Conchoid (mathematics)

The conchoid ( from Greek mussel ) is a special plane curve. It describes the motion of a point - of a fixed point ( terminal) of view - in a given curve maintains a constant distance.

  • 2.1 Features

Actual conchoid

She was known in ancient Greece and is referred to by Nicomedes as conchoid of Nicomedes. Another name is shell curve. Hence the name infers that the graph is similar to the two shells of a clam.

  • Cartesian coordinates:
  • Polar Coordinates:
  • Parametric representation:

Properties

  • The points of the conchoid of the Nikomedes are characterized by the following geometric properties: Given a straight line g ( " directrix "), a point A of the gap g has a ( where a> 0) and b is a real number ( with b > 0). Are then, for any point B of the straight line G, the two points P and P ', which lie on the straight line AB and B have the distance B of the conchoid.
  • The cases are a type of curves, which are called by the common name dogs curve, particularly for is similar to the one branch of the actual tractrix.

In the following, each case that the coordinate axes are as shown in the drawing, that is, the pole at the origin.

  • The conchoid of Nikomedes is axisymmetrical with respect to the x-axis. In general, there are three corner points on the axis of symmetry, namely, and the origin. For the origin is an isolated point.
  • For fall, two of the three points at the origin together, for the origin is a double point of the curve, ie is traversed twice, the graph has a loop. The two tangents at the origin, the equations have

Ordinary conchoid

The concept of conchoid can be generalized:

Given a curve k ( Rail), a point A ( pole) and a positive real number b. At any point B which is located on the curve C, we now look at the two points located on the straight line AB and B have the distance b. The set of all such points is called the conchoid the rail.

The simplest representation uses polar coordinates: A lies at the origin, and let, then the equation of the ordinary conchoid:

Properties

Are all ordinary Konchoiden Zissoiden, wherein the curve is a circle at the origin.

A Pascal snail is a conchoid, wherein the given curve is a circle.

General conchoid

Extending the formation rule, by not applying the distance b along the line AB, but along a straight line at point B is at a constant angle to AB, we obtain the general conchoid. In the case and results in the ordinary conchoid, otherwise it is called an oblique conchoid.

Konchoidenverzahnung in gear technology

In the transmission technique is the so-called Konchoidenverzahnung one of several techniques for teeth of gears and racks.

Comments

  • The so-called conchoid of de Sluze is actually a special cisoids
  • The so-called conchoid of Dürer is a more general construction.
Classical antiquity Nicomedes (mathematician) Parametric equation Limaçon
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