Conical Helix

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

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

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

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

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)
: $x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .$
: 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'' $\;r(\varphi)=\pm a\sqrt\;$ as floor plan.
: 3) The third example has a ''logarithmic spiral'' $\; r(\varphi)=a e^ \;$ 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 $r=ae^$, respectively.

Slope

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):

: $\tan \beta = \frac=\frac\ .$

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

* $\tan\beta=\frac\ .$

For an ''archimedean'' spiral, $n=1$, and hence its slope is$\ \tan\beta=\tfrac\ .$

* For a ''logarithmic'' spiral with $r=ae^$ the slope is $\ \tan\beta= \tfrac\$ ($\color$ ).

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

: $L=\int_^\sqrt\,\mathrm\varphi = \int_^\sqrt\,\mathrm\varphi \ .$

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

: L= \frac \left varphi\sqrt + (1+m^2)\ln \left(\varphi + \sqrt\right)\right ^\ .

For a ''logarithmic'' spiral the integral can be solved easily:

: $L=\frac(r\big(\varphi_2)-r(\varphi_1)\big)\ .$

In other cases elliptical integrals occur.

Development

For the development of a conical spiralTheodor Schmid: ''Darstellende Geometrie.'' Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229. 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:

: $\rho=\sqrt=\sqrt\;r \ ,$
: $\varphi= \sqrt\psi \ .$

Hence the polar representation of the developed conical spiral is:

* $\rho(\psi)=\sqrt\; r(\sqrt\psi)$

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

: $\rho=a\sqrt^\psi^n,$

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 $r=ae^$ the development is a logarithmic spiral:

: $\rho=a\sqrt\;e^\ .$

Tangent trace

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

* $\left( \frac\sin\varphi, -\frac\cos\varphi,0 \right)\ .$

$r=a\varphi^n$ gives $\ \tfrac=\tfrac\varphi^\$ 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 $r=a e^$ one has $\ \tfrac=\tfrac\$ and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.

References

External links

* Jamnitzer-Galerie:
3D-Spiralen.
''.
* {{MathWorld, ConicalSpiral, Conical Spiral

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Spirals