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Fermat's spiral: one branch
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

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 .[2]

In cartesian coordinatesEdit

Fermat's spiral with polar equation

 

can be described in cartesian coordinates   by the parametric representation

  •  

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

  •  

Geometric propertiesEdit

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

Division of the planeEdit

A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrary to 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  

Polar slopeEdit

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.

CurvatureEdit

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 arcsEdit

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

 
 
 
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

  •  

  depents from the difference of both the angles only and not from 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. [3]

 
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

1636 Fermat wrote a letter [4] to Marin Mersenne which contains the following special case:

Let be   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:

  • The area between two arcs of the spiral after a full turn equals the area of the circle  .

ArclengthEdit

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)

Circle inversionEdit

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

  • The image of Fermat's spiral   under the inversion at the unit circle is a Lituus spiral with polar equation  .

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

  • The tangent ( -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 AngleEdit

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[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. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.[6]

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 plantsEdit

Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants.[7]

See alsoEdit

ReferencesEdit

  1. ^ Anastasios M. Lekkas1, Andreas R. Dahl, Morten Breivik, Thor I. Fossen4: Continuous-Curvature Path GenerationUsing Fermat’s Spiral. In: Modeling, Identification and Control. Vol. 34, No. 4, 2013, p. 183–198, ISSN 1890-1328.
  2. ^ Anastasios M. Lekkas1, Andreas R. Dahl, Morten Breivik, Thor I. Fossen4: Continuous-Curvature Path GenerationUsing Fermat’s Spiral. In: Modeling, Identification and Control. Vol. 34, No. 4, 2013, S. 183–198, ISSN 1890-1328.
  3. ^ Fritz Wicke: Einführung in die höhere Mathematik. Springer-Verlag, 2013, ISBN 978-3-662-36804-6, S. 414.
  4. ^ Lettre de Fermat à Mersenne du 3 juin 1636, dans Paul Tannery. In: Oeuvres de Fermat. T. III, S. 277, Lire en ligne.
  5. ^ 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.
  6. ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8.
  7. ^ 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 readingEdit

External linksEdit