• Source: 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 and in electrical engineering for the construction of antennas.




Parameterization


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




r
=

μ

t


a


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





θ
=
t


{\displaystyle \theta =t}





z
=

μ

t


c
.


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


The projection of a conchospiral on the



(
r
,
θ
)


{\displaystyle (r,\theta )}

plane is a logarithmic spiral.
The parameter



μ


{\displaystyle \mu }

controls the opening angle of the projected spiral, while the parameter



c


{\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.




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.




References






External links


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

Kata Kunci Pencarian:

• Daftar bentuk matematika
• Conchospiral
• Conical spiral
• List of curves
• List of spirals
• List of mathematical shapes