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Conchospiral

Logarithmic spiral projected onto the surface of a cone From Wikipedia, the free encyclopedia

Conchospiral
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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]

Thumb
An example

Parameterization

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

The projection of a conchospiral on the plane is a logarithmic spiral. The parameter controls the opening angle of the projected spiral, while the parameter controls the slope of the cone on which the curve lies.

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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

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