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List of spirals

## Summary

This list of spirals includes named spirals that have been described mathematically.

Image Name First described Equation Comment
circle ${\displaystyle r=k}$ The trivial spiral
Archimedean spiral c. 320 BC ${\displaystyle r=a+b\cdot \theta }$
Euler spiral also called Cornu spiral or polynomial spiral
Fermat's spiral (also parabolic spiral) 1636[1] ${\displaystyle r^{2}=a^{2}\theta }$
hyperbolic spiral 1704 ${\displaystyle r=a/\theta }$ also reciprocal spiral
lituus 1722 ${\displaystyle r^{2}\theta =k}$
logarithmic spiral 1638[2] ${\displaystyle r=a\cdot e^{b\theta }}$ approximations of this are found in nature
Fibonacci spiral circular arcs connecting the opposite corners of squares in the Fibonacci tiling approximation of the golden spiral
golden spiral ${\displaystyle r=\varphi ^{\frac {2\theta }{\pi }}\,}$ special case of the logarithmic spiral
Spiral of Theodorus (also Pythagorean spiral) an polygonal spiral composed of contiguous right triangles, that approximates the Archimedean spiral
involute 1673
helix ${\displaystyle r(t)=1,\,}$ ${\displaystyle \theta (t)=t,\,}$ ${\displaystyle h(t)=t.\,}$ a 3-dimensional spiral
Rhumb line (also loxodrome) type of spiral drawn on a sphere
Cotes's spiral 1722
Poinsot's spirals ${\displaystyle r=a\operatorname {csch} (n\theta ),\,}$
${\displaystyle r=a\operatorname {sech} (n\theta )}$
Nielsen's spiral 1993[3] ${\displaystyle x(t)=\operatorname {ci} (t),\,}$
${\displaystyle y(t)=\operatorname {si} (t)}$
A variation of Euler spiral, using sine integral and cosine integrals
Polygonal spiral special case approximation of logarithmic spiral
Fraser's Spiral 1908 Optical illusion based on spirals
Conchospiral ${\displaystyle {\begin{cases}r=\mu ^{t}a\\\theta =t\\z=\mu ^{t}c\end{cases}}}$ three-dimensional spiral on the surface of a cone.
Calkin–Wilf spiral
Ulam spiral (also prime spiral) 1963
Sack's spiral 1994 variant of Ulam spiral and Archimedean spiral.
Seiffert's spiral spiral curve on the surface of a sphere
Tractrix spiral 1704[4] ${\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}$
Pappus spiral 1779 ${\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}}$ 3D conical spiral studied by Pappus and Pascal[5]
doppler spiral ${\displaystyle {\begin{cases}x=a(t\cos(t)+kt)\\y=at\sin(t)\end{cases}}}$ 2D projection of Pappus spiral[6]
Atzema spiral ${\displaystyle {\begin{cases}x=\sin(t)/t-2\cos(t)-t\sin(t)\\y=-\cos(t)/t-2\sin(t)+t\cos(t)\end{cases}}}$ The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.[7]
Atomic spiral 2002 ${\displaystyle r=\theta /(\theta -a)}$ This spiral has two asymptotes; one is the circle of radius 1 and the other is the line ${\displaystyle \theta =a}$[8]
Galactic spiral 2019 ${\displaystyle {\begin{cases}dx=R*{\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R*{\Bigl [}\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}{\Bigr ]}d\theta \end{cases}}{\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases}}}$ The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:${\displaystyle \rho <1,\rho =1,\rho >1}$, the spiral patterns are decided by the behavior of the parameter ${\displaystyle \rho }$. For ${\displaystyle \rho <1}$, spiral-ring pattern; ${\displaystyle \rho =1,}$ regular spiral; ${\displaystyle \rho >1,}$ loose spiral. R is the distance of spiral starting point (0, R) to the center. The calculated x and y have to be rotated backward by (${\displaystyle -\theta }$) for plotting. Please check the references for the detail[9]