where the endpoint of the string traces the involute. Consider a smooth base curve \alpha(t) in the plane, parameterized by t in some interval, with arc length s(t) = \int_{t_0}^t \|\alpha'(\tau)\| \, d\tau measured from an initial point t_0. The unit tangent vector to the base curve at \alpha(t) is T(t) = \alpha'(t) / \|\alpha'(t)\|. The involute \gamma(t) is then defined as

\gamma(t) = \alpha(t) - (c - s(t)) T(t),

where c is a positive constant representing the total initial length of the string.^[6] This formulation derives from the unwinding mechanism: at each parameter value t, the string has unwound a length s(t) along the base curve, leaving a remaining segment of length c - s(t) that extends tangentially from \alpha(t) in the direction opposite to T(t). As t increases, the parameter traces the progressive unwinding, with \gamma(t) moving away from the base curve while maintaining tangency; for t = t_0, s(t_0) = 0, so \gamma(t_0) = \alpha(t_0) - c T(t_0), starting at a distance c along the initial tangent in the negative direction. If the base curve is already parameterized by arc length (so \|\alpha'(t)\| = 1 and s(t) = t - t_0), the equation simplifies accordingly, emphasizing the role of arc length in measuring the unwound portion.^[6] An equivalent vector form for plane curves expresses the involute without explicitly introducing the constant c, by letting l(t) denote the unwound length up to t:

\gamma(t) = \alpha(t) - l(t) \cdot \frac{\alpha'(t)}{\|\alpha'(t)\|},

where l(t) can be taken as s(t) or adjusted by a constant to shift the curve family. This highlights the direct proportionality to the unit tangent (with negative sign for standard convention), with l(t) controlling the "unwinding progress" as t varies.^[6]

Geometric Construction

The geometric construction of an involute originates from the intuitive method of unwrapping a taut string from a fixed base curve, a concept developed by Christiaan Huygens in 1673 during his studies on pendulum clocks and curvature.^[7]^[1] This approach provides a physical analogy for generating the involute, highlighting its relation to the evolute as the locus of curvature centers.^[1] To construct the involute, begin by selecting a smooth base curve, such as a circle or more general plane curve, and wrapping an inextensible string of fixed total length around it, ensuring the string lies tangent to the curve at every point of contact.^[8] As the string is gradually unwrapped while kept taut, the free end of the string traces out the involute curve; at each stage, the straight segment of the string extends from the point of tangency on the base curve to the tracing point. The process starts from an initial position where the string is fully wrapped up to a certain arc length, and continues as more of the string is pulled away, with the contact point moving along the base curve. This step-by-step unwrapping ensures the tracing point follows a path that is the locus of all such end positions. In this construction, the string always aligns with the tangent to the base curve at the instantaneous contact point, which, by the definition of curvature, remains perpendicular to the radius of curvature at that point on the base curve.^[8] This perpendicularity underscores the geometric link between the tangent direction and the normal (along the radius of curvature), preserving the curve's local bending properties during unwrapping. The parameter c, representing the constant total length of the string, determines the specific member of the involute family generated, as the straight segment's length equals c minus the arc length unwrapped up to the contact point.^[8] This construction aligns with the parametric equations described elsewhere, where the position is given by the base curve point minus the scaled tangent vector.

Mathematical Properties

Cusps and Singularities

Cusps on an involute curve manifest as points where the curve undergoes a sharp reversal in direction, arising from the geometric construction in which a taut string is unwound from the base curve. These singularities occur when the string wraps around convex portions of the base curve, causing the path traced by the free end to pause momentarily before reversing, typically at the initiation of unwinding or during passage through regions of changing convexity.^[9] Cusps are classified by their order, with the standard type being a cusp of order \frac{3}{2}, corresponding to generic reversals where the first derivative vanishes but higher derivatives satisfy specific rank conditions, such as \rank(\ddot{\gamma}(t_0), \gamma'''(t_0)) = 2. Higher-order cusps, like those of order \frac{5}{2}, emerge in scenarios involving more intricate inflections on the base curve, altering the local singularity structure.^[10]^[11] Such cusps form under specific conditions on the base curve, including the presence of inflection points where the curvature vanishes (\kappa(t) = 0), or when the unwinding parameter reaches a vertex—a point of local extremal curvature—leading to a temporary halt in the parametric speed.^[9]^[12] Mathematically, singular points on the involute \gamma(t) are identified where the first derivative \gamma'(t_0) = 0, with the cusp geometry further characterized by the non-vanishing rank of subsequent derivatives, ensuring the curve is not smoother but exhibits the distinctive sharp turn.^[10]

