Image
Name
First described
Equation
Comment
circle
r
=
k
{\displaystyle r=k}
The trivial spiral
Archimedean spiral (also arithmetic spiral )c. 320 BC
r
=
a
+
b
⋅ ⋅ -->
θ θ -->
{\displaystyle r=a+b\cdot \theta }
Fermat's spiral (also parabolic spiral)1636[ 1]
r
2
=
a
2
⋅ ⋅ -->
θ θ -->
{\displaystyle r^{2}=a^{2}\cdot \theta }
Euler spiral (also Cornu 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)}
using Fresnel integrals [ 3]
hyperbolic spiral (also reciprocal spiral )1704
r
=
a
θ θ -->
{\displaystyle r={\frac {a}{\theta }}}
lituus 1722
r
2
⋅ ⋅ -->
θ θ -->
=
k
{\displaystyle r^{2}\cdot \theta =k}
logarithmic spiral (also known as equiangular spiral )1638[ 4]
r
=
a
⋅ ⋅ -->
e
b
⋅ ⋅ -->
θ 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 φ -->
2
⋅ ⋅ -->
θ θ -->
π π -->
{\displaystyle r=\varphi ^{\frac {2\cdot \theta }{\pi }}\,}
special case of the logarithmic spiral
Spiral of Theodorus (also known as Pythagorean 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 θ -->
(
t
)
=
t
,
{\displaystyle \theta (t)=t,\,}
z
(
t
)
=
t
{\displaystyle z(t)=t}
a 3-dimensional spiral
Rhumb line (also loxodrome)type of spiral drawn on a sphere
Cotes's spiral 1722
1
r
=
{
A
cosh
-->
(
k
θ θ -->
+
ε ε -->
)
A
exp
-->
(
k
θ θ -->
+
ε ε -->
)
A
sinh
-->
(
k
θ θ -->
+
ε ε -->
)
A
(
k
θ θ -->
+
ε ε -->
)
A
cos
-->
(
k
θ θ -->
+
ε 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 θ -->
)
,
{\displaystyle r=a\cdot \operatorname {csch} (n\cdot \theta ),\,}
r
=
a
⋅ ⋅ -->
sech
-->
(
n
⋅ ⋅ -->
θ 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, 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
=
μ μ -->
t
⋅ ⋅ -->
a
,
{\displaystyle r=\mu ^{t}\cdot a,\,}
θ θ -->
=
t
,
{\displaystyle \theta =t,\,}
z
=
μ μ -->
t
⋅ ⋅ -->
c
{\displaystyle z=\mu ^{t}\cdot c}
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 2000[ 6]
r
=
sn
-->
(
s
,
k
)
,
{\displaystyle r=\operatorname {sn} (s,k),\,}
θ 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 the Jacobi elliptic functions [ 7]
Tractrix spiral1704[ 8]
{
r
=
A
cos
-->
(
t
)
θ θ -->
=
tan
-->
(
t
)
− − -->
t
{\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}
Pappus spiral 1779
{
r
=
a
θ θ -->
ψ ψ -->
=
k
{\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}}
3D conical spiral studied by Pappus and Pascal [ 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 a catacaustic forming a circle. Approximates the Archimedean spiral.[ 11]
Atomic spiral 2002
r
=
θ θ -->
θ θ -->
− − -->
a
{\displaystyle r={\frac {\theta }{\theta -a}}}
This spiral has two asymptotes ; one is the circle of radius 1 and the other is the line
θ θ -->
=
a
{\displaystyle \theta =a}
[ 12]
Galactic spiral 2019
{
d
x
=
R
⋅ ⋅ -->
y
x
2
+
y
2
d
θ θ -->
d
y
=
R
⋅ ⋅ -->
[
ρ ρ -->
(
θ θ -->
)
− − -->
x
x
2
+
y
2
]
d
θ 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 ρ -->
<
1
,
ρ ρ -->
=
1
,
ρ ρ -->
>
1
{\displaystyle \rho <1,\rho =1,\rho >1}
ρ ρ -->
{\displaystyle \rho }
ρ ρ -->
<
1
{\displaystyle \rho <1}
ρ ρ -->
=
1
,
{\displaystyle \rho =1,}
ρ ρ -->
>
1
,
{\displaystyle \rho >1,}
− − -->
θ θ -->
{\displaystyle -\theta }
[ 13] [predatory publisher
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47cba127b869ed1f803e339d0dd8f979c175be01)
