Involute

The involute (also involute ) is a term from the mathematical branch differential geometry. Every rectifiable curve is associated with a host of other curves as its involute, resulting from the " settlement " of the tangent.

Can be graphically represented as the involute thread line: A shallow body, one side surface has the shape of the output curve is placed on a sheet of paper. About the output curve a thin thread is taut. At the outer end of the thread, a pin is attached, the tip is resting on the paper. Then the yarn is withdrawn slowly from the curve, wherein it is kept taut. The curve produced on the paper, is an involute.

Since the thread may initially be of any length, there are infinitely many involutes to every curve, which are all parallel to each other, that is: If two involutes given, then every normal one also normal to the other, and all these are normal between the two involutes of equal length. Each Normal Normal is therefore an involute involute all. The normal to the involutes are just the tangents to the given curve. This is the envelope ( envelope ) of the Evolventennormalen. Most is meant by the involute involute; However, this is only a special case of the general involute.

Example

As a starting curve here neilsche a parabola ( marked in black ) will be used.

P1E1, P2E2, ... ( gray marked ) are tangents to the parabola neilsche; SE is a left side tangent. Here The track has a length of 0.5. The following routes, ... exceed the length of the arc, ... The end points E1, E2, ... of the tangent sections form the involute of neil between parabola ( purple marked ), here is a normal parabola. ( The left branch of the parabola arises when the " settlement " to right ). If you choose a different length than 0.5 so arise to parallel parabolas more involute (dashed). Plotted is the parabola

Involute and evolute

The output curve from which the involute is created, called the evolute. In the above example, the evolute is the Neilsche parabola, involute when it is the circle. Because of this interdependence, the involute is sometimes also called involute.

Applications

In the art, the involute has a great significance especially in the construction of gears and racks. With the involute frequently used the cross-section of a tooth flank is part of an involute. This ensures that the teeth are in engagement with contact along a straight line of action ( the tangent to the base circle ). The involute shape is easier to manufacture than the cycloidal also used the tooth flank.

• Geometric curve
• Gear
• Transmission teaching
• Elementary Differential Geometry