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Conical spiral

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Plane spiral projected onto the surface of a cone
Conical spiral with an archimedean spiral as floor projection
Floor projection: Fermat's spiral
Floor projection: logarithmic spiral
Floor projection: hyperbolic 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 {\displaystyle x} - y {\displaystyle y} -plane a spiral with parametric representation

x = r ( φ ) cos ⁡ φ   , y = r ( φ ) sin ⁡ φ {\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }

a third coordinate z ( φ ) {\displaystyle 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 {\displaystyle \;m^{2}(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}  :

  • x = r ( φ ) cos ⁡ φ   , y = r ( φ ) sin ⁡ φ   , z = z 0 + m r ( φ )   . {\displaystyle 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 {\displaystyle m} is the slope of the cone's lines with respect to the x {\displaystyle x} - y {\displaystyle 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 ( φ ) = a φ {\displaystyle \;r(\varphi )=a\varphi \;} gives the conical spiral (see diagram)
x = a φ cos ⁡ φ   , y = a φ sin ⁡ φ   , z = z 0 + m a φ   , φ ≥ 0   . {\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 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 ( φ ) = ± a φ {\displaystyle \;r(\varphi )=\pm a{\sqrt {\varphi }}\;} as floor plan.
3) The third example has a logarithmic spiral r ( φ ) = a e k φ {\displaystyle \;r(\varphi )=ae^{k\varphi }\;} as floor plan. Its special feature is its constant slope (see below).
Introducing the abbreviation K = e k {\displaystyle K=e^{k}} gives the description: r ( φ ) = a K φ {\displaystyle r(\varphi )=aK^{\varphi }} .
4) Example 4 is based on a hyperbolic spiral r ( φ ) = a / φ {\displaystyle \;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 φ → 0 {\displaystyle \varphi \to 0} .

Properties

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

Slope

Slope angle at a point of a conical spiral

The slope at a point of a conical spiral is the slope of this point's tangent with respect to the x {\displaystyle x} - y {\displaystyle y} -plane. The corresponding angle is its slope angle (see diagram):

tan ⁡ β = z ′ ( x ′ ) 2 + ( y ′ ) 2 = m r ′ ( r ′ ) 2 + r 2   . {\displaystyle \tan \beta ={\frac {z'}{\sqrt {(x')^{2}+(y')^{2}}}}={\frac {mr'}{\sqrt {(r')^{2}+r^{2}}}}\ .}

A spiral with r = a φ n {\displaystyle r=a\varphi ^{n}} gives:

  • tan ⁡ β = m n n 2 + φ 2   . {\displaystyle \tan \beta ={\frac {mn}{\sqrt {n^{2}+\varphi ^{2}}}}\ .}

For an archimedean spiral, n = 1 {\displaystyle n=1} , and hence its slope is   tan ⁡ β = m 1 + φ 2   . {\displaystyle \ \tan \beta ={\tfrac {m}{\sqrt {1+\varphi ^{2}}}}\ .}

  • For a logarithmic spiral with r = a e k φ {\displaystyle r=ae^{k\varphi }} the slope is   tan ⁡ β = m k 1 + k 2   {\displaystyle \ \tan \beta ={\tfrac {mk}{\sqrt {1+k^{2}}}}\ } (  constant! {\displaystyle \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 = ∫ φ 1 φ 2 ( x ′ ) 2 + ( y ′ ) 2 + ( z ′ ) 2 d φ = ∫ φ 1 φ 2 ( 1 + m 2 ) ( r ′ ) 2 + r 2 d φ   . {\displaystyle 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 = a 2 [ φ ( 1 + m 2 ) + φ 2 + ( 1 + m 2 ) ln ⁡ ( φ + ( 1 + m 2 ) + φ 2 ) ] φ 1 φ 2   . {\displaystyle L={\frac {a}{2}}\left_{\varphi _{1}}^{\varphi _{2}}\ .}

For a logarithmic spiral the integral can be solved easily:

L = ( 1 + m 2 ) k 2 + 1 k ( r ( φ 2 ) − r ( φ 1 ) )   . {\displaystyle 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

Development(green) of a conical spiral (red), right: a side view. The plane containing the development is designed by π {\displaystyle \pi } . Initially the cone and the plane touch at the purple line.

