v.

退行する

関: involution、involutional、regress、regression、retrogression

WordNet

(of some shells) closely coiled so that the axis is obscured
especially of petals or leaves in bud; having margins rolled inward (同)rolled
(psychiatry) a defense mechanism in which you flee from reality by assuming a more infantile state
returning to a former state (同)regress, reversion, retrogression, retroversion
the relation between selected values of x and observed values of y (from which the most probable value of y can be predicted for any value of x) (同)simple regression, regression toward the mean, statistical regression
reduction in size of an organ or part (as in the return of the uterus to normal size after childbirth)
the action of enfolding something (同)enfolding
a long and intricate and complicated grammatical construction
go back to a statistical means
get worse or fall back to a previous condition (同)retrograde, retrogress
the reasoning involved when you assume the conclusion is true and reason backward to the evidence (同)reasoning backward

PrepTutorEJDIC

うず巻状の,らせん状の / 入り組んだ,複雑な
後退;退化,退歩
巻き(包み)込むこと,巻き込まれた状態 / 複雑なもの,込み入ったもの
後退する,退行する

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/06/13 00:34:25」(JST)

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In the differential geometry of curves, an involute (also known as evolvent) is a curve obtained from another given curve 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. For example, an involute approximates the path followed by a tetherball as the connecting tether is wound around the center pole. If the center pole has a circular cross-section, then the curve is an involute of a circle.

Alternatively, another way to construct the involute of a curve is to replace the taut string by a line segment that is tangent to the curve on one end, while the other end traces out the involute. The length of the line segment is changed by an amount equal to the arc length traversed by the tangent point as it moves along 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 the function is a natural parametrization of the curve (i.e., for all s), then : parametrizes the involute.

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).^[1]

Parametric curve[edit]

Equations of an involute curve for a parametrically defined function ( x(t), y(t) ) are:

Examples[edit]

Involute of a circle[edit]

The involute of a circle

(in reverse, by unwinding)

The involute of a circle resembles an Archimedean spiral.

Its successive turns are parallel curves with constant separation distance, a property which is often (somewhat misleadingly) ascribed to the Archimedean spiral.

In Cartesian coordinates the involute of a circle has the parametric equation:

where is the radius of the circle and is an angle parameter in radians () .

In polar coordinates the involute of a circle has the parametric equation:

where is the radius of the circle and is a parameter in the interval

With a different parameter with it can be written in the form:

.

Curve length[edit]

The arc length of the above curve for is

Application[edit]

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[edit]

The involute of a catenary, a tractrix.

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

Where: t is a parameter and sech is the hyperbolic secant (1/cosh(t))

Derivative

With

we have

and .

Substitute

to get .

Involute of a cycloid[edit]

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

Where t is the angle and r is the radius

Application[edit]

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 can be built based on this shape. Scroll compressors make less sound than conventional compressors, and have proven to be quite efficient.

References[edit]

^ John McCleary – Geometry from a Differentiable Viewpoint / Cambridge University Press, 1995 / pg. 73

External links[edit]

Mathworld

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The current approach to the diagnosis of vascular anomalies of the head and neck: A pictorial essay.

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Japanese Journal

コニカルギヤのかみ合い効率解析

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日本機械学会論文集 C編 79(804), 2874-2883, 2013
We have been investigating small and high-strengthened single threaded worm gear. Previous our studies have shown that reducing thickness of worm tooth on addendum face enlarges elastic deformation of …
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