Fermat's spiral

A Fermat's spiral or parabolic spiral is a

Their applications include curvature continuous blending of curves,[1] 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

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

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

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

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.

Geometric properties

Division of the plane

A complete Fermat's spiral (both branches) is a smooth

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

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

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

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

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]

Special case due to Fermat

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.

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

The arc length of the positive branch of the Fermat's spiral from the origin can also be defined by

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

Circle inversion

The

• 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.

The golden ratio and the golden angle

In disc

is

$${\displaystyle {\begin{aligned}r&=c{\sqrt {n}},\\\theta &=n\times 137.508^{\circ },\end{aligned}}}$$

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

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.^[7]

See also

• List of spirals
• Patterns in nature
• Spiral of Theodorus

References