Cycloid

A cycloid ( from the Latin or Greek heterocycle κύκλος Kyklos = circle and ειδής Oath = similar), also cyclic curve, Wheel ( running ) - or rolling curve is the path that a circuit point when rolling a circle on a trajectory, for example of a line describes. The use of cycloid when drawing ornaments found by the toy Spirograph widespread.

Mathematical representation of the cycloid

A cycloid can be represented as an analytical equation and parametric functions. The parametric representation is

Wherein the radius of the circle and the parameter ( " roll angle "), respectively. From this, the parameters can be eliminated. The analytical equation is

Any cycloid can be calculated by the following parametric representation:

Where the distance of the generating point indicating the center. With cycloid be shortened cycloid with are called extended. However, this arbitrary cycloid can no longer be all represented in an analytical form.

Properties of the cycloid

An ordinary cycloid is formed when a circle rolls on a straight line. Intuitively, a point moves on a tire of a moving bicycle ( approximately the valve ) on an ordinary cycloid. The Catacaustic, the evolute and the involute of the cycloid are themselves cycloid. The centers of the circles of curvature of a cycloid are completely on its evolute.

A shortened cycloid formed when the web is viewed in a point inside the circle, as clearly the side reflector in the bicycle. An extended cycloid presupposes, however, that a point outside the rolling circle moves together with the circle. These two curves are also called trochoid (Greek τροχός trochos " wheel ").

• Ordinary cycloids be points on the tread of a car tire or other impellers (railway, funicular ) and described by the points along the tread rolling marbles.
• Shortened cycloid described by points with a radius smaller than the tread, about points of bicycle spokes or the starting points of the connecting rods in a steam locomotive.
• Extended cycloid describe points of a radius greater than that of the tread; in the case of railways would be the all points of the flange.

In the form of an ordinary cycloid resembles a succession of further arcs extended cycloid has at the apexes between the arcs or loops, while the tips are rounded in the condensed cycloids.

A brachistochrone or tautochrone created by mirroring a cycloid about the x -axis.

The Tautochronie the cycloid

Assuming that air resistance and friction are negligible, reaches a freely movable mass point from any starting point on an inverted cycloid always in the same time at the lowest point. This property is also called Tautochronie ( line of equal fall time; ταὐτό Greek tauto the same χρόνος chronos time).

Epi - and hypocycloid

Roll the circle outside on a circuit different from, arise epicycloids (Greek ἐπίκυκλος epíkyklos, " secondary circuit "). Its radius is equal to the sum of the radius of Leitkreises and the diameter of the moving loop. Historically, attempts were made to explain the epicycle the observed planetary orbits with some strange -looking loops. Rolls from a circle around a fixed smaller circle, also arise epicycloids. To distinguish the formation mechanism epicycloids these are also called Perizykloiden.

Roll the circle on the other hand inside the other circle from, arise hypocycloids curves with peaks. If, instead of a point on the intrinsic circle a point outside used and rotated along the route from that point to the center of the inner circle, then flowery describes this point sounding curves, so-called hypotrochoids. This effect is also a toy than a Spirograph marketed in the form of gears made ​​of plastic, which contain in their interior holes for inserting a pencil tip. The " trajectory " is pinned (in the form of a large wheel with cut gear inside ) on a sheet of paper, and thereafter as long as the inserted pin moves the rolling gear until there is a closed curve.

Both epicycloids and hypocycloids are accurate then closed curves when the ratio of the radii can be written as a fraction of two integers, so if it is rational. In the technical implementation in the form of gears The number of teeth is instrumental, so that always make closed curves.

For the number of " loops " of an outer circle having a radius and an inner circle formed by the radius of the correlation is epicycloids

For hypocycloids this formula is reduced to

Applies to natural k> 1, then the hypocycloid therefore has exactly k peaks.

• For k = 2 ( Cardanische circles) results in a straight hypocycloid, all of whose points lie on a diameter.
• For k = 3 results in a deltoids ( hypocycloid with three peaks)
• For k = 4 results in a astroid: the " diamonds " as we know it from playing cards.

The cardioid is a special case of Epicycloid.

In cycloidal gear technology

In the transmission technique is the cycloidal one of several techniques for teeth of gears and racks. In Zykloidgetrieben the contour of the cam plates equidistant follows a cycloid.

Key findings from the 16th century

The 17th century is considered the " Golden Age of Analysis ," was also relevant for the study of the cycloid. So the best mathematicians and scientists employed with this particular aesthetic curve.

The first publication was in 1570 to cycloids by Gerolamo Cardano, who here describes among other things the gimbal circles. Galileo Galilei in 1598 undertook further geometrical studies of Cycloid. The first area and length calculation of a cycloid succeeded in 1629 the Italian Bonaventura Cavalieri. Further research impulses delivered in the same year the Frenchman Marin Mersenne.

Further progress by quadratures managed Gilles Personne de Roberval in 1634 and 1635 René Descartes and Pierre de Fermat. Roberval succeeded in 1638 a tangent construction, in 1641 succeeded in doing so also Evangelista Torricelli. Torricelli developed a quadrature to 1643 in relation to the helix. The Englishman Christopher Wren in 1658 showed that the length of a cycloid is four times the diameter of the generated circuit.

In a contest of Newton from 1658 through Blaise Pascal created the rectification, squaring, the center of gravity determination, and the cubature. A quadrature over an infinite series was made in 1664 by Isaac Newton. Gottfried Wilhelm Leibniz developed quadrature 1673 on the quadratrix. The Dutchman Christiaan Huygens in 1673 managed the Evolutenbestimmung and Tautochronie.

By the Leibniz integral representation was completed in 1686. The last important finding was the Brachistochroneneigenschaft 1697 by Johann Bernoulli.