What is "involute"

Involute \In"vo*lute\, Involuted \In"vo*lu`ted\, a. [L. involutus, p. p. of involvere. See Involve.]

1. (Bot.) Rolled inward from the edges; -- said of leaves in vernation, or of the petals of flowers in [ae]stivation.
   --Gray.

2. (Zo["o]l.)

   1. Turned inward at the margin, as the exterior lip of the Cyprea.

   2. Rolled inward spirally.

Involute \In"vo*lute\, n. (Geom.) A curve traced by the end of a string wound upon another curve, or unwound from it; -- called also evolvent. See Evolute.

early 15c., from Latin involutus "rolled up, intricate, obscure," past participle of involvere (see involve).

1. 1 (context formal English) difficult to understand; complicated. 2 (context botany English) Having the edges rolled with the adaxial side outward. 3 (context biology of shells English) Having a complex pattern of coils. 4 (context biology English) Turned inward at the margin, like the exterior lip of the Cyprea. 5 (context biology English) Rolled inward spirally. n. (context geometry English) A curve that cuts all tangents of another curve at right angles; traced by a point on a string that unwinds from a curved object. v

2. To roll or curl inwards.

1. adj. especially of petals or leaves in bud; having margins rolled inward [syn: rolled]

2. (of some shells) closely coiled so that the axis is obscured

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 $r:\mathbb R\to\mathbb R^n$ is a natural parametrization of the curve (i.e., ∣r(s)∣ = 1 for all s), then
s ↦ r(s) − sr(s)
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).