Editing Hyperbolic spiral (section)

== Properties ==
=== Asymptotes ===
The hyperbolic spiral approaches the origin as an asymptotic point.{{r|shikin}} Because
<math display=block>\lim_{\varphi\to 0}x = a\lim_{\varphi\to 0} \frac{\cos \varphi} \varphi =\infty,\qquad
\lim_{\varphi\to 0}y = a\lim_{\varphi\to 0} \frac{\sin \varphi} \varphi = a,</math>
the curve has an [[asymptotic line]] with {{nowrap|equation <math>y=a</math>.{{r|polezhaev}}}}

=== Pitch angle ===
[[File:Sektor-steigung-pk-def.svg|thumb|Definition of sector (light blue) and pitch angle {{mvar|α}}]]
From [[polar coordinate system#Vector calculus|vector calculus in polar coordinates]] one gets the formula <math>\tan\alpha=\tfrac{r'}{r}</math> for the [[Pitch angle of a spiral|pitch angle]] <math>\alpha</math> between the tangent of any curve and the tangent of its corresponding polar circle.{{r|kepr}} For the hyperbolic spiral <math>r=\tfrac{a}{\varphi}</math> the pitch angle is{{r|scott-noland}}
<math display=block>\alpha=\tan^{-1}\left(-\frac{1}{\varphi}\right).</math>

=== Curvature ===
The [[curvature]] of any curve with polar equation <math>r=r(\varphi)</math> is{{r|curvature}}
<math display=block>\kappa = \frac{r^2 + 2(r')^2 - r\, r''}{\left(r^2+(r')^2\right)^{3/2}} .</math>
From the equation <math>r=a/\varphi</math> and its derivatives <math>r'=-a/\varphi^2</math> and <math>r''=2a/\varphi^3</math> one gets the curvature of a hyperbolic spiral, in terms of the radius <math>r</math> or of the angle <math>\varphi</math> of any of its points:{{r|ganguli}}
<math display=block>\kappa = \frac{\varphi^4}{a \left(\varphi^2 + 1\right)^{3/2}}
= \frac{a^3}{r(a^2+r^2)^{3/2}}.
</math>

=== Arc length ===
The length of the arc of a hyperbolic spiral <math>r=a/\varphi</math> between the points <math>(r(\varphi_1),\varphi_1)</math> and <math>(r(\varphi_2),\varphi_2)</math> can be calculated by the integral:{{r|polezhaev}}
<math display=block>\begin{align}
L&=a \int_{\varphi_1}^{\varphi_2}\frac{\sqrt{1+\varphi^2}}{\varphi^2}\,d\varphi \\
&= a\left[-\frac{\sqrt{1+\varphi^2}}{\varphi}+\ln\left(\varphi+\sqrt{1+\varphi^2}\right)\right]_{\varphi_1}^{\varphi_2} .
\end{align}</math>
Here, the bracket notation means to calculate the formula within the brackets for both <math>\varphi_1</math> and <math>\varphi_2</math>, and to subtract the result for <math>\varphi_1</math> from the result for <math>\varphi_2</math>.

=== Sector area ===
The area of a sector (see diagram above) of a hyperbolic spiral with equation <math>r=a/\varphi</math> is:{{r|polezhaev}}
<math display=block>\begin{align}
A&=\frac12\int_{\varphi_1}^{\varphi_2} r(\varphi)^2\, d\varphi\\
&=\frac{a}{2}\bigl(r(\varphi_1)-r(\varphi_2)\bigr) .
\end{align}</math>
That is, the area is proportional to the difference in radii, with constant of proportionality {{nowrap|<math>a/2</math>.{{r|cotes|polezhaev}}}}