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
In cylindrical coordinates, the conchospiral is described by the parametric equations:
- [math]\displaystyle{ r=\mu^t a }[/math]
- [math]\displaystyle{ \theta=t }[/math]
- [math]\displaystyle{ z=\mu^t c. }[/math]
The projection of a conchospiral on the [math]\displaystyle{ (r,\theta) }[/math] plane is a logarithmic spiral. The parameter [math]\displaystyle{ \mu }[/math] controls the opening angle of the projected spiral, while the parameter [math]\displaystyle{ 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
- ↑ 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, https://books.google.com/books?id=aEMYAAAAYAAJ&pg=PA23
- ↑ 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, https://books.google.com/books?id=6DG09xOjbkMC&pg=PA142
- ↑ Balanis, Constantine A. (2015), "11.3.2 Conical spiral", Antenna Theory: Analysis and Design, John Wiley & Sons, p. 598, ISBN 9781119178989, https://books.google.com/books?id=PTFcCwAAQBAJ&pg=PA598
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