Tangent and Curvature Properties

A fundamental property of the involute curve is that the line segment connecting a point on the involute to its corresponding point of tangency on the base curve lies along the tangent to the base curve at that point, with the length of this segment given by |c - s| in parametric constructions of the family of involutes, where c parameterizes the specific involute and s is the arc length along the base. This varying length reflects the taut-string analogy in the unwrapping process.^[13] The curvature \kappa of the involute at the point corresponding to arc length parameter s along the base curve is given by

\kappa(s) = \frac{1}{|c - s|},

a standard property arising because the radius of curvature equals the length of the tangent segment from the point to the evolute.^[14] This formula shows that the curvature decreases as |c - s| increases, with the radius of curvature \rho = 1/\kappa = |c - s| increasing monotonically along the involute, reflecting the expanding tangent segments and smoother profile as the curve develops away from the base. Involutes also possess notable orthogonal trajectory properties with respect to the evolute, where the family of involutes intersects the tangent lines to the evolute at right angles, establishing a duality in their geometric interplay.^[8]^[15]

Relation to Evolute

Evolute Definition

The evolute of a given curve \alpha is defined as the locus of the centers of curvature of \alpha, or equivalently, the envelope of the family of its normal lines. For a plane curve \alpha(s) parametrized by arc length s, the position of the center of curvature at parameter s is given by the parametric equation

\mathbf{e}(s) = \alpha(s) + \rho(s) \mathbf{N}(s),

where \rho(s) = 1/\kappa(s) is the radius of curvature, \kappa(s) is the curvature, and \mathbf{N}(s) is the unit principal normal vector.^[16] This construction traces the path followed by the center of the osculating circle as it moves along \alpha.^[17] The concept of the evolute traces back to ancient Greek geometry, where Apollonius of Perga (c. 200 BCE) investigated properties of conic sections, including the locus of centers of curvature and related normal constructions in his seminal work Conics. It was later formalized in the modern sense by Christiaan Huygens in the 1670s, particularly in his optical studies of caustics and wave propagation, as detailed in Traité de la lumière (1690), where evolutes describe the envelopes of reflected or refracted rays.^[18] Key properties of the evolute include the presence of cusps at parameter values corresponding to stationary points of the curvature \kappa(s) on the base curve \alpha, where the derivative d\kappa/ds = 0 (known as vertices of \alpha); at these points, the evolute exhibits sharp turns as the center of curvature reverses direction.^[19] Additionally, the arc length of the evolute between two such cusps equals the absolute difference in the radii of curvature \rho at the corresponding endpoints on \alpha, reflecting how the evolute's progression measures changes in \rho; this length relates to the tangent segment lengths in the dual involute construction, where unwinding along the evolute generates tangents of that measure.^[12] For plane curves, the parametric equations of the evolute are derived using the Frenet-Serret frame \{\mathbf{T}(s), \mathbf{N}(s)\}, where \mathbf{T}(s) = \alpha'(s) is the unit tangent. The Frenet-Serret formulas for plane curves yield \mathbf{T}'(s) = \kappa(s) \mathbf{N}(s) and \mathbf{N}'(s) = -\kappa(s) \mathbf{T}(s). Differentiating the evolute equation gives

\mathbf{e}'(s) = \rho'(s) \mathbf{N}(s),

since the tangential component vanishes due to \rho \kappa = 1. Thus, the arc length element of the evolute is ds_e = |\rho'(s)| ds = |d\rho|, confirming the integrated length as the variation in \rho. This framework extends naturally to space curves via the full Frenet-Serret apparatus including the binormal \mathbf{B}(s) and torsion, though for plane curves torsion vanishes.^[16]^[17]

