Fermat's spiral

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Fermat's spiral: one branch Spiral-fermat-1.svg
Fermat's spiral: one branch
Fermat's spiral: both branches Spiral-fermat-2.svg
Fermat's spiral: both branches

A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. [1] Its polar coordinate representation is given by

Contents

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. [1]


In cartesian coordinates

Fermat's spiral with polar equation

can be described in cartesian coordinates by the parametric representation

From the parametric representation and one gets a representation by an equation:

Geometric properties

A Fermat's spiral divides the plane into two connected regions (diagram: black and white) Fermat's spiral connected components.svg
A Fermat's spiral divides the plane into two connected regions (diagram: black and white)

Division of the plane

A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedian 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
a
{\displaystyle \alpha } Sektor-steigung-pk-def.svg
Definition of sector (light blue) and polar slope angle

Polar slope

From vector calculus in polar coordinates one gets the formula

for the polar slope and its angle between the tangent of a curve and the corresponding polar circle (s. diagram).

For Fermat's spiral one gets

Hence the slope angle is monotonely decreasing.

Curvature

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

Area between arcs

The area of a sector of Fermat's spiral between two points and is

Fermat's spiral:
area between neighbored arcs Spiral-fermat-area.svg
Fermat's spiral:
area between neighbored arcs

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. [2]

the regions in between (white, blue, yellow) have all the same area, which is equal to the area of the drawn circle. Fermat's spiral area.svg
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 [3] to Marin Mersenne which contains the following special case:

Let , then the area of the black region (see diagram) is half of the area of the circle with radius . The regions between neighboring curves (white, blue, yellow) have the same area Hence:

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.

The inversion of Fermat's spiral (green) is a lituus spiral (blue) Invers-fermats-lits.svg
The inversion of Fermat's spiral (green) is a lituus spiral (blue)

Circle inversion

The inversion at the unit circle has in polar coordinates the simple description: .

For both curves intersect at a fixpoint on the unit circle.

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 spiralsideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979 [4] 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. [5]

Sunflower spiral.png
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.

Solar plants

Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants. [6]

See also

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References

  1. 1 2 Anastasios M. Lekkas, Andreas R. Dahl, Morten Breivik, Thor I. Fossen: "Continuous-Curvature Path Generation Using Fermat's Spiral". In: Modeling, Identification and Control. Vol. 34, No. 4, 2013, pp. 183–198, ISSN   1890-1328.
  2. Fritz Wicke: Einführung in die höhere Mathematik. Springer-Verlag, 2013, ISBN   978-3-662-36804-6, S. 414.
  3. Lettre de Fermat à Mersenne du 3 juin 1636, dans Paul Tannery. In: Oeuvres de Fermat. T. III, S. 277, Lire en ligne.
  4. Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences. 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4.
  5. Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp.  101–107. ISBN   978-0-387-97297-8.
  6. Noone, Corey J.; Torrilhon, Manuel; Mitsos, Alexander (December 2011). "Heliostat Field Optimization: A New Computationally Efficient Model and Biomimetic Layout". Solar Energy. doi:10.1016/j.solener.2011.12.007.

Further reading