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]

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.^[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]