Image Name First described Equation Comment Circle r = k {\displaystyle r=k} The trivial spiral Archimedean spiral (alsoarithmetic spiral )c. 320 BC r = a + b ⋅ θ {\displaystyle r=a+b\cdot \theta } Fermat's spiral (also parabolic spiral)1636[ 1] r 2 = a 2 ⋅ θ {\displaystyle r^{2}=a^{2}\cdot \theta } Euler spiral (alsoCornu spiral or polynomial spiral)1696[ 2] x ( t ) = C ( t ) , {\displaystyle x(t)=\operatorname {C} (t),\,} y ( t ) = S ( t ) {\displaystyle y(t)=\operatorname {S} (t)} UsingFresnel integrals [ 3] Hyperbolic spiral (alsoreciprocal spiral )1704 r = a θ {\displaystyle r={\frac {a}{\theta }}} Lituus 1722 r 2 ⋅ θ = k {\displaystyle r^{2}\cdot \theta =k} Logarithmic spiral (also known asequiangular spiral )1638[ 4] r = a ⋅ e b ⋅ θ {\displaystyle 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 tilingApproximation of the golden spiral Golden spiral r = φ 2 ⋅ θ π {\displaystyle r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,} Special case of the logarithmic spiral Spiral of Theodorus (also known asPythagorean spiral )c. 500 BC 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 ) ) , {\displaystyle x(t)=r(\cos(t+a)+t\sin(t+a)),} y ( t ) = r ( sin ( t + a ) − t cos ( t + a ) ) {\displaystyle y(t)=r(\sin(t+a)-t\cos(t+a))}
Involutes of a circle appear like Archimedean spirals Helix r ( t ) = 1 , {\displaystyle r(t)=1,\,} θ ( t ) = t , {\displaystyle \theta (t)=t,\,} z ( t ) = t {\displaystyle z(t)=t} A three-dimensional spiral Rhumb line (also loxodrome)Type of spiral drawn on a sphere Cotes's spiral 1722 1 r = { A cosh ( k θ + ε ) A exp ( k θ + ε ) A sinh ( k θ + ε ) A ( k θ + ε ) A cos ( k θ + ε ) {\displaystyle {\frac {1}{r}}={\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 Poinsot's spirals r = a ⋅ csch ( n ⋅ θ ) , {\displaystyle r=a\cdot \operatorname {csch} (n\cdot \theta ),\,} r = a ⋅ sech ( n ⋅ θ ) {\displaystyle r=a\cdot \operatorname {sech} (n\cdot \theta )} Nielsen's spiral 1993[ 5] x ( t ) = ci ( t ) , {\displaystyle x(t)=\operatorname {ci} (t),\,} y ( t ) = si ( t ) {\displaystyle y(t)=\operatorname {si} (t)} A variation of Euler spiral, usingsine 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 = μ t ⋅ a , {\displaystyle r=\mu ^{t}\cdot a,\,} θ = t , {\displaystyle \theta =t,\,} z = μ t ⋅ c {\displaystyle 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[ 6] r = sn ( s , k ) , {\displaystyle r=\operatorname {sn} (s,k),\,} θ = k ⋅ s {\displaystyle \theta =k\cdot s} z = cn ( s , k ) {\displaystyle z=\operatorname {cn} (s,k)} Spiral curve on the surface of a sphere using theJacobi elliptic functions [ 7] Tractrix spiral1704[ 8] { r = A cos ( t ) θ = tan ( t ) − t {\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}} Pappus spiral 1779 { r = a θ ψ = k {\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}} 3D conical spiral studied byPappus andPascal [ 9] Doppler spiral x = a ⋅ ( t ⋅ cos ( t ) + k ⋅ t ) , {\displaystyle x=a\cdot (t\cdot \cos(t)+k\cdot t),\,} y = a ⋅ t ⋅ sin ( t ) {\displaystyle y=a\cdot t\cdot \sin(t)} 2D projection of Pappus spiral[ 10] Atzema spiral x = sin ( t ) t − 2 ⋅ cos ( t ) − t ⋅ sin ( t ) , {\displaystyle x={\frac {\sin(t)}{t}}-2\cdot \cos(t)-t\cdot \sin(t),\,} y = − cos ( t ) t − 2 ⋅ sin ( t ) + t ⋅ cos ( t ) {\displaystyle y=-{\frac {\cos(t)}{t}}-2\cdot \sin(t)+t\cdot \cos(t)} The curve that has acatacaustic forming a circle. Approximates the Archimedean spiral.[ 11] Atomic spiral 2002 r = θ θ − a {\displaystyle r={\frac {\theta }{\theta -a}}} This spiral has twoasymptotes ; one is the circle of radius 1 and the other is the lineθ = a {\displaystyle \theta =a} [ 12] Galactic spiral 2019 { d x = R ⋅ y x 2 + y 2 d θ d y = R ⋅ [ ρ ( θ ) − x x 2 + y 2 ] d θ { x = ∑ d x y = ∑ d y + R {\displaystyle {\begin{cases}dx=R\cdot {\frac {y}{\sqrt {x^{2}+y^{2}}}}d\theta \\dy=R\cdot \left[\rho (\theta )-{\frac {x}{\sqrt {x^{2}+y^{2}}}}\right]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:ρ < 1 , ρ = 1 , ρ > 1 {\displaystyle \rho <1,\rho =1,\rho >1} ρ {\displaystyle \rho } ρ < 1 {\displaystyle \rho <1} ρ = 1 , {\displaystyle \rho =1,} ρ > 1 , {\displaystyle \rho >1,} − θ {\displaystyle -\theta } [ 13] [predatory publisher 