For the development of a conical spiral the distance ρ ( φ ) {\displaystyle \rho (\varphi )} of a curve point ( x , y , z ) {\displaystyle (x,y,z)} to the cone's apex ( 0 , 0 , z 0 ) {\displaystyle (0,0,z_{0})} and the relation between the angle φ {\displaystyle \varphi } and the corresponding angle ψ {\displaystyle \psi } of the development have to be determined:

ρ = x 2 + y 2 + ( z − z 0 ) 2 = 1 + m 2 r   , {\displaystyle \rho ={\sqrt {x^{2}+y^{2}+(z-z_{0})^{2}}}={\sqrt {1+m^{2}}}\;r\ ,}
φ = 1 + m 2 ψ   . {\displaystyle \varphi ={\sqrt {1+m^{2}}}\psi \ .}

Hence the polar representation of the developed conical spiral is:

  • ρ ( ψ ) = 1 + m 2 r ( 1 + m 2 ψ ) {\displaystyle \rho (\psi )={\sqrt {1+m^{2}}}\;r({\sqrt {1+m^{2}}}\psi )}

In case of r = a φ n {\displaystyle r=a\varphi ^{n}} the polar representation of the developed curve is

ρ = a 1 + m 2 n + 1 ψ n , {\displaystyle \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 {\displaystyle n=-1} ) the development is congruent to the floor plan spiral.

In case of a logarithmic spiral r = a e k φ {\displaystyle r=ae^{k\varphi }} the development is a logarithmic spiral:

ρ = a 1 + m 2 e k 1 + m 2 ψ   . {\displaystyle \rho =a{\sqrt {1+m^{2}}}\;e^{k{\sqrt {1+m^{2}}}\psi }\ .}

Tangent trace

The trace (purple) of the tangents of a conical spiral with a hyperbolic spiral as floor plan. The black line is the asymptote of the hyperbolic spiral.

The collection of intersection points of the tangents of a conical spiral with the x {\displaystyle x} - y {\displaystyle y} -plane (plane through the cone's apex) is called its tangent trace.

For the conical spiral

( r cos ⁡ φ , r sin ⁡ φ , m r ) {\displaystyle (r\cos \varphi ,r\sin \varphi ,mr)}

the tangent vector is

( r ′ cos ⁡ φ − r sin ⁡ φ , r ′ sin ⁡ φ + r cos ⁡ φ , m r ′ ) T {\displaystyle (r'\cos \varphi -r\sin \varphi ,r'\sin \varphi +r\cos \varphi ,mr')^{T}}

and the tangent:

x ( t ) = r cos ⁡ φ + t ( r ′ cos ⁡ φ − r sin ⁡ φ )   , {\displaystyle x(t)=r\cos \varphi +t(r'\cos \varphi -r\sin \varphi )\ ,}
y ( t ) = r sin ⁡ φ + t ( r ′ sin ⁡ φ + r cos ⁡ φ )   , {\displaystyle y(t)=r\sin \varphi +t(r'\sin \varphi +r\cos \varphi )\ ,}
z ( t ) = m r + t m r ′   . {\displaystyle z(t)=mr+tmr'\ .}

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

  • ( r 2 r ′ sin ⁡ φ , − r 2 r ′ cos ⁡ φ , 0 )   . {\displaystyle \left({\frac {r^{2}}{r'}}\sin \varphi ,-{\frac {r^{2}}{r'}}\cos \varphi ,0\right)\ .}

r = a φ n {\displaystyle r=a\varphi ^{n}} gives   r 2 r ′ = a n φ n + 1   {\displaystyle \ {\tfrac {r^{2}}{r'}}={\tfrac {a}{n}}\varphi ^{n+1}\ } and the tangent trace is a spiral. In the case n = − 1 {\displaystyle n=-1} (hyperbolic spiral) the tangent trace degenerates to a circle with radius a {\displaystyle a} (see diagram). For r = a e k φ {\displaystyle r=ae^{k\varphi }} one has   r 2 r ′ = r k   {\displaystyle \ {\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.

Snail shells (Neptunea angulata left, right: Neptunea despecta

References

  1. "Conical helix". MATHCURVE.COM. Retrieved 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.

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