Coordinate equations

The hyperbolic spiral has the polar equation

$${\displaystyle r={\frac {a}{\varphi }},\quad \varphi >0}$$

for polar coordinates ${\displaystyle (r,\varphi )}$ and scale coefficient ${\displaystyle a}$. It can be represented in Cartesian coordinates by applying the standard polar-to-Cartesian conversions ${\displaystyle x=r\cos \varphi }$ and ${\displaystyle y=r\sin \varphi }$, obtaining a parametric equation for the Cartesian coordinates of this curve that treats ${\displaystyle \varphi }$ as a parameter rather than as a coordinate:^[15]

$${\displaystyle x=a{\frac {\cos \varphi }{\varphi }},\qquad y=a{\frac {\sin \varphi }{\varphi }},\quad \varphi >0.}$$

Relaxing the constraint that ${\displaystyle \varphi >0}$ to ${\displaystyle \varphi \neq 0}$ and using the same equations produces a reflected copy of the spiral, and some sources treat these two copies as branches of a single curve.^[9][16]

Hyperbolic spiral: branch for φ > 0

Hyperbolic spiral: both branches

The hyperbolic spiral is a transcendental curve, meaning that it cannot be defined from a polynomial equation of its Cartesian coordinates.^[15] However, one can obtain a trigonometric equation in these coordinates by starting with its polar defining equation in the form ${\displaystyle r\varphi =a}$ and replacing its variables according to the Cartesian-to-polar conversions ${\displaystyle \varphi =\tan ^{-1}{\tfrac {y}{x}}}$ and ${\textstyle r={\sqrt {x^{2}+y^{2}}}}$, giving:^[17]

$${\displaystyle {\sqrt {x^{2}+y^{2}}}\tan ^{-1}{\frac {y}{x}}=a.}$$

Inversion

Hyperbolic spiral (blue) as image of an Archimedean spiral (green) by inversion through a circle (red)

Circle inversion through the unit circle is a transformation of the plane that, in polar coordinates, maps the point ${\displaystyle (r,\varphi )}$ (excluding the origin) to ${\displaystyle ({\tfrac {1}{r}},\varphi )}$ and vice versa.^[18] The image of an Archimedean spiral ${\displaystyle r={\tfrac {\varphi }{a}}}$ under this transformation (its inverse curve) is the hyperbolic spiral with equation ${\displaystyle r={\tfrac {a}{\varphi }}}$.^[11]

Central projection of a helix

Hyperbolic spiral as central projection of a helix

The central projection of a helix onto a plane perpendicular to the axis of the helix describes the view that one would see of the guardrail of a spiral staircase, looking up or down from a viewpoint on the axis of the staircase.^[1] To model this projection mathematically, consider the central projection from point ${\displaystyle (0,0,d)}$ onto the image plane ${\displaystyle z=0}$. This will map a point ${\displaystyle (x,y,z)}$ to the point ${\displaystyle {\tfrac {d}{d-z}}(x,y)}$.^[19]

The image under this projection of the helix with parametric representation

$${\displaystyle (r\cos t,r\sin t,ct),\quad c\neq 0,}$$

is the curve

$${\displaystyle {\frac {dr}{d-ct}}(\cos t,\sin t)}$$

with the polar equation

$${\displaystyle \rho ={\frac {dr}{d-ct}},}$$

which describes a hyperbolic spiral.^[19]

Asymptotes

The hyperbolic spiral approaches the origin as an asymptotic point.^[17] Because

$${\displaystyle \lim _{\varphi \to 0}x=a\lim _{\varphi \to 0}{\frac {\cos \varphi }{\varphi }}=\infty ,\qquad \lim _{\varphi \to 0}y=a\lim _{\varphi \to 0}{\frac {\sin \varphi }{\varphi }}=a,}$$

the curve has an asymptotic line with equation ${\displaystyle y=a}$.^[15]

Pitch angle

Definition of sector (light blue) and pitch angle α

From vector calculus in polar coordinates one gets the formula ${\displaystyle \tan \alpha ={\tfrac {r'}{r}}}$ for the pitch angle ${\displaystyle \alpha }$ between the tangent of any curve and the tangent of its corresponding polar circle.^[20] For the hyperbolic spiral ${\displaystyle r={\tfrac {a}{\varphi }}}$ the pitch angle is^[14]

$${\displaystyle \alpha =\tan ^{-1}\left(-{\frac {1}{\varphi }}\right).}$$

Curvature

The curvature of any curve with polar equation ${\displaystyle r=r(\varphi )}$ is^[21]

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

From the equation ${\displaystyle r=a/\varphi }$ and its derivatives ${\displaystyle r'=-a/\varphi ^{2}}$ and ${\displaystyle r''=2a/\varphi ^{3}}$ one gets the curvature of a hyperbolic spiral:^[22]

$${\displaystyle \kappa ={\frac {\varphi ^{4}}{a\left(\varphi ^{2}+1\right)^{3/2}}}={\frac {a^{3}}{r(a^{2}+r^{2})^{3/2}}}.}$$

Arc length

The length of the arc of a hyperbolic spiral between the points ${\displaystyle (r(\varphi _{1}),\varphi _{1})}$ and ${\displaystyle (r(\varphi _{2}),\varphi _{2})}$ can be calculated by the integral:^[15]

$${\displaystyle {\begin{aligned}L&=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,d\varphi =\cdots \\&=a\int _{\varphi _{1}}^{\varphi _{2}}{\frac {\sqrt {1+\varphi ^{2}}}{\varphi ^{2}}}\,d\varphi \\&=a\left[-{\frac {\sqrt {1+\varphi ^{2}}}{\varphi }}+\ln \left(\varphi +{\sqrt {1+\varphi ^{2}}}\right)\right]_{\varphi _{1}}^{\varphi _{2}}.\end{aligned}}}$$

Sector area

The area of a sector (see diagram above) of a hyperbolic spiral with equation ${\displaystyle r=a/\varphi }$ is:^[15]

$${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\,d\varphi \\&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi ^{2}}}\,d\varphi \\&={\frac {a}{2}}\left({\frac {a}{\varphi _{1}}}-{\frac {a}{\varphi _{2}}}\right)\\&={\frac {a}{2}}{\bigl (}r(\varphi _{1})-r(\varphi _{2}){\bigr )}.\end{aligned}}}$$

That is, the area is proportional to the difference in radii, with constant of proportionality ${\displaystyle a/2}$.^[15]