Fermat's spiral

A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by

$r=\pm a{\sqrt {\varphi }},\quad \varphi \geq 0$ which describes a parabola with horizontal axis.

Fermat's spiral is similar to the Archimedean spiral. But an Archimedean spiral has always the same distance between neighboring arcs, which is not true for Fermat's spiral.

Like other spirals Fermat's spiral is used for curvature continuous blending of curves.

In Cartesian coordinates

Fermat's spiral with polar equation

$r=a{\sqrt {\varphi }},\quad \varphi \geq 0$ can be described in Cartesian coordinates (x = r cos φ, y = r sin φ) by the parametric representation

$x=a{\sqrt {\varphi }}\cos \varphi ,\qquad y=a{\sqrt {\varphi }}\sin \varphi ,\quad \varphi \geq 0\ .$ From the parametric representation and φ = r2/a2, r = x2 + y2 one gets a representation by an equation:

${\frac {y}{x}}=\tan \left({\frac {x^{2}+y^{2}}{a^{2}}}\right)\ .$ Geometric properties

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. 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.

Polar slope

From vector calculus in polar coordinates one gets the formula

$\tan \alpha ={\frac {r'}{r}}$ 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 = aφ one gets

$\tan \alpha ={\frac {1}{2\varphi }}.$ Hence the slope angle is monotonely decreasing.

Curvature

From the formula

$\kappa ={\frac {r^{2}+2(r')^{2}-r\,r''}{\left(r^{2}+(r')^{2}\right)^{\frac {3}{2}}}}$ for the curvature of a curve with polar equation r = r(φ) and its derivatives

{\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:

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

Area between arcs

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

{\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}} After raising both angles by 2π one gets

${\overline {A}}={\frac {a^{2}}{4}}\left(\varphi _{2}+\varphi _{1}+4\pi \right)\left(\varphi _{2}-\varphi _{1}\right)={\underline {A}}+a^{2}\pi \left(\varphi _{2}-\varphi _{1}\right).$ Hence the area A of the region between two neighboring arcs is

$A=a^{2}\pi \left(\varphi _{2}-\varphi _{1}\right).$ 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. the regions in between (white, blue, yellow) have all the same area, which is equal to the area of the drawn circle.

Special case due to Fermat

In 1636, Fermat wrote a letter  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.

Arclength

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

{\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.

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 = aφ under the inversion at the unit circle is a lituus spiral with polar equation
$\;r={\frac {1}{a{\sqrt {\varphi }}}}.$ 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.

The golden ratio and the golden angle

In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H. Vogel in 1979 is

{\begin{aligned}r&=c{\sqrt {n}},\\\theta &=n\times 137.508^{\circ },\end{aligned}} where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.

The pattern of florets produced by Vogel's model (central image). The other two images show the patterns for slightly different values of the angle.

The resulting spiral pattern of unit disks should be distinguished from the Doyle spirals, patterns formed by tangent disks of geometrically increasing radii placed on logarithmic spirals.

Solar plants

Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants.