Английская Википедия:Fermat's spiral
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat.[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.
Coordinate representation
Polar
The representation of the Fermat spiral in polar coordinates Шаблон:Math is given by the equation <math display=block>r=\pm a\sqrt{\varphi}</math> for Шаблон:Math.
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.
Cartesian
The Fermat spiral with polar equation <math display=block>r=\pm a\sqrt{\varphi}</math> can be converted to the Cartesian coordinates Шаблон:Math by using the standard conversion formulas Шаблон:Math and Шаблон:Math. Using the polar equation for the spiral to eliminate Шаблон:Mvar from these conversions produces parametric equations for one branch of the curve:
- <math>\begin{cases}
x(\varphi) = + a\sqrt{\varphi} \cos(\varphi) \\
y(\varphi) = + a\sqrt{\varphi} \sin(\varphi) \end{cases}</math>
and the second one
- <math>\begin{cases}
x(\varphi) = - a\sqrt{\varphi} \cos(\varphi) \\
y(\varphi) = - a\sqrt{\varphi} \sin(\varphi) \end{cases}</math>
They generate the points of branches of the curve as the parameter Шаблон:Mvar ranges over the positive real numbers.
For any Шаблон:Math generated in this way, dividing Шаблон:Mvar by Шаблон:Mvar cancels the Шаблон:Math parts of the parametric equations, leaving the simpler equation Шаблон:Math. From this equation, substituting Шаблон:Mvar by Шаблон:Math (a rearranged form of the polar equation for the spiral) and then substituting Шаблон:Mvar by Шаблон:Math (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only Шаблон:Mvar and Шаблон:Mvar: <math display="block">\frac{x}{y}=\cot\left( \frac{x^2 + y^{2}}{a^{2}} \right).</math> Because the sign of Шаблон:Mvar is lost when it is squared, this equation covers both branches of the curve.
Geometric properties
Division of the plane
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.
Polar slope
From vector calculus in polar coordinates one gets the formula
- <math>\tan\alpha = \frac{r'}{r}</math>
for the polar slope and its angle Шаблон:Mvar between the tangent of a curve and the corresponding polar circle (see diagram).
For Fermat's spiral Шаблон:Math one gets
- <math>\tan\alpha=\frac{1}{2\varphi}.</math>
Hence the slope angle is monotonely decreasing.
Curvature
From the formula
- <math>\kappa = \frac{r^2 + 2(r')^2 - r\,r}{\left(r^2+(r')^2\right)^\frac32}</math>
for the curvature of a curve with polar equation Шаблон:Math and its derivatives
- <math>\begin{align}
r' &= \frac{a}{2\sqrt{\varphi}}=\frac{a^2}{2r}\\ r&= -\frac{a}{4\sqrt{\varphi}^3}=-\frac{a^4}{4r^3} \end{align}</math> one gets the curvature of a Fermat's spiral: <math display="block">\kappa(r) = \frac{2r\left(4r^4+3a^4\right)}{\left(4r^4+a^4\right)^\frac32}.</math>
At the origin the curvature is 0. Hence the complete curve has at the origin an inflection point and the Шаблон:Mvar-axis is its tangent there.
Area between arcs
The area of a sector of Fermat's spiral between two points Шаблон:Math and Шаблон:Math is
- <math>\begin{align}
\underline A&=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} r(\varphi)^2\, d\varphi \\ &=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} a^2 \varphi\, d\varphi \\ &=\frac{a^2}{4}\left(\varphi_2^2-\varphi_1^2\right) \\ &=\frac{a^2}{4}\left(\varphi_2+\varphi_1\right)\left(\varphi_2-\varphi_1\right). \end{align}</math>
After raising both angles by Шаблон:Math one gets
- <math> \overline A= \frac{a^2}{4}\left(\varphi_2+\varphi_1 +4\pi\right)\left(\varphi_2-\varphi_1\right)=\underline A + a^2\pi\left(\varphi_2-\varphi_1\right).</math>
Hence the area Шаблон:Mvar of the region between two neighboring arcs is <math display="block">A=a^2\pi\left(\varphi_2-\varphi_1\right).</math> Шаблон:Mvar 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: Шаблон:Math.
This property is used in electrical engineering for the construction of variable capacitors.[2]
Special case due to Fermat
In 1636, Fermat wrote a letter [3] to Marin Mersenne which contains the following special case:
Let Шаблон:Math; then the area of the black region (see diagram) is Шаблон:Math, which is half of the area of the circle Шаблон:Math with radius Шаблон:Math. The regions between neighboring curves (white, blue, yellow) have the same area Шаблон:Math. Hence:
- The area between two arcs of the spiral after a full turn equals the area of the circle Шаблон:Math.
Arc length
The length of the arc of Fermat's spiral between two points Шаблон:Math can be calculated by the integral:
<math display="block">\begin{align} L&=\int_{\varphi_1}^{\varphi_2}\sqrt{\left(r^\prime(\varphi)\right)^2+r^2(\varphi)}\,d\varphi \\ &=\frac{a}{2}\int_{\varphi_1}^{\varphi_2}\sqrt{\frac{1}{\varphi}+4\varphi}\,d\varphi . \end{align}</math>
This integral leads to an elliptical integral, which can be solved numerically.
The arc length of the positive branch of the Fermat's spiral from the origin can also be defined by hypergeometric functions Шаблон:Math and the incomplete beta function Шаблон:Math:[4]
<math display="block">\begin{align} L &= a \cdot \sqrt{\varphi} \cdot \operatorname{_{2}F_{1}}\left( -\tfrac12,\, \tfrac14;\, \tfrac54;\, -4 \cdot \varphi^{2} \right)\\ &= a \cdot\frac{1 - i}{8} \cdot \operatorname{B}\left( -4 \cdot \varphi^{2};\, \tfrac14,\, \tfrac32 \right)\\ \end{align}</math>
Circle inversion
The inversion at the unit circle has in polar coordinates the simple description Шаблон:Math.
- The image of Fermat's spiral Шаблон:Math under the inversion at the unit circle is a lituus spiral with polar equation <math display="block">r=\frac{1}{a\sqrt{\varphi}}.</math> When Шаблон:Math, both curves intersect at a fixed point on the unit circle.
- The tangent (Шаблон:Mvar-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 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
<math display="block">\begin{align} r &= c \sqrt{n},\\ \theta &= n \times 137.508^\circ, \end{align}</math>
where Шаблон:Mvar is the angle, Шаблон:Mvar is the radius or distance from the center, and Шаблон:Mvar is the index number of the floret and Шаблон:Mvar is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.[6] Шаблон:Wide image
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.[7] Шаблон:Clear right
See also
References
Further reading
External links
- Шаблон:Springer
- Online exploration using JSXGraph (JavaScript)
- Fermat's Natural Spirals, in sciencenews.org
Шаблон:Spirals Шаблон:Pierre de Fermat
