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 (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 $x$-$y$-plane a spiral with parametric representation

$x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi$

a third coordinate $z(\varphi )$ can be added such that the space curve lies on the cone with equation $\;m^{2}(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;$ :

• $x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .$

Such curves are called conical spirals. They were known to Pappos.

Parameter $m$ is the slope of the cone's lines with respect to the $x$-$y$-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 $\;r(\varphi )=a\varphi \;$ gives the conical spiral (see diagram)

$x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .$

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 $\;r(\varphi )=\pm a{\sqrt {\varphi }}\;$ as floor plan.

3) The third example has a logarithmic spiral $\;r(\varphi )=ae^{k\varphi }\;$ as floor plan. Its special feature is its constant slope (see below).

Introducing the abbreviation $K=e^{k}$gives the description: $r(\varphi )=aK^{\varphi }$.

4) Example 4 is based on a hyperbolic spiral $\;r(\varphi )=a/\varphi \;$. Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for $\varphi \to 0$.

Properties

The following investigation deals with conical spirals of the form $r=a\varphi ^{n}$ and $r=ae^{k\varphi }$, 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 $x$-$y$-plane. The corresponding angle is its slope angle (see diagram):

$\tan \beta ={\frac {z'}{\sqrt {(x')^{2}+(y')^{2}}}}={\frac {mr'}{\sqrt {(r')^{2}+r^{2}}}}\ .$

A spiral with $r=a\varphi ^{n}$ gives:

• $\tan \beta ={\frac {mn}{\sqrt {n^{2}+\varphi ^{2}}}}\ .$

For an archimedean spiral is $n=1$ and hence its slope is$\ \tan \beta ={\tfrac {m}{\sqrt {1+\varphi ^{2}}}}\ .$

• For a logarithmic spiral with $r=ae^{k\varphi }$ the slope is $\ \tan \beta ={\tfrac {mk}{\sqrt {1+k^{2}}}}\$ ($\color {red}{\text{ constant!}}$ ).

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

$L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {(x')^{2}+(y')^{2}+(z')^{2}}}\,\mathrm {d} \varphi =\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {(1+m^{2})(r')^{2}+r^{2}}}\,\mathrm {d} \varphi \ .$

For an archimedean spiral the integral can be solved with help of a table of integrals, analogously to the planar case:

$L={\frac {a}{2}}\left[\varphi {\sqrt {(1+m^{2})+\varphi ^{2}}}+(1+m^{2})\ln \left(\varphi +{\sqrt {(1+m^{2})+\varphi ^{2}}}\right)\right]_{\varphi _{1}}^{\varphi _{2}}\ .$

For a logarithmic spiral the integral can be solved easily:

$L={\frac {\sqrt {(1+m^{2})k^{2}+1}}{k}}(r{\big (}\varphi _{2})-r(\varphi _{1}){\big )}\ .$

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 $\pi$. Initially the cone and the plane touch at the purple line.

For the development of a conical spiral the distance $\rho (\varphi )$ of a curve point $(x,y,z)$ to the cone's apex $(0,0,z_{0})$ and the relation between the angle $\varphi$ and the corresponding angle $\psi$ of the development have to be determined:

$\rho ={\sqrt {x^{2}+y^{2}+(z-z_{0})^{2}}}={\sqrt {1+m^{2}}}\;r\ ,$

$\varphi ={\sqrt {1+m^{2}}}\psi \ .$

Hence the polar representation of the developed conical spiral is:

• $\rho (\psi )={\sqrt {1+m^{2}}}\;r({\sqrt {1+m^{2}}}\psi )$

In case of $r=a\varphi ^{n}$ the polar representation of the developed curve is

$\rho =a{\sqrt {1+m^{2}}}^{\,n+1}\psi ^{n},$

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 ($n=-1$) the development is congruent to the floor plan spiral.

In case of a logarithmic spiral $r=ae^{k\varphi }$ the development is a logarithmic spiral:

$\rho =a{\sqrt {1+m^{2}}}\;e^{k{\sqrt {1+m^{2}}}\psi }\ .$

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 $x$-$y$-plane (plane through the cone's apex) is called its tangent trace.

For the conical spiral

$(r\cos \varphi ,r\sin \varphi ,mr)$

the tangent vector is

$(r'\cos \varphi -r\sin \varphi ,r'\sin \varphi +r\cos \varphi ,mr')^{T}$

and the tangent:

$x(t)=r\cos \varphi +t(r'\cos \varphi -r\sin \varphi )\ ,$

$y(t)=r\sin \varphi +t(r'\sin \varphi +r\cos \varphi )\ ,$

$z(t)=mr+tmr'\ .$

The intersection point with the $x$-$y$-plane has parameter $t=-r/r'$ and the intersection point is

• $\left({\frac {r^{2}}{r'}}\sin \varphi ,-{\frac {r^{2}}{r'}}\cos \varphi ,0\right)\ .$

$r=a\varphi ^{n}$ gives $\ {\tfrac {r^{2}}{r'}}={\tfrac {a}{n}}\varphi ^{n+1}\$ and the tangent trace is a spiral. In the case $n=-1$ (hyperbolic spiral) the tangent trace degenerates to a circle with radius $a$ (see diagram). For $r=ae^{k\varphi }$ one has $\ {\tfrac {r^{2}}{r'}}={\tfrac {r}{k}}\$ 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

External links

• Jamnitzer-Galerie: 3D-Spiralen. Archived 2021-07-02 at the Wayback Machine.
• Weisstein, Eric W. "Conical Spiral". MathWorld.