Conical spiral explained

In mathematics, a conical spiral, also known as a conical helix,^[1] 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 representationx = r(\varphi)\cos\varphi , 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 , m> 0} ^{0 )} :x = r(\varphi)\cos\varphi , y = r(\varphi)\sin\varphi , \color{red}{z = z _{0} + mr(\varphi)} .

Such curves are called conical spirals.^[2] 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 , y = a\varphi\sin\varphi , z = z _{0} + ma\varphi , \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{\varphi} as floor plan.

3) The third example has a logarithmic spiral

r(\varphi) = a e ^{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

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 = (z')/(\sqrt{(x') ^{2+(y')} ^{2}) }=\frac\ .

A spiral with

r = a\varphi ^{n} gives:\tan\beta = (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}{ 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 _{2}} _{\varphi _{1}} \sqrt{(x') ^{2+(y')} ^{2+(z')} ^{2}d\varphi =} \int ^{\varphi _{2}} _{\varphi _{1}} \sqrt{( 1+ m ^{2)(r')} ^{2+r} ^{2}d\varphi} .

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

L = (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 _{2}} _{\varphi _{1}} .

For a logarithmic spiral the integral can be solved easily:

L = (\sqrt{( 1+ m ^{2)k} ^{2+1})/() }(r\big(\varphi_2)-r(\varphi_1)\big)\ .

In other cases elliptical integrals occur.

Development

For the development of a conical spiral^[3] 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 } = \sqrt{ 1+ m} ^{0 )} ^{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 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( (r ^{2})/(r') \sin\varphi, - (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 = a e ^{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.

References

1. Web site: Conical helix . MATHCURVE.COM . 2022-03-03.
2. Siegmund Günther, Anton Edler von Braunmühl, Heinrich Wieleitner: Geschichte der mathematik. G. J. Göschen, 1921, p. 92.
3. Theodor Schmid: Darstellende Geometrie. Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229.

External links

• Jamnitzer-Galerie: 3D-Spiralen. .