Involute-Evolute Duality

The involute-evolute duality refers to the reciprocal relationship between these two curve constructions in differential geometry, where the involutes of a given curve \gamma are precisely the curves whose evolute is \gamma, and conversely, the evolute of any involute of \gamma recovers \gamma as the original base curve. This symmetry underscores that involutes and evolutes are inverse operations in the geometry of plane curves with non-vanishing curvature.^[20]^[21] A proof outline using the tangent construction proceeds as follows. Parameterize the base curve \gamma by arc length s, with unit tangent T(s) and principal normal N(s). An involute I of \gamma is given by I(s) = \gamma(s) - (c - s) T(s), where c is a constant determining the specific involute in the one-parameter family. The tangent to I at s is perpendicular to the vector from \gamma(s) to I(s), which aligns with N(s); thus, the normals to I coincide with the tangents to \gamma. The evolute of I, being the envelope of these normals, is therefore the locus of intersection points of consecutive tangents to \gamma, which is \gamma itself. This demonstrates that applying the evolute operation to an involute returns the base curve.^[21]^[20] Geometrically, this duality manifests in the string unwinding model: the involute traces the path of the endpoint of a taut string unwinding from the evolute (the base curve), with the string length equal to the arc length difference. Conversely, the centers of curvature along the involute—points where the osculating circles are centered—trace out the original base curve, reinforcing the paired reciprocity. For example, the involute of a circle is a spiral curve whose evolute is the original circle.^[22] (pp. 88, 291) In variations, non-convex base curves can yield involutes with multiple branches per unwinding direction, arising from self-intersections or regions where the string wraps differently along concave portions. For singular cases like (n, m)-cusp curves, the duality preserves a transformed cusp type, such as an (n, m)-cusp mapping to an (m, 2m - n)-cusp on the involute, with the evolute recovering the original.^[22] (pp. 99, 121)^[23]

Specific Examples

Involute of a Circle

The involute of a circle of radius a is generated by unwrapping a taut string from the circumference of the circle, tracing the path of the free end. This curve, often parameterized by the unwinding angle \theta, takes the form

\gamma(\theta) = a \left( \cos \theta + \theta \sin \theta, \, \sin \theta - \theta \cos \theta \right),

where \theta \geq 0 measures the rotation in radians.^[24] This parameterization arises from integrating the unit tangent vector of the base circle, scaled by the arc length unwound, starting from the point of tangency at \theta = 0.^[25] In polar coordinates centered at the circle's origin, the involute exhibits a spiral form with radial distance r = a \sqrt{1 + \theta^2}.^[26] This equation reflects the curve's asymptotic behavior, approximating an Archimedean spiral for large \theta, as the radius grows roughly linearly with the angular parameter due to constant angular unwinding speed. The curve progresses uniformly without cusps or singularities, maintaining smooth tangency with the base circle at every point, unlike involutes of certain non-circular curves.^[12] A key property is the length of the tangent segment from the point of tangency on the base circle to the corresponding point on the involute, which equals c = a \theta. This length directly corresponds to the arc length unwound from the circle, ensuring the string remains taut.^[27] Furthermore, the evolute of this involute curve coincides with the original base circle, confirming the involute-evolute duality where applying the evolute operation reverses the construction.^[12]

Involute of a Catenary

The involute of a catenary is the tractrix, a curve that arises as the path traced by a point pulled along a straight line by a taut string of constant length.^[28] The catenary, serving as the base curve, can be parametrized in the plane as \alpha(t) = (t, \cosh t).^[29] The corresponding involute, derived from unwrapping a taut string along this catenary, has parametric equations x = t - \tanh t, y = \sech t (for the unscaled case).^[30] In its standard form, scaled by a positive constant a representing the constant tangent length, the tractrix is given by

x = a (t - \tanh t), \quad y = \frac{a}{\cosh t}.

This parametrization describes the curve starting from the point (0, a) and approaching the x-axis asymptotically as t \to \infty.^[28] A defining property of the tractrix is that the segment of the tangent line from any point on the curve to its intersection with the asymptote (the x-axis) has fixed length a.^[28] Geometrically, this makes the tractrix a classical pursuit curve, modeling the trajectory of an object (such as a dog on a leash) drawn toward a point moving uniformly along a straight line, with the pursuing path ensuring the connector remains taut at length a.^[31] The catenary and tractrix form an involute-evolute pair, where the catenary is the evolute of the tractrix, and vice versa, highlighting their reciprocal geometric relationship in differential geometry.^[28]

Involute of a Cycloid

A cycloid is generated as the roulette curve traced by a point on the circumference of a circle of radius a rolling without slipping along a straight line. Its parametric equations are given by

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

where \theta is the parameter representing the rotation angle.^[32] The involute of a cycloid is another cycloid congruent to the original, translated within the plane. This self-involute property means that unwrapping a taut string from the cycloid produces a path that is an identical copy of the cycloid, shifted by a vector such as (0, 2a) to align successive arches. For the standard cycloid with a cusp at the origin and arches oriented upward, one such involute has parametric equations

x(\theta) = a (\theta + \sin \theta), \quad y(\theta) = a (3 + \cos \theta).

