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Summary

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

Image Name First described Equation Comment circle $r=k$ The trivial spiral Archimedean spiral c. 320 BC $r=a+b\cdot \theta$  Euler spiral also called Cornu spiral or polynomial spiral Fermat's spiral (also parabolic spiral) 1636 $r^{2}=a^{2}\theta$  hyperbolic spiral 1704 $r=a/\theta$ also reciprocal spiral lituus 1722 $r^{2}\theta =k$  logarithmic spiral 1638 $r=a\cdot e^{b\theta }$ 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 $r=\varphi ^{\frac {2\theta }{\pi }}\,$ 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 $r(t)=1,\,$ $\theta (t)=t,\,$ $h(t)=t.\,$ a 3-dimensional spiral Rhumb line (also loxodrome) type of spiral drawn on a sphere Cotes's spiral 1722 Poinsot's spirals $r=a\operatorname {csch} (n\theta ),\,$ $r=a\operatorname {sech} (n\theta )$  Nielsen's spiral 1993 $x(t)=\operatorname {ci} (t),\,$ $y(t)=\operatorname {si} (t)$ 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 ${\begin{cases}r=\mu ^{t}a\\\theta =t\\z=\mu ^{t}c\end{cases}}$ 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 ${\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}$ Pappus spiral 1779 ${\begin{cases}r=a\theta \\\psi =k\end{cases}}$ 3D conical spiral studied by Pappus and Pascal doppler spiral ${\begin{cases}x=a(t\cos(t)+kt)\\y=at\sin(t)\end{cases}}$ 2D projection of Pappus spiral Atzema spiral ${\begin{cases}x=\sin(t)/t-2\cos(t)-t\sin(t)\\y=-\cos(t)/t-2\sin(t)+t\cos(t)\end{cases}}$ The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral. Atomic spiral 2002 $r=\theta /(\theta -a)$ This spiral has two asymptotes; one is the circle of radius 1 and the other is the line $\theta =a$  Galactic spiral 2019 ${\begin{cases}dx=R*{\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R*{\Bigl [}\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}{\Bigr ]}d\theta \end{cases}}{\begin{cases}x=\sum dx\\\\\\y=\sum dy+R\end{cases}}$ The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:$\rho <1,\rho =1,\rho >1$ , the spiral patterns are decided by the behavior of the parameter $\rho$ . For $\rho <1$ , spiral-ring pattern; $\rho =1,$ regular spiral; $\rho >1,$ 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 ($-\theta$ ) for plotting. Please check the references for the detail