# List of spirals

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

ImageNameFirst describedEquationComment circle $r=k$ The trivial spiral Archimedean spiral c.320 BC$r=a+b\cdot \theta$  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 tilingapproximation of the golden spiral golden spiral $r=\varphi ^{\theta {\frac {2}{\pi }}}\,$ 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 helix $r(t)=1,\,$ $\theta (t)=t,\,$ $h(t)=t.\,$ a 3-dimensional spiral 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 Conchospiral ${\begin{cases}r=\mu ^{t}a\\\theta =t\\z=\mu ^{t}c\end{cases}}$ three-dimensional spiral on the surface of a cone.
Seiffert's spiral spiral curve on the surface of a sphere Tractrix spiral1704  ${\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}$ 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${\begin{cases}dx=R(y/({\sqrt {(}}x^{2}+y^{2}))d\theta \\dy=R(\rho (\theta )-x/{\sqrt {(}}x^{2}+y^{2}))d\theta \end{cases}}{\begin{cases}x=\sum dx\\y=\sum dy+R\end{cases}}$ 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 

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1. "Fermat spiral - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 18 February 2019.
2. Weisstein, Eric W. "Logarithmic Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
3. Weisstein, Eric W. "Nielsen's Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
4. "Tractrix spiral". www.mathcurve.com. Retrieved 2019-02-23.
5. "Conical spiral of Pappus". www.mathcurve.com. Retrieved 28 February 2019.
6. "Doppler spiral". www.mathcurve.com. Retrieved 28 February 2019.
7. "Atzema spiral". www.2dcurves.com. Retrieved 11 March 2019.
8. "atom-spiral". www.2dcurves.com. Retrieved 11 March 2019.
9. Pan, Hongjun. "New spiral" (PDF). www.arpgweb.com. Retrieved 5 March 2021.