^[33] This translation ensures that the involute's arches align with those of the original, preserving the periodic structure. Prolate and curtate cycloids represent elongated and shortened variants of the base cycloid, obtained by adjusting the distance b of the tracing point from the center of the rolling circle relative to the radius a. For b > a, the prolate form elongates the curve without cusps, while for b < a, the curtate form shortens it, producing retrograde loops but remaining smooth. The general parametric equations for these variants are x(\theta) = a \theta - b \sin \theta, y(\theta) = a - b \cos \theta, and their involutes follow analogous constructions with parametric adjustments incorporating b, resulting in translated versions that maintain the core geometric duality but adapted to the variant's scale and form.^[34]^[35] The cusps in the involute of the standard cycloid occur at the vertices of the original curve's arches and exhibit an order of 3/2, consistent with the semicubical parabolic approximation near these points. The involute's cusp structure mirrors that of the base cycloid, with singularities arising periodically every $2\pi a along the curve. This involute property plays a key role in theoretical studies of roulette curves, enabling constructions like the tautochrone for pendulum motion independent of amplitude.^[36]

Involute of a Semicubic Parabola

The semicubical parabola is a cuspidal cubic curve, expressible in implicit form as y^2 = x^3 (up to affine transformation) or parametrically as \alpha(t) = (t^2, t^3).^[37] The involute of this curve is obtained by considering the locus traced by the endpoint of a taut string unwound from the base curve, yielding a parametric representation \gamma(t) that incorporates integrals of the tangent vector components to account for arc length and direction. This derivation highlights the geometric construction central to classical differential geometry, where the involute satisfies \gamma(t) = \alpha(t) - s(t) \mathbf{T}(t), with s(t) the arc length and \mathbf{T}(t) the unit tangent. A defining property of this involute is its cusp of order $5/2 at the origin, manifesting as a singularity akin to y^2 = x^5, which arises from the interaction between the base curve's cusp and the unwinding process. This higher-order cusp exemplifies the amplification of singularities in involute constructions and is a standard illustration in the study of curve singularities. The involute-evolute relation for the semicubical parabola underscores an algebraic duality: the evolute of the involute recovers the original semicubical parabola, mirroring how the evolute of a parabola yields a semicubical form, thus pairing these curves in a reciprocal geometric transformation.^[38]

Applications

Involute Gears

Involute gears are mechanical components where the tooth profiles are generated as involutes of a base circle, a geometric curve that ensures conjugate action during meshing. This profile is created by unwinding a taut string from the base circle, with the path traced by the string's end point defining the tooth flank. As two such gears mesh, the point of contact moves along a straight line of action tangent to both base circles, maintaining a constant velocity ratio between the gears regardless of the specific contact position. This fundamental property allows for smooth power transmission without variations in speed, making involute profiles the standard for spur, helical, and other gear types in precision machinery.^[39] The primary advantages of involute gears stem from their conjugate action, which permits controlled sliding between teeth without interference or undercutting, provided the base circle is appropriately sized relative to the gear's pitch circle. Unlike cycloidal profiles, involutes exhibit a constant pressure angle—the angle between the line of action and the tangent to the pitch circle—typically 14.5°, 20°, or 25°, which remains invariant throughout meshing and facilitates predictable load distribution and force transmission. This invariance simplifies design and manufacturing, as gears can tolerate minor center distance variations while preserving the velocity ratio, reducing sensitivity to alignment errors in assemblies. Additionally, the involute's ease of generation using standard hobbing or shaping tools contributes to cost-effective production at scale.^[2]^[40]^[41] Key parameters in involute gear design include the base circle radius r_b, which serves as the foundation for the tooth curve and is calculated as r_b = r \cos \phi, where r is the pitch radius and \phi is the standard pressure angle. The addendum represents the radial distance from the pitch circle to the tooth tip, typically 1.0 times the module m (a measure of tooth size), while the dedendum is the distance from the pitch circle to the root, usually 1.25m to provide clearance. The parametric equation for the tooth curve in polar coordinates, with the pressure angle \phi as the parameter, is given by

r(\phi) = \frac{r_b}{\cos \phi},

where \phi varies from the base circle tangency to the addendum limit, ensuring the curve extends appropriately for meshing. These parameters allow engineers to optimize gear strength, contact ratio, and backlash for specific applications.^[39]^[40] The involute profile was adopted in the 19th century during the Industrial Revolution, with early proposals like that of German engineer Hoppe in 1873 highlighting its benefits for varying tooth counts and pressure angles. It gained prominence for precision machinery as manufacturing techniques advanced, and was further refined in the early 20th century through analytical works by designers such as Earle Buckingham, whose 1922 treatise on spur gears established foundational design principles still used today.^[41]^[42]

