Examples

Some of the most important sorts of two-dimensional spirals include:

• The Archimedean spiral: ${\displaystyle r=a\varphi }$
• The hyperbolic spiral: ${\displaystyle r=a/\varphi }$
• Fermat's spiral: ${\displaystyle r=a\varphi ^{1/2}}$
• The lituus: ${\displaystyle r=a\varphi ^{-1/2}}$
• The logarithmic spiral: ${\displaystyle r=ae^{k\varphi }}$
• The Cornu spiral or clothoid
• The Fibonacci spiral and golden spiral
• The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles
• The involute of a circle, used twice on each tooth of almost every modern gear

• 
  Archimedean spiral
• 
  hyperbolic spiral
• 
  Fermat's spiral
• 
  lituus
• 
  logarithmic spiral
• 
  Cornu spiral
• 
  spiral of Theodorus
• 
  Fibonacci Spiral (golden spiral)
• 
  The involute of a circle (black) is not identical to the Archimedean spiral (red).

Hyperbolic spiral as central projection of a helix

An Archimedean spiral is, for example, generated while coiling a carpet.^[6]

A hyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).^[7]

The name logarithmic spiral is due to the equation ${\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}}$. Approximations of this are found in nature.

Spirals which do not fit into this scheme of the first 5 examples:

A Cornu spiral has two asymptotic points.
The spiral of Theodorus is a polygon.
The Fibonacci Spiral consists of a sequence of circle arcs.
The involute of a circle looks like an Archimedean, but is not: see Involute#Examples.

Geometric properties

The following considerations are dealing with spirals, which can be described by a polar equation ${\displaystyle r=r(\varphi )}$, especially for the cases ${\displaystyle r(\varphi )=a\varphi ^{n}}$ (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral ${\displaystyle r=ae^{k\varphi }}$.

Definition of sector (light blue) and polar slope angle ${\displaystyle \alpha }$

Polar slope angle

The angle ${\displaystyle \alpha }$ between the spiral tangent and the corresponding polar circle (see diagram) is called angle of the polar slope and ${\displaystyle \tan \alpha }$ the polar slope.

From vector calculus in polar coordinates one gets the formula

${\displaystyle \tan \alpha ={\frac {r'}{r}}\ .}$

Hence the slope of the spiral ${\displaystyle \;r=a\varphi ^{n}\;}$ is

• ${\displaystyle \tan \alpha ={\frac {n}{\varphi }}\ .}$

In case of an Archimedean spiral (${\displaystyle n=1}$) the polar slope is ${\displaystyle \;\tan \alpha ={\tfrac {1}{\varphi }}\ .}$

The logarithmic spiral is a special case, because of ${\displaystyle \ \tan \alpha =k\ }$ constant !

curvature

The curvature ${\displaystyle \kappa }$ of a curve with polar equation ${\displaystyle r=r(\varphi )}$ is

${\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .}$

For a spiral with ${\displaystyle r=a\varphi ^{n}}$ one gets

• ${\displaystyle \kappa =\dotsb ={\frac {1}{a\varphi ^{n-1}}}{\frac {\varphi ^{2}+n^{2}+n}{(\varphi ^{2}+n^{2})^{3/2}}}\ .}$

In case of ${\displaystyle n=1}$ (Archimedean spiral) ${\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}}$.
Only for ${\displaystyle -1<n<0}$ the spiral has an inflection point.

The curvature of a logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.}$

Sector area

The area of a sector of a curve (see diagram) with polar equation ${\displaystyle r=r(\varphi )}$ is

${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .}$

For a spiral with equation ${\displaystyle r=a\varphi ^{n}\;}$ one gets

• ${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},}$

${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .}$

The formula for a logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}$

Arc length

The length of an arc of a curve with polar equation ${\displaystyle r=r(\varphi )}$ is

${\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}$

For the spiral ${\displaystyle r=a\varphi ^{n}\;}$ the length is

• ${\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .}$

Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by elliptic integrals only.

The arc length of a logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}$

Circle inversion

The inversion at the unit circle has in polar coordinates the simple description: ${\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }$.

• The image of a spiral ${\displaystyle \ r=a\varphi ^{n}\ }$ under the inversion at the unit circle is the spiral with polar equation ${\displaystyle \ r={\tfrac {1}{a}}\varphi ^{-n}\ }$. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.

A logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is mapped onto the logarithmic spiral ${\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\;.}$

Bounded spirals

Bounded spirals:
${\displaystyle r=a\arctan(k\varphi )}$ (left),
${\displaystyle r=a(\arctan(k\varphi )+\pi /2)}$ (right)

The function ${\displaystyle r(\varphi )}$ of a spiral is usually strictly monotonic, continuous and unbounded. For the standard spirals ${\displaystyle r(\varphi )}$ is either a power function or an exponential function. If one chooses for ${\displaystyle r(\varphi )}$ a bounded function, the spiral is bounded, too. A suitable bounded function is the arctan function:

Example 1

Setting ${\displaystyle \;r=a\arctan(k\varphi )\;}$ and the choice ${\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;}$ gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius ${\displaystyle \;r=a\pi /2\;}$ (diagram, left).

Example 2

For ${\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;}$ and ${\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;}$ one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius ${\displaystyle \;r=a\pi \;}$ (diagram, right).

Conical spirals

If in the ${\displaystyle x}$-${\displaystyle y}$-plane a spiral with parametric representation

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

is given, then there can be added a third coordinate ${\displaystyle z(\varphi )}$, such that the now space curve lies on the cone with equation ${\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}$:

• ${\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}$

Spirals based on this procedure are called conical spirals.

Example

Starting with an archimedean spiral ${\displaystyle \;r(\varphi )=a\varphi \;}$ one gets the conical spiral (see diagram)

${\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}$

Spherical spiral with ${\displaystyle c=8}$

Spherical spirals

If one represents a sphere of radius ${\displaystyle r}$ by:

${\displaystyle {\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos \varphi \\y&=&r\cdot \sin \theta \cdot \sin \varphi \\z&=&r\cdot \cos \theta \end{array}}}$

and sets the linear dependency ${\displaystyle \;\varphi =c\theta ,\;c>2\;,}$ for the angle coordinates, one gets a spherical curve called spherical spiral^[11] with the parametric representation (with ${\displaystyle c}$ equal to twice the number of turns)

• ${\displaystyle {\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos {\color {red}c\theta }\\y&=&r\cdot \sin \theta \cdot \sin {\color {red}c\theta }\\z&=&r\cdot \cos \theta \qquad \qquad 0\leq \theta \leq \pi \ .\end{array}}}$

Spherical spirals were known to Pappus, too.

Remark: a rhumb line is not a spherical spiral in this sense.

• 
  Spherical spiral
• 
  Loxodrome

A rhumb line (also known as a loxodrome or "spherical spiral") is the curve on a sphere traced by a ship with constant bearing (e.g., travelling from one pole to the other while keeping a fixed angle with respect to the meridians). The loxodrome has an infinite number of revolutions, with the separation between them decreasing as the curve approaches either of the poles, unlike an Archimedean spiral which maintains uniform line-spacing regardless of radius.