In mathematics, an **involute** (also known as an **evolvent**) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.^{ [1] }

- Involute of a parameterized curve
- Properties of involutes
- Examples
- Involutes of a circle
- Involutes of a semicubic parabola
- Involutes of a catenary
- Involutes of a cycloid
- Involute and evolute
- Application
- See also
- References
- External links

It is a class of curves coming under the roulette family of curves.

The evolute of an involute is the original curve.

The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled * Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae * (1673).^{ [2] }

Let be a regular curve in the plane with its curvature nowhere 0 and , then the curve with the parametric representation

is an *involute* of the given curve.

Proof The string acts as a tangent to the curve . Its length is changed by an amount equal to the arc length traversed as it winds or unwinds. Arc length of the curve traversed in the interval is given by where is the starting point from where the arc length is measured. Since the tangent vector depicts the taut string here, we get the string vector as The vector corresponding to the end point of the string () can be easily calculated using vector addition, and one gets |

Adding an arbitrary but fixed number to the integral results in an involute corresponding to a string extended by (like a ball of wool yarn having some length of thread already hanging before it is unwound). Hence, the involute can be varied by constant and/or adding a number to the integral (see Involutes of a semicubic parabola).

If one gets

In order to derive properties of a regular curve it is advantageous to suppose the arc length to be the parameter of the given curve, which lead to the following simplifications: and , with the curvature and the unit normal. One gets for the involute:

- and

and the statement:

- At point the involute is
*not regular*(because ),

and from follows:

- The normal of the involute at point is the tangent of the given curve at point .
- The involutes are parallel curves, because of and the fact, that is the unit normal at .

For a circle with parametric representation , one has . Hence , and the path length is .

Evaluating the above given equation of the involute, one gets

for the parametric equation of the involute of the circle.

The term is optional; it serves to set the start location of the curve on the circle. The figure shows involutes for (green), (red), (purple) and (light blue). The involutes look like Archimedean spirals, but they are actually not.

The arc length for and of the involute is

The parametric equation describes a semicubical parabola. From one gets and . Extending the string by extensively simplifies further calculation, and one gets

Eliminating t yields showing that this involute is a parabola.

The other involutes are thus parallel curves of a parabola, and are not parabolas, as they are curves of degree six (See Parallel curve § Further examples).

For the catenary , the tangent vector is , and, as its length is . Thus the arc length from the point (0, 1) is

Hence the involute starting from (0, 1) is parametrized by

and is thus a tractrix.

The other involutes are not tractrices, as they are parallel curves of a tractrix.

The parametric representation describes a cycloid. From , one gets (after having used some trigonometric formulas)

and

Hence the equations of the corresponding involute are

which describe the shifted red cycloid of the diagram. Hence

- The involutes of the cycloid are parallel curves of the cycloid

(Parallel curves of a cycloid are not cycloids.)

The evolute of a given curve consists of the curvature centers of . Between involutes and evolutes the following statement holds: ^{ [3] }^{ [4] }

*A curve is the evolute of any of its involutes.*

The involute has some properties that makes it extremely important to the gear industry: If two intermeshed gears have teeth with the profile-shape of involutes (rather than, for example, a traditional triangular shape), they form an involute gear system. Their relative rates of rotation are constant while the teeth are engaged. The gears also always make contact along a single steady line of force. With teeth of other shapes, the relative speeds and forces rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern gear teeth bear the involute shape.^{ [5] }

The involute of a circle is also an important shape in gas compressing, as a scroll compressor can be built based on this shape. Scroll compressors make less sound than conventional compressors and have proven to be quite efficient.

The High Flux Isotope Reactor uses involute-shaped fuel elements, since these allow a constant-width channel between them for coolant.

In physics and geometry, a **catenary** is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity *e*, a number ranging from *e =* 0 to *e* = 1.

In mathematics, a **hyperbola** is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In geometry, a **cycloid** is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.

In mathematics, **hyperbolic functions** are analogues of the ordinary trigonometric functions defined for the hyperbola rather than on the circle: just as the points (cos *t*, sin *t*) form a circle with a unit radius, the points (cosh *t*, sinh *t*) form the right half of the equilateral hyperbola.

In mathematics, **de Moivre's formula ** states that for any real number x and integer n it holds that

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In geometry, a **hyperboloid of revolution**, sometimes called a **circular hyperboloid**, is the surface generated by rotating a hyperbola around one of its principal axes. A **hyperboloid** is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

A **tautochrone** or **isochrone curve** is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.

A **cardioid** is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.

In the differential geometry of curves, the **evolute** of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve.

The **pedal curve** results from the orthogonal projection of a fixed point on the tangent lines of a given curve. More precisely, for a plane curve *C* and a given fixed *pedal point**P*, the **pedal curve** of *C* is the locus of points *X* so that the line *PX* is perpendicular to a tangent *T* to the curve passing through the point *X*. Conversely, at any point *R* on the curve *C*, let *T* be the tangent line at that point *R*; then there is a unique point *X* on the tangent *T* which forms with the pedal point *P* a line perpendicular to the tangent *T* – the pedal curve is the set of such points *X*, called the *foot* of the perpendicular to the tangent *T* from the fixed point *P*, as the variable point *R* ranges over the curve *C*.

**Projectile motion** is a form of motion experienced by an object or particle that is projected near the Earth's surface and moves along a curved path under the action of gravity only. This curved path was shown by Galileo to be a parabola, but may also be a line in the special case when it is thrown directly upwards. The study of such motions is called ballistics, and such a trajectory is a ballistic trajectory. The only force of significance that acts on the object is gravity, which acts downward, thus imparting to the object a downward acceleration. Because of the object's inertia, no external horizontal force is needed to maintain the horizontal velocity component of the object. Taking other forces into account, such as friction from aerodynamic drag or internal propulsion such as in a rocket, requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose subsequent course is governed by the laws of classical mechanics.

The **Gödel metric** is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant. It is also known as the **Gödel solution** or **Gödel universe**.

In geometry, the **elliptic coordinate system** is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

**Toroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.

**Elliptic cylindrical coordinates** are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

**Prolate spheroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.

In geometry, **focal conics** are a pair of curves consisting of either

- ↑ Rutter, J.W. (2000).
*Geometry of Curves*. CRC Press. pp. 204. ISBN 9781584881667. - ↑ McCleary, John (2013).
*Geometry from a Differentiable Viewpoint*. Cambridge University Press. pp. 89. ISBN 9780521116077. - ↑ K. Burg, H. Haf, F. Wille, A. Meister:
*Vektoranalysis: Höhere Mathematik für Ingenieure, Naturwissenschaftler und ...*, Springer-Verlag, 2012, ISBN 3834883468, S. 30. - ↑ R. Courant:
*Vorlesungen über Differential- und Integralrechnung, 1. Band*, Springer-Verlag, 1955, S. 267. - ↑ V. G. A. Goss (2013) "Application of analytical geometry to the shape of gear teeth", Resonance 18(9): 817 to 31 Springerlink (subscription required).

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