Conical spiral

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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

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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

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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

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

Slope

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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,  , 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

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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

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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

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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

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  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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