Conical spiral

File:Spiral-cone-arch-s.svg

Conical spiral with an archimedean spiral as floor projection

File:Spiral-cone-ferm-s.svg

Floor projection: Fermat's spiral

File:Spiral-cone-log.svg

Floor projection: logarithmic spiral

File:Spiral-con-hyperb.svg

Floor projection: hyperbolic 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).

Parametric representation

In the $x$-$y$-plane a spiral with parametric representation

$$\begin{gathered} x=r(\phi )\cos \phi \\ ,y=r(\phi )\sin \phi \end{gathered}$$

a third coordinate $z(\phi )$ can be added such that the space curve lies on the cone with equation $$\begin{gathered} m^2(x^2+y^2)=(z-z_0{)}^2 \\ , \\ m>0 \end{gathered}$$ :

• $$\begin{gathered} x=r(\phi )\cos \phi \\ ,y=r(\phi )\sin \phi \\ ,z=z_0+mr(\phi ) \\ . \end{gathered}$$

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 $r(\phi )=a\phi$ gives the conical spiral (see diagram)

$$\begin{gathered} x=a\phi \cos \phi \\ ,y=a\phi \sin \phi \\ ,z=z_0+ma\phi \\ ,\phi \ge 0 \\ . \end{gathered}$$

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

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

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

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

Properties

The following investigation deals with conical spirals of the form $r=a{\phi }^n$ and $r=a{e}^{k\phi }$, respectively.

Slope

File:Spiral-cone-slope.svg

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):

$$\begin{gathered} \tan \beta =\frac{z^{\prime }}{\sqrt{(x^{\prime }{)}^2+(y^{\prime }{)}^2}}=\frac{m{r}^{\prime }}{\sqrt{(r^{\prime }{)}^2+r^2}} \\ . \end{gathered}$$

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

• $$\begin{gathered} \tan \beta =\frac{mn}{\sqrt{n^2+{\phi }^2}} \\ . \end{gathered}$$

For an archimedean spiral, $n=1$, and hence its slope is$$\begin{gathered} \tan \beta =\frac{m}{\sqrt{1+{\phi }^2}} \\ . \end{gathered}$$

• For a logarithmic spiral with $r=a{e}^{k\phi }$ the slope is $\tan \beta =\frac{mk}{\sqrt{1+k^2}}$ ($\text{ constant!}$ ).

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

$$\begin{gathered} L=\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}\sqrt{(x^{\prime }{)}^2+(y^{\prime }{)}^2+(z^{\prime }{)}^2}\mathrm{d}\phi =\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}\sqrt{(1+m^2)(r^{\prime }{)}^2+r^2}\mathrm{d}\phi \\ . \end{gathered}$$

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

$$\begin{gathered} L=\frac{a}{2}{[\phi \sqrt{(1+m^2)+{\phi }^2}+(1+m^2)\ln (\phi +\sqrt{(1+m^2)+{\phi }^2})]}_{{\phi }_1}^{{\phi }_2} \\ . \end{gathered}$$

For a logarithmic spiral the integral can be solved easily:

$$\begin{gathered} L=\frac{\sqrt{(1+m^2)k^2+1}}{k}(r({\phi }_2)-r({\phi }_1)) \\ . \end{gathered}$$

In other cases elliptical integrals occur.

Development

File:Kegel-spirale-abw-12.svg

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 $\rho (\phi )$ of a curve point $(x,y,z)$ to the cone's apex $(0,0,z_0)$ and the relation between the angle $\phi$ and the corresponding angle $\psi$ of the development have to be determined:

$$\begin{gathered} \rho =\sqrt{x^2+y^2+(z-z_0{)}^2}=\sqrt{1+m^2}r \\ , \end{gathered}$$

$$\begin{gathered} \phi =\sqrt{1+m^2}\psi \\ . \end{gathered}$$

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 $r=a{\phi }^n$ 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 $r=a{e}^{k\phi }$ the development is a logarithmic spiral:

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

Tangent trace

File:Spiral-con-hyperb-ts-12.svg

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

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

the tangent vector is

$(r^{\prime }\cos \phi -r\sin \phi ,r^{\prime }\sin \phi +r\cos \phi ,m{r}^{\prime }{)}^T$

and the tangent:

$$\begin{gathered} x(t)=r\cos \phi +t(r^{\prime }\cos \phi -r\sin \phi ) \\ , \end{gathered}$$

$$\begin{gathered} y(t)=r\sin \phi +t(r^{\prime }\sin \phi +r\cos \phi ) \\ , \end{gathered}$$

$$\begin{gathered} z(t)=mr+tm{r}^{\prime } \\ . \end{gathered}$$

The intersection point with the $x$-$y$-plane has parameter $t=-r/r^{\prime }$ and the intersection point is

• $$\begin{gathered} (\frac{r^2}{r^{\prime }}\sin \phi ,-\frac{r^2}{r^{\prime }}\cos \phi ,0) \\ . \end{gathered}$$

$r=a{\phi }^n$ gives $\frac{r^2}{r^{\prime }}=\frac{a}{n}{\phi }^{n+1}$ 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 $r=a{e}^{k\phi }$ one has $\frac{r^2}{r^{\prime }}=\frac{r}{k}$ and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.

File:Neptunea - links&rechts gewonden.jpg

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.