Conchospiral

In mathematics, a conchospiral a specific type of three-dimensional spiral on the surface of a cone (a conical spiral), whose floor projection is a logarithmic spiral.

An example

Parameterization

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

${\displaystyle r=\mu ^{t}a}$

${\displaystyle \theta =t}$

${\displaystyle z=\mu ^{t}c.}$

The projection of a conchospiral on the ${\displaystyle (r,\theta )}$ plane is a logarithmic spiral. The parameter ${\displaystyle \mu }$ controls the opening angle of the projected spiral, while the parameter ${\displaystyle c}$ 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.[1]

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.[2][3]

References

1. Blake, John Frederick (1882), A Monograph of the British Fossil Cephalopoda, Part 1, J. Van Voorst, p. 23
2. Burberry, R. A. (1992), "8.2.4 Conical spiral", VHF and UHF Antennas, Institution of Electrical Engineers Electromagnetic Waves Series, 35, IET, p. 142, ISBN 9780863412691
3. Balanis, Constantine A. (2015), "11.3.2 Conical spiral", Antenna Theory: Analysis and Design, John Wiley & Sons, p. 598, ISBN 9781119178989

External links

• Weisstein, Eric W. "Concho-Spiral". MathWorld.