Английская Википедия:Conical spiral
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 <math>x</math>-<math>y</math>-plane a spiral with parametric representation
- <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi</math>
a third coordinate <math>z(\varphi)</math> can be added such that the space curve lies on the cone with equation <math>\;m^2(x^2+y^2)=(z-z_0)^2\ ,\ m>0\;</math> :
- <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\ , \qquad \color{red}{z=z_0 + mr(\varphi)} \ .</math>
Such curves are called conical spirals.[2] They were known to Pappos.
Parameter <math> m </math> is the slope of the cone's lines with respect to the <math>x</math>-<math>y</math>-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 <math>\;r(\varphi)=a\varphi\;</math> gives the conical spiral (see diagram)
- <math>x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .</math>
- 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 <math>\;r(\varphi)=\pm a\sqrt{\varphi}\;</math> as floor plan.
- 3) The third example has a logarithmic spiral <math>\; r(\varphi)=a e^{k\varphi} \; </math> as floor plan. Its special feature is its constant slope (see below).
- Introducing the abbreviation <math>K=e^k</math>gives the description: <math>r(\varphi)=aK^\varphi</math>.
- 4) Example 4 is based on a hyperbolic spiral <math>\; r(\varphi)=a/\varphi\; </math>. Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for <math> \varphi \to 0</math>.
Properties
The following investigation deals with conical spirals of the form <math>r=a\varphi^n</math> and <math>r=ae^{k\varphi}</math>, respectively.
Slope
The slope at a point of a conical spiral is the slope of this point's tangent with respect to the <math>x</math>-<math>y</math>-plane. The corresponding angle is its slope angle (see diagram):
- <math>\tan \beta = \frac{z'}{\sqrt{(x')^2+(y')^2}}=\frac{mr'}{\sqrt{(r')^2+r^2}}\ .</math>
A spiral with <math>r=a\varphi^n</math> gives:
- <math>\tan\beta=\frac{mn}{\sqrt{n^2+\varphi^2}}\ .</math>
For an archimedean spiral is <math>n=1</math> and hence its slope is<math>\ \tan\beta=\tfrac{m}{\sqrt{1+\varphi^2}}\ .</math>
- For a logarithmic spiral with <math>r=ae^{k\varphi}</math> the slope is <math>\ \tan\beta= \tfrac{mk}{\sqrt{1+k^2}}\ </math> (<math>\color{red}{\text{ constant!}}</math> ).
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
- <math>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 \ .</math>
For an archimedean spiral the integral can be solved with help of a table of integrals, analogously to the planar case:
- <math>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}\ .</math>
For a logarithmic spiral the integral can be solved easily:
- <math>L=\frac{\sqrt{(1+m^2)k^2+1}}{k}(r\big(\varphi_2)-r(\varphi_1)\big)\ .</math>
In other cases elliptical integrals occur.
Development
For the development of a conical spiral[3] the distance <math>\rho(\varphi)</math> of a curve point <math>(x,y,z)</math> to the cone's apex <math>(0,0,z_0)</math> and the relation between the angle <math>\varphi</math> and the corresponding angle <math>\psi</math> of the development have to be determined:
- <math>\rho=\sqrt{x^2+y^2+(z-z_0)^2}=\sqrt{1+m^2}\;r \ ,</math>
- <math>\varphi= \sqrt{1+m^2}\psi \ .</math>
Hence the polar representation of the developed conical spiral is:
- <math>\rho(\psi)=\sqrt{1+m^2}\; r(\sqrt{1+m^2}\psi)</math>
In case of <math>r=a\varphi^n</math> the polar representation of the developed curve is
- <math>\rho=a\sqrt{1+m^2}^{\,n+1}\psi^n,</math>
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 (<math>n=-1</math>) the development is congruent to the floor plan spiral.
In case of a logarithmic spiral <math>r=ae^{k\varphi}</math> the development is a logarithmic spiral:
- <math>\rho=a\sqrt{1+m^2}\;e^{k\sqrt{1+m^2}\psi}\ .</math>
Tangent trace
The collection of intersection points of the tangents of a conical spiral with the <math>x</math>-<math>y</math>-plane (plane through the cone's apex) is called its tangent trace.
For the conical spiral
- <math>(r\cos\varphi, r\sin\varphi,mr)</math>
the tangent vector is
- <math>(r'\cos\varphi-r\sin\varphi,r'\sin\varphi+r\cos\varphi,mr')^T</math>
and the tangent:
- <math>x(t)=r\cos\varphi+t(r'\cos\varphi-r\sin\varphi)\ ,</math>
- <math>y(t)=r\sin\varphi +t(r'\sin\varphi+r\cos\varphi)\ ,</math>
- <math>z(t)=mr+tmr'\ .</math>
The intersection point with the <math>x</math>-<math>y</math>-plane has parameter <math>t=-r/r'</math> and the intersection point is
- <math> \left( \frac{r^2}{r'}\sin\varphi, -\frac{r^2}{r'}\cos\varphi,0 \right)\ .</math>
<math>r=a\varphi^n</math> gives <math>\ \tfrac{r^2}{r'}=\tfrac{a}{n}\varphi^{n+1}\ </math> and the tangent trace is a spiral. In the case <math>n=-1</math> (hyperbolic spiral) the tangent trace degenerates to a circle with radius <math>a</math> (see diagram). For <math> r=a e^{k\varphi} </math> one has <math>\ \tfrac{r^2}{r'}=\tfrac{r}{k}\ </math> 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
- ↑ Шаблон:Cite web
- ↑ Siegmund Günther, Anton Edler von Braunmühl, Heinrich Wieleitner: Geschichte der mathematik. G. J. Göschen, 1921, p. 92.
- ↑ Theodor Schmid: Darstellende Geometrie. Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229.
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