List of spirals

Template:Short description Template:Expand list 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 (also arithmetic spiral)	Template:Sort	$r=a+b\cdot \theta$
	Fermat's spiral (also parabolic spiral)	1636Template:R	$r^2=a^2\cdot \theta$
	Euler spiral (also Template:Em or polynomial spiral)	1696	$x(t)=\mathrm{C}(t),$$y(t)=\mathrm{S}(t)$	Using Fresnel integrals
	Hyperbolic spiral (also reciprocal spiral)	1704	$r=\frac{a}{\theta }$
	Lituus	1722	$r^2\cdot \theta =k$
	Logarithmic spiral (also known as equiangular spiral)	1638Template:R	$r=a\cdot e^{b\cdot \theta }$	Approximations of this are found in nature
	Fibonacci spiral		Circular arcs connecting the opposite corners of squares in the Fibonacci tiling	Approximation of the golden spiral
	Golden spiral		$r={\phi }^{\frac{2\cdot \theta }{\pi }}$	Special case of the logarithmic spiral
	Spiral of Theodorus (also known as Pythagorean spiral)	Template:Sort	Contiguous right triangles composed of one leg with unit length and the other leg being the hypotenuse of the prior triangle	Approximates the Archimedean spiral
	Involute	1673	$x(t)=r(\cos (t+a)+t\sin (t+a)),$ $y(t)=r(\sin (t+a)-t\cos (t+a))$	Involutes of a circle appear like Archimedean spirals
	Helix		$r(t)=1,$ $\theta (t)=t,$ $z(t)=t$	A three-dimensional spiral
	Rhumb line (also loxodrome)			Type of spiral drawn on a sphere
	Cotes's spiral	1722	$\frac{1}{r}=\{\begin{array}{c}A\cosh (k\theta +\epsilon )\\ A\exp (k\theta +\epsilon )\\ A\sinh (k\theta +\epsilon )\\ A(k\theta +\epsilon )\\ A\cos (k\theta +\epsilon )\end{array}$	Solution to the two-body problem for an inverse-cube central force
	Poinsot's spirals		$r=a\cdot \mathrm{csch}(n\cdot \theta ),$ $r=a\cdot \mathrm{sech}(n\cdot \theta )$
	Nielsen's spiral	1993Template:R	$x(t)=\mathrm{ci}(t),$ $y(t)=\mathrm{si}(t)$	A variation of Euler spiral, using sine integral and cosine integrals
	Polygonal spiral			Special case approximation of arithmetic or logarithmic spiral
	Fraser's Spiral	1908		Optical illusion based on spirals
	Conchospiral		$r={\mu }^t\cdot a,$$\theta =t,$$z={\mu }^t\cdot c$	A three-dimensional spiral on the surface of a cone.
	Calkin–Wilf spiral
	Ulam spiral (also prime spiral)	1963
	Sacks spiral	1994		Variant of Ulam spiral and Archimedean spiral.
	Seiffert's spiral	2000	$r=\mathrm{sn}(s,k),$$\theta =k\cdot s$$z=\mathrm{cn}(s,k)$	Spiral curve on the surface of a sphere using the Jacobi elliptic functions
	Tractrix spiral	1704Template:R	$\{\begin{array}{c}r=A\cos (t)\\ \theta =\tan (t)-t\end{array}$
	Pappus spiral	1779	$\{\begin{array}{c}r=a\theta \\ \psi =k\end{array}$	3D conical spiral studied by Pappus and PascalTemplate:R
	Doppler spiral		$x=a\cdot (t\cdot \cos (t)+k\cdot t),$$y=a\cdot t\cdot \sin (t)$	2D projection of Pappus spiralTemplate:R
	Atzema spiral		$x=\frac{\sin (t)}{t}-2\cdot \cos (t)-t\cdot \sin (t),$$y=-\frac{\cos (t)}{t}-2\cdot \sin (t)+t\cdot \cos (t)$	The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.Template:R
	Atomic spiral	2002	$r=\frac{\theta }{\theta -a}$	This spiral has two asymptotes; one is the circle of radius 1 and the other is the line $\theta =a$Template:R
	Galactic spiral	2019	$\{\begin{array}{c}dx=R\cdot \frac{y}{\sqrt{x^2+y^2}}d\theta \\ dy=R\cdot [\rho (\theta )-\frac{x}{\sqrt{x^2+y^2}}]d\theta \end{array}\{\begin{array}{c}x=\sum dx\\ \\ \\ y=\sum dy+R\end{array}$	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.Template:RTemplate:Pred

See also

• Catherine wheel (firework)
• List of spiral galaxies
• Parker spiral
• Spirangle
• Spirograph