A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by 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.
From the parametric representation and one gets a representation by an equation:
Geometric properties
Division of the plane
A complete Fermat's spiral is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. It divides the plane 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.
From the formula for the curvature of a curve with polar equation and its derivatives and one gets the curvature of a Fermat's spiral:
At the origin the curvature is. 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 and is After raising both angles by one gets Hence the area of the region between two neighboring arcs is
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:. This property is used in electrical engineering for the construction of variable capacitors. ; Special case due to Fermat In 1636, Fermat wrote a letter to Marin Mersenne which contains the following special case: Let, then the area of the black region is half of the area of the circle with radius. The regions between neighboring curves have the same area Hence:
The area between two arcs of the spiral after a full turn equals the area of the circle.
Arclength
The length of the arc of Fermat's spiral between two points can be calculated by the integral: This integral leads to an elliptical integral, which can be solved numerically.
In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence 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 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 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.
