Polar

The representation of the Fermat spiral in polar coordinates (r, φ) is given by the equation

$${\displaystyle r=\pm a{\sqrt {\varphi }}}$$

for φ ≥ 0.

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

$${\displaystyle r=\pm a{\sqrt {\varphi }}}$$

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:

${\displaystyle {\begin{cases}x(\varphi )=+a{\sqrt {\varphi }}\cos(\varphi )\\y(\varphi )=+a{\sqrt {\varphi }}\sin(\varphi )\end{cases}}}$

and the second one

${\displaystyle {\begin{cases}x(\varphi )=-a{\sqrt {\varphi }}\cos(\varphi )\\y(\varphi )=-a{\sqrt {\varphi }}\sin(\varphi )\end{cases}}}$

They generate the points of branches of the curve as the parameter φ ranges over the positive real numbers.

For any (x, y) generated in this way, dividing x by y cancels the a√φ parts of the parametric equations, leaving the simpler equation x/y = cot φ. From this equation, substituting φ by φ = r²/a² (a rearranged form of the polar equation for the spiral) and then substituting r by r = √x² + y² (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only x and y:

$${\displaystyle {\frac {x}{y}}=\cot \left({\frac {x^{2}+y^{2}}{a^{2}}}\right).}$$

Because the sign of a is lost when it is squared, this equation covers both branches of the curve.

Division of the plane

A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. Like a line or circle or parabola, it divides the plane into two connected regions.

Definition of sector (light blue) and polar slope angle α

Polar slope

From vector calculus in polar coordinates one gets the formula

${\displaystyle \tan \alpha ={\frac {r'}{r}}}$

for the polar slope and its angle α between the tangent of a curve and the corresponding polar circle (see diagram).

For Fermat's spiral r = a√φ one gets

${\displaystyle \tan \alpha ={\frac {1}{2\varphi }}.}$

Hence the slope angle is monotonely decreasing.

Curvature

From the formula

${\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\,r''}{\left(r^{2}+(r')^{2}\right)^{\frac {3}{2}}}}}$

for the curvature of a curve with polar equation r = r(φ) and its derivatives

${\displaystyle {\begin{aligned}r'&={\frac {a}{2{\sqrt {\varphi }}}}={\frac {a^{2}}{2r}}\\r''&=-{\frac {a}{4{\sqrt {\varphi }}^{3}}}=-{\frac {a^{4}}{4r^{3}}}\end{aligned}}}$

one gets the curvature of a Fermat's spiral:

$${\displaystyle \kappa (r)={\frac {2r\left(4r^{4}+3a^{4}\right)}{\left(4r^{4}+a^{4}\right)^{\frac {3}{2}}}}.}$$

At the origin the curvature is 0. Hence the complete curve has at the origin an inflection point and the x-axis is its tangent there.

Area between arcs

The area of a sector of Fermat's spiral between two points (r(φ₁), φ₁) and (r(φ₂), φ₂) is

${\displaystyle {\begin{aligned}{\underline {A}}&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\,d\varphi \\&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi \,d\varphi \\&={\frac {a^{2}}{4}}\left(\varphi _{2}^{2}-\varphi _{1}^{2}\right)\\&={\frac {a^{2}}{4}}\left(\varphi _{2}+\varphi _{1}\right)\left(\varphi _{2}-\varphi _{1}\right).\end{aligned}}}$

Fermat's spiral: area between neighbored arcs

After raising both angles by 2π one gets

${\displaystyle {\overline {A}}={\frac {a^{2}}{4}}\left(\varphi _{2}+\varphi _{1}+4\pi \right)\left(\varphi _{2}-\varphi _{1}\right)={\underline {A}}+a^{2}\pi \left(\varphi _{2}-\varphi _{1}\right).}$

Hence the area A of the region between two neighboring arcs is

$${\displaystyle A=a^{2}\pi \left(\varphi _{2}-\varphi _{1}\right).}$$

A only depends on the difference of the two angles, not on the angles themselves.

For the example shown in the diagram, all neighboring stripes have the same area: A₁ = A₂ = A₃.

This property is used in electrical engineering for the construction of variable capacitors.^[2]

the regions in between (white, blue, yellow) have all the same area, which is equal to the area of the drawn circle.

Arc length

The length of the arc of Fermat's spiral between two points (r(φ_i), φ_i) can be calculated by the integral:

$${\displaystyle {\begin{aligned}L&=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,d\varphi \\&={\frac {a}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {1}{\varphi }}+4\varphi }}\,d\varphi .\end{aligned}}}$$

This integral leads to an elliptical integral, which can be solved numerically.

The arc length of the positive branch of the Fermat's spiral from the origin can also be defined by hypergeometric functions ₂F₁(a, b; c; z) and the incomplete beta function B(z; a, b):^[4]

$${\displaystyle {\begin{aligned}L&=a\cdot {\sqrt {\varphi }}\cdot \operatorname {_{2}F_{1}} \left(-{\tfrac {1}{2}},\,{\tfrac {1}{4}};\,{\tfrac {5}{4}};\,-4\cdot \varphi ^{2}\right)\\&=a\cdot {\frac {1-i}{8}}\cdot \operatorname {B} \left(-4\cdot \varphi ^{2};\,{\tfrac {1}{4}},\,{\tfrac {3}{2}}\right)\\\end{aligned}}}$$

The inversion of Fermat's spiral (green) is a lituus (blue)

Circle inversion

The inversion at the unit circle has in polar coordinates the simple description (r, φ) ↦ (1/r, φ).

• The image of Fermat's spiral r = a√φ under the inversion at the unit circle is a lituus spiral with polar equation
  $${\displaystyle r={\frac {1}{a{\sqrt {\varphi }}}}.}$$

  
  When φ = 1/a², both curves intersect at a fixed point on the unit circle.
• The tangent (x-axis) at the inflection point (origin) of Fermat's spiral is mapped onto itself and is the asymptotic line of the lituus spiral.