Conchospiral
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
[edit]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.
History
[edit]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
[edit]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
[edit]- ^ New Scientist
- ^ Conchospirals in the Flight of Insects
- ^ John D. Dyson: The Equiangular Spiral Antenna. In: IRE Transactions on Antennas and Propagation. Vol. 7, 1959, pp. 181–187.
- ^ T. A. Kozlovskaya: The Concho-Spiral on the Cone. Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 11:2 (2011), pp. 65–76.
- ^ Blake, John Frederick (1882), A Monograph of the British Fossil Cephalopoda, Part 1, J. Van Voorst, p. 23
- ^ Burberry, R. A. (1992), "8.2.4 Conical spiral", VHF and UHF Antennas, Institution of Electrical Engineers Electromagnetic Waves Series, vol. 35, IET, p. 142, ISBN 9780863412691
- ^ Balanis, Constantine A. (2015), "11.3.2 Conical spiral", Antenna Theory: Analysis and Design, John Wiley & Sons, p. 598, ISBN 9781119178989