Conical spiral

In the ${\displaystyle x}$-${\displaystyle y}$-plane a spiral with parametric representation

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

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

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

Such curves are called conical spirals.^[2] They were known to Pappos.

Parameter ${\displaystyle m}$ is the slope of the cone's lines with respect to the ${\displaystyle x}$-${\displaystyle y}$-plane.

A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone.

1) Starting with an archimedean spiral ${\displaystyle \;r(\varphi )=a\varphi \;}$ gives the conical spiral (see diagram)

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

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

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

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

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

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

The slope at a point of a conical spiral is the slope of this point's tangent with respect to the ${\displaystyle x}$-${\displaystyle y}$-plane. The corresponding angle is its slope angle (see diagram):

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

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

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

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

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

Because of this property a conchospiral is called an equiangular conical spiral.

The length of an arc of a conical spiral can be determined by

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

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

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

For a logarithmic spiral the integral can be solved easily:

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

In other cases elliptical integrals occur.

For the development of a conical spiral^[3] the distance ${\displaystyle \rho (\varphi )}$ of a curve point ${\displaystyle (x,y,z)}$ to the cone's apex ${\displaystyle (0,0,z_{0})}$ and the relation between the angle ${\displaystyle \varphi }$ and the corresponding angle ${\displaystyle \psi }$ of the development have to be determined:

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

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

Hence the polar representation of the developed conical spiral is:

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

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

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

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

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

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

The collection of intersection points of the tangents of a conical spiral with the ${\displaystyle x}$-${\displaystyle y}$-plane (plane through the cone's apex) is called its tangent trace.

For the conical spiral

${\displaystyle (r\cos \varphi ,r\sin \varphi ,mr)}$

the tangent vector is

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

and the tangent:

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

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

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

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

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

${\displaystyle r=a\varphi ^{n}}$ gives ${\displaystyle \ {\tfrac {r^{2}}{r'}}={\tfrac {a}{n}}\varphi ^{n+1}\ }$ and the tangent trace is a spiral. In the case ${\displaystyle n=-1}$ (hyperbolic spiral) the tangent trace degenerates to a circle with radius ${\displaystyle a}$ (see diagram). For ${\displaystyle r=ae^{k\varphi }}$ one has ${\displaystyle \ {\tfrac {r^{2}}{r'}}={\tfrac {r}{k}}\ }$ and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.