mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a conchospiral a specific type of

space spiral In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimension ...

on the surface of a

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

(a '' conical spiral''), whose floor projection is a

logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewi ...

. Conchospirals are used in biology for modelling snail shells, and flight paths of insects and in

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

for the construction of

antennas In radio-frequency engineering, an antenna (American English) or aerial (British English) is an electronic device that converts an alternating electric current into radio waves (transmitting), or radio waves into an electric current (receivi ...

Parameterization

cylindrical coordinates A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...

, the conchospiral is described by the parametric equations: :

r=\mu^t a

\theta=t

z=\mu^t c.

The projection of a conchospiral on the

(r,\theta)

plane is a

. The parameter

\mu

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

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 antenna In radio-frequency engineering, an antenna (American English) or aerial (British English) is an electronic device that converts an alternating electric current into radio waves (transmitting), or radio waves into an electric current (receivi ...

s. 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

*{{mathworld, title=Concho-Spiral, urlname=Concho-Spiral Spirals