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
{\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 (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
{\displaystyle r={\frac {a}{\theta ))}
lituus
1722
r
2
⋅
θ
=
k
{\displaystyle r^{2}\cdot \theta =k}
logarithmic spiral (also known as equiangular 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 tiling
approximation 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 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
(
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
+
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, using sine integral and cosine integrals
Polygonal spiral
special case approximation of 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}
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
=
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 spiral
1704[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 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
−
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
=
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}
, the spiral patterns are decided by the behavior of the parameter
ρ
{\displaystyle \rho }
. For
ρ
<
1
{\displaystyle \rho <1}
, spiral-ring pattern;
ρ
=
1
,
{\displaystyle \rho =1,}
regular spiral;
ρ
>
1
,
{\displaystyle \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 (
−
θ
{\displaystyle -\theta }
) for plotting.[13]