Fermat's spiral

A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. It divides the plane (like a line or circle or parabola) into two connected regions. But this division is less obvious than the division by a line or circle or parabola. It is not obvious to which side a chosen point belongs.

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

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.

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

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

one gets the curvature of a Fermat's spiral:

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

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.

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

In 1636, Fermat wrote a letter [3] to Marin Mersenne which contains the following special case:

Let φ₁ = 0, φ₂ = 2π; then the area of the black region (see diagram) is A₀ = a²π², which is half of the area of the circle K₀ with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a²π². Hence:

• The area between two arcs of the spiral after a full turn equals the area of the circle K₀.

The length of the arc of Fermat's spiral between two points (r(φ₁), φ₁) can be calculated by the integral:

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

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

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

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.