Conical spiral

 ( Spirals ) 

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).

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

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

a third coordinate $z(\backslash varphi)$ can be added such that the space curve lies on the cone with equation $\backslash ;m^2(x^2+y^2)=(z-z\_0)^2\backslash \; ,\backslash \; m>0\backslash ;$ :

• $x=r(\backslash varphi)\backslash cos\backslash varphi\; \backslash \; ,\backslash qquad\; y=r(\backslash varphi)\backslash sin\backslash varphi\backslash \; ,\; \backslash qquad\; \backslash color\{red\}\{z=z\_0\; +\; mr(\backslash varphi)\}\; \backslash \; .$

Such curves are called conical spirals.Siegmund Günther, Anton Edler von Braunmühl, Heinrich Wieleitner: Geschichte der mathematik. G. J. Göschen, 1921, p. 92. 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.

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Examples

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

$x=a\backslash varphi\backslash cos\backslash varphi\; \backslash \; ,\backslash qquad\; y=a\backslash varphi\backslash sin\backslash varphi\backslash \; ,\; \backslash qquad\; z=z\_0\; +\; ma\backslash varphi\; \backslash \; ,\backslash quad\; \backslash varphi\; \backslash ge\; 0\; \backslash \; .$

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

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

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

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

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

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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):

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

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

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

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

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

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

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

$L=\backslash int\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\backslash sqrt\{(x\text{'})^2+(y\text{'})^2+(z\text{'})^2\}\backslash ,\backslash mathrm\{d\}\backslash varphi$

= \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:

$L=\; \backslash frac\{a\}\{2\}\; \backslash left\backslash varphi\backslash sqrt\{(1+m^2)\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\backslash \; .$

For a logarithmic spiral the integral can be solved easily:

$L=\backslash frac\{\backslash sqrt\{(1+m^2)k^2+1\}\}\{k\}(r\backslash big(\backslash varphi\_2)-r(\backslash varphi\_1)\backslash big)\backslash \; .$

In other cases elliptical integrals occur.

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Development For the development of a conical spiralTheodor Schmid: Darstellende Geometrie. Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229. the distance $\backslash rho(\backslash varphi)$ of a curve point $(x,y,z)$ to the cone's apex $(0,0,z\_0)$ and the relation between the angle $\backslash varphi$ and the corresponding angle $\backslash psi$ of the development have to be determined:

$\backslash rho=\backslash sqrt\{x^2+y^2+(z-z\_0)^2\}=\backslash sqrt\{1+m^2\}\backslash ;r\; \backslash \; ,$

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

Hence the polar representation of the developed conical spiral is:

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

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

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

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

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

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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

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

the tangent vector is

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

and the tangent:

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

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

$z(t)=mr+tmr\text{'}\backslash \; .$

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

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

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

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External links

• Wenzel Jamnitzer-Galerie: 3D-Spiralen. .

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