Fermat's spiral
plane curve
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat.
Their applications include curvature continuous blending of curves, modeling plant growth and the shapes of certain spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotrons.
Coordinate representation
Polar
The representation of the Fermat spiral in polar coordinates (r, φ) is given by the equation
r
=
±
a
φ
{\displaystyle r=\pm a{\sqrt {\varphi }}}
for φ ≥ 0.
The parameter
a
{\displaystyle a}
is a scaling factor affecting the size of the spiral but not its shape.
The two choices of sign give the two branches of the spiral, which meet smoothly at the origin. If the same variables were reinterpreted as Cartesian coordinates, this would be the equation of a parabola with horizontal axis, which again has two branches above and below the axis, meeting at the origin.
Cartesian
The Fermat spiral with polar equation
r
=
±
a
φ
{\displaystyle r=\pm a{\sqrt {\varphi }}}
can be converted to the Cartesian coordinates (x, y) by using the standard conversion formulas x = r cos φ and y = r sin φ. Using the polar equation for the spiral to eliminate r from these conversions produces parametric equations for one branch of the curve:
{
x
(
φ
)
=
+
a
φ
cos
(
φ
)
y
(
φ
)
=
+
a
φ
sin
(
φ
)
{\displaystyle {\begin{cases}x(\varphi )=+a{\sqrt {\varphi }}\cos(\varphi )\\y(\varphi )=+a{\sqrt {\varphi }}\sin(\varphi )\end{cases}}}
and the second one
{
x
(
φ
)
=
−
a
φ
cos
(
φ
)
y
(
φ
)
=
−
a
φ
sin
(
φ
)
{\displaystyle {\begin{cases}x(\varphi )=-a{\sqrt {\varphi }}\cos(\varphi )\\y(\varphi )=-a{\sqrt {\varphi }}\sin(\varphi )\end{cases}}}
They generate the points of branches of the curve as the parameter φ ranges over the positive real numbers.
This brief starts where responsible research should: with the source description of “Fermat's spiral” as plane curve. Everything that follows is an evidence route, not borrowed authority.
Why this record matters
The subject matters to the general reference register because the source frames it as plane curve. Its deeper value depends on whether names, dates, institutions and citations support that framing.
Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Aug 30, 2026. The linked authority identifier is Q907869. None of the 0 selected statements returned an explicit reference.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
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The closest primary source, responsible institution and strongest cited specialist reference.
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Questions for further research
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This entry incorporates text from “Fermat's spiral” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.