Conical spiral

In mathematics, a conical spiral, also known as a conical helix,[1] 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 (from conch).

Conical spiral with an archimedean spiral as floor projection
Floor projection: Fermat's spiral
Floor projection: logarithmic spiral
Floor projection: hyperbolic spiral

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.[2] 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

1) Starting with an archimedean spiral gives the conical spiral (see diagram)
In this case the conical spiral can be seen as the intersection curve of the cone with a helicoid.
2) The second diagram shows a conical spiral with a Fermat's spiral as floor plan.
3) The third example has a logarithmic spiral as floor plan. Its special feature is its constant slope (see below).
Introducing the abbreviation gives the description: .
4) Example 4 is based on a hyperbolic spiral . Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for .

Properties

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

Slope

Slope angle at a point of a conical spiral

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 (see diagram):

A spiral with gives:

For an archimedean spiral is 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

Development(green) of a conical spiral (red), right: a side view. The plane containing the development is designed by . Initially the cone and the plane touch at the purple line.

For the development of a conical spiral[3] 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 hyperbolic spiral () the development is congruent to the floor plan spiral.

In case of a logarithmic spiral the development is a logarithmic spiral:

Tangent trace

The trace (purple) of the tangents of a conical spiral with a hyperbolic spiral as floor plan. The black line is the asymptote of the hyperbolic spiral.

The collection of intersection points of the tangents of a conical spiral with the --plane (plane through the cone's apex) 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 (hyperbolic spiral) the tangent trace degenerates to a circle with radius (see diagram). 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.

Snail shells (Neptunea angulata left, right: Neptunea despecta

References

  1. "Conical helix". MATHCURVE.COM. Retrieved 2022-03-03.
  2. Siegmund Günther, Anton Edler von Braunmühl, Heinrich Wieleitner: Geschichte der mathematik. G. J. Göschen, 1921, p. 92.
  3. Theodor Schmid: Darstellende Geometrie. Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229.
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