Image
Name
First described
Equation
Comment
circle
r
=
k
{\displaystyle r=k}
The trivial spiral
Archimedean spiral c. 320 BC
r
=
a
+
b
⋅
θ
{\displaystyle r=a+b\cdot \theta }
Euler spiral also called Cornu spiral or polynomial spiral
Fermat's spiral (also parabolic spiral)1636[1]
r
2
=
a
2
θ
{\displaystyle r^{2}=a^{2}\theta }
hyperbolic spiral 1704
r
=
a
/
θ
{\displaystyle r=a/\theta }
also reciprocal spiral
lituus 1722
r
2
θ
=
k
{\displaystyle r^{2}\theta =k}
logarithmic spiral 1638[2]
r
=
a
⋅
e
b
θ
{\displaystyle r=a\cdot e^{b\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
θ
2
π
{\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}
special case of the logarithmic spiral
Spiral of Theodorus (also Pythagorean spiral )an polygonal spiral composed of contiguous right triangles, that approximates the Archimedean spiral
involute 1673
helix
r
(
t
)
=
1
,
{\displaystyle r(t)=1,\,}
θ
(
t
)
=
t
,
{\displaystyle \theta (t)=t,\,}
h
(
t
)
=
t
.
{\displaystyle h(t)=t.\,}
a 3-dimensional spiral
Rhumb line (also loxodrome)type of spiral drawn on a sphere
Cotes's spiral 1722
Poinsot's spirals
r
=
a
csch
(
n
θ
)
,
{\displaystyle r=a\operatorname {csch} (n\theta ),\,}
r
=
a
sech
(
n
θ
)
{\displaystyle r=a\operatorname {sech} (n\theta )}
Nielsen's spiral 1993[3]
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
θ
=
t
z
=
μ
t
c
{\displaystyle {\begin{cases}r=\mu ^{t}a\\\theta =t\\z=\mu ^{t}c\end{cases}}}
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 spiral curve on the surface of a sphere
Tractrix spiral1704[4]
{
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 [5]
doppler spiral
{
x
=
a
(
t
cos
(
t
)
+
k
t
)
y
=
a
t
sin
(
t
)
{\displaystyle {\begin{cases}x=a(t\cos(t)+kt)\\y=at\sin(t)\end{cases}}}
2D projection of Pappus spiral[6]
Atzema spiral
{
x
=
sin
(
t
)
/
t
−
2
cos
(
t
)
−
t
sin
(
t
)
y
=
−
cos
(
t
)
/
t
−
2
sin
(
t
)
+
t
cos
(
t
)
{\displaystyle {\begin{cases}x=\sin(t)/t-2\cos(t)-t\sin(t)\\y=-\cos(t)/t-2\sin(t)+t\cos(t)\end{cases}}}
The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral.[7]
Atomic spiral 2002
r
=
θ
/
(
θ
−
a
)
{\displaystyle r=\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}
[8]
