Conical spiral


In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor projection is a plane spiral. If the floor projection is a logarithmic spiral, it is called conchospiral.

Parametric representation

In the --plane a spiral with parametric representation
a third coordinate can be added such that the space curve lies on the cone with equation :
Such curves are called conical spirals. They were known to Pappos.
Parameter is the slope of the cone's lines with respect to the --plane.
A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone.

Examples

Properties

The following investigation deals with conical spirals of the form and, respectively.

Slope

The slope at a point of a conical spiral is the slope of this point's tangent with respect to the --plane. The corresponding angle is its slope angle :
A spiral with gives:
For an archimedean spiral,, and hence its slope is
  • For a logarithmic spiral with the slope is .
Because of this property a conchospiral is called an equiangular conical spiral.

Arclength

The length of an arc of a conical spiral can be determined by
For an archimedean spiral the integral can be solved with help of a table of integrals, analogously to the planar case:
For a logarithmic spiral the integral can be solved easily:
In other cases elliptical integrals occur.

Development

For the development of a conical spiral the distance of a curve point to the cone's apex and the relation between the angle and the corresponding angle of the development have to be determined:
Hence the polar representation of the developed conical spiral is:
In case of the polar representation of the developed curve is
which describes a spiral of the same type.
  • If the floor plan of a conical spiral is an archimedean spiral than its development is an archimedean spiral.
In case of a logarithmic spiral the development is a logarithmic spiral:

Tangent trace

The collection of intersection points of the tangents of a conical spiral with the --plane is called its tangent trace.
For the conical spiral
the tangent vector is
and the tangent:
The intersection point with the --plane has parameter and the intersection point is
gives and the tangent trace is a spiral. In the case the tangent trace degenerates to a circle with radius . For one has and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.
File:Neptunea_-_links&rechts_gewonden.jpg|thumb|Snail shells (Neptunea angulata left, right: ''Neptunea despecta''