Conical spiral



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 $$\;m^2(x^2+y^2)=(z-z_0)^2\ ,\ m>0\;$$ :


 * $$x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\, \qquad \color{red}{z=z_0 + mr(\varphi)} \ .$$

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 $$\;r(\varphi)=\pm a\sqrt{\varphi}\;$$ 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):


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

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


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

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 $$\ \tan\beta= \tfrac{mk}{\sqrt{1+k^2}}\ $$ ($$\color{red}{\text{ constant!}}$$ ).

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_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:


 * $$L= \frac{a}{2} \left[\varphi\sqrt{(1+m^2) + \varphi^2} + (1+m^2)\ln \left(\varphi + \sqrt{(1+m^2) + \varphi^2}\right)\right ]_{\varphi_1}^{\varphi_2}\ .$$

For a logarithmic spiral the integral can be solved easily:


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

In other cases elliptical integrals occur.

Development
For the development of a conical spiral 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:


 * $$\rho=\sqrt{x^2+y^2+(z-z_0)^2}=\sqrt{1+m^2}\;r \ ,$$
 * $$\varphi= \sqrt{1+m^2}\psi \ .$$

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


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

$$r=a\varphi^n$$ 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.