This list of spirals includes named spirals that have been described mathematically.
Image | Name | First described | Equation | Comment | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
r=k | The trivial spiral | ||||||||||
Archimedean spiral (also arithmetic spiral) | r=a+b ⋅ \theta | ||||||||||
Fermat's spiral (also parabolic spiral) | 1636 | r2=a2 ⋅ \theta | |||||||||
Euler spiral (also or polynomial spiral) | 1696[1] | x(t)=\operatorname{C}(t), y(t)=\operatorname{S}(t) | using Fresnel integrals[2] | ||||||||
hyperbolic spiral (also reciprocal spiral) | 1704 | r=
| |||||||||
1722 | r2 ⋅ \theta=k | ||||||||||
logarithmic spiral (also known as equiangular spiral) | 1638 | r=a ⋅ eb | Approximations of this are found in nature | ||||||||
circular arcs connecting the opposite corners of squares in the Fibonacci tiling | approximation of the golden spiral | ||||||||||
r=
| special case of the logarithmic spiral | ||||||||||
Spiral of Theodorus (also known as Pythagorean spiral) | 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 | |||||||||
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 | |||||||||
r(t)=1, \theta(t)=t, z(t)=t | a 3-dimensional spiral | ||||||||||
Rhumb line (also loxodrome) | type of spiral drawn on a sphere | ||||||||||
1722 |
=\begin{cases} A\cosh(k\theta+\varepsilon)\\ A\exp(k\theta+\varepsilon)\\ A\sinh(k\theta+\varepsilon)\\ A(k\theta+\varepsilon)\\ A\cos(k\theta+\varepsilon)\\ \end{cases} | Solution to the two-body problem for an inverse-cube central force | |||||||||
r=a ⋅ \operatorname{csch}(n ⋅ \theta), r=a ⋅ \operatorname{sech}(n ⋅ \theta) | |||||||||||
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 arithmetic or logarithmic spiral | ||||||||||
1908 | Optical illusion based on spirals | ||||||||||
r=\mut ⋅ a, \theta=t, z=\mut ⋅ c | three-dimensional spiral on the surface of a cone. | ||||||||||
Ulam spiral (also prime spiral) | 1963 | ||||||||||
1994 | variant of Ulam spiral and Archimedean spiral. | ||||||||||
2000[3] | r=\operatorname{sn}(s,k), \theta=k ⋅ s z=\operatorname{cn}(s,k) | spiral curve on the surface of a sphereusing the Jacobi elliptic functions[4] | |||||||||
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 | x=a ⋅ (t ⋅ \cos(t)+k ⋅ t), y=a ⋅ t ⋅ \sin(t) | 2D projection of Pappus spiral | |||||||||
Atzema spiral | x=
-2 ⋅ \cos(t)-t ⋅ \sin(t), y=-
-2 ⋅ \sin(t)+t ⋅ \cos(t) | The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral. | |||||||||
Atomic spiral | 2002 | r=
| 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 ⋅
| 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 \rho \rho<1 \rho=1, \rho>1, -\theta | ||||||||
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