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involute

 
Dictionary: in·vo·lute   (ĭn'və-lūt') pronunciation
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involute
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involute

=cosθ + θ sinθ
=sinθ - θ cosθ
(Academy Artworks)
adj.
  1. Intricate; complex.
  2. Botany.
    1. Having the margins rolled inward.
    2. Having whorls that obscure the axis or other volutions, as the shell of a cowrie.
intr.v., -lut·ed, -lut·ing, -lutes.
  1. To curl inward.
  2. To return to a normal or former condition.
n.
The curve traced by a point on a taut, inextensible string as it unwinds from another curve.

[Latin involūtus, past participle of involvere, to enwrap. See involve.]

involutely in'vo·lute'ly adv.

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Sci-Tech Encyclopedia: Involute
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A term applied to a curve C′ that cuts at right angles all tangents of a curve C (see illustration). Each curve C has infinitely many involutes and the distance between corresponding points of any two involutes is constant. Let a length of string be coincident with a curve C, with one end fastened at a point P0 of C. If the string is unwound, remaining taut, the other end of the string traces an involute C′ of C. By varying the length of the string, all involutes of C are obtained. See also Analytic geometry.

An involute <i>C</i>′ of curve <i>C</i>.
An involute C′ of curve C.

Thesaurus: involute
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adjective

    Difficult to understand because of intricacy: byzantine, complex, complicated, convoluted, daedal, Daedalian, elaborate, intricate, involved, knotty, labyrinthine, tangled. See simple/complex.

Dental Dictionary: involute
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(in'vəlōōt)
v

To decrease normally, in size and functional activity, an organ whose role in the body economy is temporary or confined to certain periods of life. Involute should be distinguished from atrophy, which means to waste away from abnormal causes.

Architecture: involute
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1. A curve traced by a point at the end of a string as the string is unwound from a stationary cylinder.
2. Curved spirally.

Wikipedia: Involute
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In the differential geometry of curves, an involute (also known as evolvent) of a smooth curve is another curve, obtained by attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve; or in reverse, unwound. It is a roulette wherein the rolling curve is a straight line containing the generating point.

Alternativelly, another way to imagine the involute of a curve is to replace the taut string by a line, and tracing a given point at the line as the line is rolled on the curve, all the time keeping the line as a tangent to the curve.

The evolute of an involute is the original curve, less portions of zero or undefined curvature. Compare Media:Evolute2.gif and Media:Involute.gif

If function r:\mathbb R\to\mathbb R^n is a natural parametrization of the curve (i.e. |r^\prime(s)|=1 for all s), then :t\mapsto r(t)-tr^\prime(t) parametrizes the involute.

Contents

Parametric curve

Equations of an involute of a parametrically defined curve are:

X[x,y]=x-\frac{x'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}

Y[x,y]=y-\frac{y'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}

Examples

Involut cir.jpg
The involute of a circle
(in reverse, by unwinding)
The involute of a catenary, a tractrix.


Involute of a circle

The involute of a circle forms a shape which resembles an Archimedean spiral.

\, r=a\sec\alpha
\, \theta = \tan\alpha - \alpha

where \, a is the radius of the circle and \, \alpha is a parameter

Leonhard Euler proposed to use the involute of the circle for the shape of the teeth of toothwheel gear, a design which is the prevailing one in current use, called involute gear.

Involute of a catenary

The involute of a catenary through its vertex is a tractrix. In cartesian coordinates the curve follows:

x=t-\tanh(t)\,
y=\rm sech(t)\,
Where: t is the angle and sech is the hyperbolic secant (1/cosh(x)) Derivative

With r(s)=(\sinh^{-1}(s),\cosh(\sinh^{-1}(s)))\,

we have r^\prime(s)=(1,s)/\sqrt{1+s^2}\,

and r(t)-tr^\prime(t)=(\sinh^{-1}(t)-t/\sqrt{1+t^2},1/\sqrt{1+t^2}).

Substitute t=\sqrt{1-y^2}/y

to get ({\rm sech}^{-1}(y)-\sqrt{1-y^2},y).

Involute of a cycloid

One involute of a cycloid is a congruent cycloid. In cartesian coordinates the curve follows:

x=r(t-sin(t))\,
y=r(1-cos(t))\,

Where t is the angle and r is the radius

Application

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 "classic" triangular shape), they form an involute gear system. Their relative rates of rotation are constant while the teeth are engaged, and also, the gears 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.

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

See also

External links

v • d • e
Differential transforms of plane curves

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.  Read more
Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.  Read more
Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.  Read more
Dental Dictionary. Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved.  Read more
Architecture. McGraw-Hill Dictionary of Architecture and Construction. Copyright © 2003 by McGraw-Hill Companies, Inc. All rights reserved.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Involute".  Read more

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