Fermat's spiral

[1] 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.

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

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.

The Fermat spiral with polar equation

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: and the second one 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:

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

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.

For Fermat's spiral r = a√φ one gets Hence the slope angle is monotonely decreasing.

From the formula for the curvature of a curve with polar equation r = r(φ) and its derivatives one gets the curvature of a Fermat's spiral:

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(φ2), φ2) is After raising both angles by 2π one gets Hence the area A of the region between two neighboring arcs is

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

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]

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.

The full model proposed by H. Vogel in 1979[5] is

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.

[6] 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.

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