# Fermat's spiral

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

## Contents

$r=\pm a{\sqrt {\varphi }}\ ,\ \varphi \geq 0$ 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. 

## In cartesian coordinates

Fermat's spiral with polar equation

$r=a{\sqrt {\varphi }}\ ,\quad \varphi \geq 0$ can be described in cartesian coordinates $(x=r\cos \varphi ,\;y=r\sin \varphi )$ by the parametric representation

• $x=a{\sqrt {\varphi }}\cos \varphi ,\qquad y=a{\sqrt {\varphi }}\sin \varphi ,\quad \varphi \geq 0\ .$ From the parametric representation and $\ \varphi ={\tfrac {r^{2}}{a^{2}}},\ r={\sqrt {x^{2}+y^{2}}}\$ one gets a representation by an equation:

• ${\frac {y}{x}}=\tan {\big (}{\frac {x^{2}+y^{2}}{a^{2}}}{\big )}\ .$ ## Geometric properties

### 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.

### Polar slope

From vector calculus in polar coordinates one gets the formula

$\tan \alpha ={\frac {r'}{r}}$ for the polar slope and its angle $\alpha$ between the tangent of a curve and the corresponding polar circle (s. diagram).

For Fermat's spiral $r=a{\sqrt {\varphi }}$ one gets

• $\tan \alpha ={\frac {1}{2\varphi }}.$ Hence the slope angle is monotonely decreasing.

### Curvature

From the formula

$\kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}$ for the curvature of a curve with polar equation $\;r=r(\varphi )\;$ and its derivatives $\;r'={\tfrac {a}{2{\sqrt {\varphi }}}}={\tfrac {a^{2}}{2r}}\;$ and $\;r''=-{\tfrac {a}{4{\sqrt {\varphi }}^{3}}}=-{\tfrac {a^{4}}{4r^{3}}}\;$ one gets the curvature of a Fermat's spiral:

• $\kappa (r)={\frac {2r(4r^{4}+3a^{4})}{(4r^{4}+a^{2})^{3/2}}}.$ At the origin the curvature is $0$ . Hence the complete curve has

### Area between arcs

The area of a sector of Fermat's spiral between two points $\;(r(\varphi _{1}),\varphi _{1})\;$ and $\;(r(\varphi _{2}),\varphi _{2})\;$ is

${\underline {A}}={\tfrac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi ={\tfrac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi \;d\varphi ={\frac {a^{2}}{4}}{\big (}\varphi _{2}^{2}-\varphi _{1}^{2}{\big )}$ $\quad ={\frac {a^{2}}{4}}{\big (}\varphi _{2}+\varphi _{1}{\big )}{\big (}\varphi _{2}-\varphi _{1}{\big )}\;.$ After raising both angles by $2\pi$ one gets

${\overline {A}}={\frac {a^{2}}{4}}{\big (}\varphi _{2}+\varphi _{1}+4\pi {\big )}{\big (}\varphi _{2}-\varphi _{1}{\big )}={\underline {A}}+a^{2}\pi {\big (}\varphi _{2}-\varphi _{1}{\big )}\;.$ Hence the area $A$ of the region between two neighboring arcs is

• $A=a^{2}\pi {\big (}\varphi _{2}-\varphi _{1}{\big )}\;.$ $A$ 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: $A_{1}=A_{2}=A_{3}$ .

This property is used in electrical engineering for the construction of variable capacitors. 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  to Marin Mersenne which contains the following special case:

Let $\varphi _{1}=0,\varphi _{2}=2\pi$ , then the area of the black region (see diagram) is $A_{0}=a^{2}\pi ^{2}=$ half of the area of the circle $K_{0}$ with radius $r(2\pi )$ . The regions between neighboring curves (white, blue, yellow) have the same area $A={\color {red}2}a^{2}\pi ^{2}$ Hence:

• The area between two arcs of the spiral after a full turn equals the area of the circle $K_{0}$ .

### Arclength

The length of the arc of Fermat's spiral between two points $\;(r(\varphi _{1}),\varphi _{1}),(r(\varphi _{2}),\varphi _{2})\;$ can be calculated by the integral:

$L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi =\cdots ={\frac {a}{2}}\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {1}{\varphi }}+4\varphi }}\,\mathrm {d} \varphi \;.$ This integral leads to an elliptical integral, which can be solved numerically.

### Circle inversion

The inversion at the unit circle has in polar coordinates the simple description: $\ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\$ .

• The image of Fermat's spiral $\ r=a{\sqrt {\varphi }}\$ under the inversion at the unit circle is a Lituus spiral with polar equation $\;r={\frac {1}{a{\sqrt {\varphi }}}}\;$ .

For $\;\varphi ={\tfrac {1}{a^{2}}}\;$ both curves intersect at a fixpoint 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 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  is

$r=c{\sqrt {n}},$ $\theta =n\times 137.508^{\circ },$ 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 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. 

## Related Research Articles In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians. A sphere is a geometrical object in three-dimensional space that is the surface of a ball. A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north. A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. In geometry, a nephroid is a specific plane curve whose name means 'kidney-shaped'. Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Proctor in 1878.

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. Among these are the following Arc length is the distance between two points along a section of a curve. In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined by an equation of the form In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle. Space-oblique Mercator projection is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic.

The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions. In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. In geometry, the spiral of Theodorus is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene. In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin, which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases are also known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them. In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. In mathematics, a conical spiral is a curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral.

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