Other Engineering and Geometric Uses

In optics, involute aspheric surfaces represent a specialized class of axially symmetrical rotation aspheres designed to correct aberrations in optical systems. These surfaces exhibit second-order aberration characteristics, providing designers with an additional degree of freedom to minimize distortions while reducing the overall number of optical elements required. By leveraging the involute profile, such lenses achieve favorable aberration correction properties, enabling compact and efficient imaging systems.^[43] The generation of an involute profile aligns conceptually with ray tracing in optics, where the curve's parametric form—derived from unwinding a taut string around a base curve—facilitates precise control over light paths to mitigate spherical and other aberrations. This approach has practical value in applications demanding high image quality, such as advanced lens systems.^[43] In robotics and mechanisms, string-pulley systems approximate involute paths to maintain constant tension in actuators, particularly when strings unwind from circular pulleys. The involute trajectory ensures the string remains taut throughout the motion, delivering uniform force transmission essential for lightweight, compliant designs like those in tensegrity robots. This property supports efficient energy use and precise control in tendon-driven mechanisms.^[44] In modern engineering, computational methods generate involute profiles in CAD software for applications like cam designs, where the curve ensures smooth, constant-velocity motion in rotary actuators. Involute cams, for example, transmit torque efficiently in hydraulic servomechanisms with oscillations under one revolution. To handle complex geometries, involute curves are approximated using B-splines or polynomial functions, improving accuracy and processing efficiency in CAD systems compared to traditional methods. These spline approximations reduce computational errors while preserving the curve's essential parametric integrity for manufacturing.^[45]^[46]

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Prolate Cycloid -- from Wolfram MathWorld
The path traced out by a fixed point at a radius b>a , where a is the radius of a rolling circle, also sometimes called an extended cycloid.
[37]
Curtate Cycloid -- from Wolfram MathWorld
A curtate cycloid, sometimes also called a contracted cycloid, is the path traced out by a fixed point at a radius b<a , where a is the radius of a rolling ...Missing: involute | Show results with:involute
[38]
Cycloid - MacTutor History of Mathematics
Both the evolute and involute of a cycloid is an identical cycloid. In fact the evolute was studied by Huygens and because of his work on the cycloid ...
[39]
Semicubical Parabola -- from Wolfram MathWorld
It has parametric equations x = t^2 (2) y = at^3, (3) and the polar equation r=(tan^2thetasectheta)/a. (4) The evolute of the parabola is a particular case ...
[40]
Semicubical Parabola Involute -- from Wolfram MathWorld
For a semicubical parabola with parametric equations x = t^2 (1) y = at^3, (2) the involute is given by x_i = (t^2)/3-8/(27a^2) (3) y_i = -(4t)/(9a), ...
[41]
Construction and design of involute gears | tec-science
Oct 31, 2018 · An involute is constructed by rolling a so-called rolling line around a base circle. The resulting trajectory curve describes the shape of the involute.Missing: method | Show results with:method
[42]
Involute Gear Profile | KHK
The definition of involute curve is the curve traced by a point on a straight line which rolls without slipping on the circle. The circle is called the base ...
[43]
Involute Gear - an overview | ScienceDirect Topics
Involute gears appeared during the First Industrial Revolution. The application of involute gears greatly improved the quality of mechanical transmission, being ...
[44]
Involute spur gears., by Earle Buckingham - The Online Books Page
Title: Involute spur gears. Author: Buckingham, Earle, 1887-. Note: Niles-Bement-Pond Co., 1922. Link: page images at ...Missing: Earl | Show results with:Earl
[45]
https://apps.dtic.mil/sti/trecms/pdf/AD0626652.pdf
[46]
[PDF] Impedance Controlled Twisted String Actuators for Tensegrity Robots
Abstract: We are developing impedance controlled twisted string actuators (TSA) for use in tensegrity robots, as an alternative to traditional spooled cable ...Missing: involute pulley
[47]
[PDF] DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Apr 26, 2021 · Find the involute of the cycloid ˛.t/ D .t C sin t;1 cos t/, t 2 Œ ; , using t D 0 as your starting point. Give a geometric description of ...
[48]
[PDF] The Involute Cam Rotary Actuator for Hydraulic Servomechanisms
This actuator design is adaptable to any hydraulically powered mechanism requiring an efficient transmission of torque through an oscillation of less than one- ...
[49]
(PDF) Approximation of involute curves for CAD-system processing
Aug 5, 2025 · ... arc length, unit tangent vector and curvature of the circle involute ... parametric representation,. even as a transcendental equation ...