Archimedean spiral

The Archimedean spiral (also known as arithmetic spiral) is the simplest of all spirals. It occurs when during a rotational movement of the radius increases in proportion to the angle of rotation, that is, it applies with radius, rotation angle and.

Properties

The representation as a parametric curve in Cartesian coordinates is as follows:

The length of a piece of sheet to

The total length of the spiral is thus to

The area which is included in the first rotation is

While for the n th revolution of the surface

Is also included.

Turn spacing

Everyone from the origin O (0 | 0) outgoing beam intersects successive turns of the Archimedean spiral in points with the constant spacing (see figure to the right ). Hence the designation comes as " arithmetic spiral".

This particular property of the Archimedean spiral is often expressed by saying that her turn spacing is constant. This speech, however, can easily be misunderstood, because it is not dealing with a constant distance between curves in the sense of parallel curves. A spiral whose turns have actually constant distance in the latter sense, would be the involute.

Historical

Archimedes described the spiral named after him in 225 BC in his treatise " On Spirals ", she was, however, already his friend and contemporary Conon of Samos known, regarded as its discoverer. In the 4th century AD, it was examined by Pappus. The general provision of the spiral length succeeded Isaac Barrow in 1670.

Generalizations

There are various generalizations of the originally described by Archimedes spiral, for those in the literature also often Archimedean spirals is used as a collective term. Here, the original equation is extended to with. Is obtained for the usual spiral of Archimedes, that is referred to as a Fermat spiral. In general, these spirals can differ significantly in properties and appearance of the original Archimedean spiral.

Applications

Many storage devices use the principle of the Archimedean spiral, so rollers memory tapes ( eg, audio and video cassettes) in the form of a spiral on. Marks on vinyl records or CDs are also arranged in the form of an Archimedean spiral, this allows the read head to read without being interrupted by a lane change as much data as linear ( sequential).

Hard disk drives for random access use the other hand since the beginning blocks / segments of a circle on concentrically arranged circles.