Английская Википедия:Conchospiral

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Шаблон:Short description

An example
An example

In mathematics, a conchospiral a specific type of space spiral on the surface of a cone (a conical spiral), whose floor projection is a logarithmic spiral. Conchospirals are used in biology for modelling snail shells, and flight paths of insects [1][2] and in electrical engineering for the construction of antennas.[3][4]

Parameterization

In cylindrical coordinates, the conchospiral is described by the parametric equations:

<math>r=\mu^t a</math>
<math>\theta=t</math>
<math>z=\mu^t c.</math>

The projection of a conchospiral on the <math>(r,\theta)</math> plane is a logarithmic spiral. The parameter <math>\mu</math> controls the opening angle of the projected spiral, while the parameter <math>c</math> controls the slope of the cone on which the curve lies.

History

The name "conchospiral" was given to these curves by 19th-century German mineralogist Georg Amadeus Carl Friedrich Naumann, in his study of the shapes of sea shells.[5]

Applications

The conchospiral has been used in the design for radio antennas. In this application, it has the advantage of producing a radio beam in a single direction, towards the apex of the cone.[6][7]

References

Шаблон:Reflist

External links

  1. New Scientist
  2. Conchospirals in the Flight of Insects
  3. John D. Dyson: The Equiangular Spiral Antenna. In: IRE Transactions on Antennas and Propagation. Vol. 7, 1959, pp. 181–187.
  4. T. A. Kozlovskaya: The Concho-Spiral on the Cone. Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 11:2 (2011), pp. 65–76.
  5. Шаблон:Citation
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