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Sinusoidal spiral

In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates

Sinusoidal spirals (rn = –1n cos(), θ = π/2) in polar coordinates and their equivalents in rectangular coordinates:
  n = −2: Equilateral hyperbola
  n = −1: Line
  n = −1/2: Parabola
  n = 1/2: Cardioid
  n = 1: Circle

where a is a nonzero constant and n is a rational number other than 0. With a rotation about the origin, this can also be written

The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:

The curves were first studied by Colin Maclaurin.

Equations edit

Differentiating

 

and eliminating a produces a differential equation for r and θ:

 .

Then

 

which implies that the polar tangential angle is

 

and so the tangential angle is

 .

(The sign here is positive if r and cos nθ have the same sign and negative otherwise.)

The unit tangent vector,

 ,

has length one, so comparing the magnitude of the vectors on each side of the above equation gives

 .

In particular, the length of a single loop when   is:

 

The curvature is given by

 .

Properties edit

The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a rectangular hyperbola.

The isoptic, pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals.

One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.

When n is an integer, and n points are arranged regularly on a circle of radius a, then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate.

References edit

  • Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Spiral" p. 213–214
  • "Sinusoidal spiral" at www.2dcurves.com
  • "Sinusoidal Spirals" at The MacTutor History of Mathematics
  • Weisstein, Eric W. "Sinusoidal Spiral". MathWorld.

sinusoidal, spiral, algebraic, geometry, sinusoidal, spirals, family, curves, defined, equation, polar, coordinatess, polar, coordinates, their, equivalents, rectangular, coordinates, equilateral, hyperbola, line, parabola, cardioid, circle, lemniscate, bernou. In algebraic geometry the sinusoidal spirals are a family of curves defined by the equation in polar coordinatesSinusoidal spirals rn 1n cos n8 8 p 2 in polar coordinates and their equivalents in rectangular coordinates n 2 Equilateral hyperbola n 1 Line n 1 2 Parabola n 1 2 Cardioid n 1 Circle n 2 Lemniscate of Bernoulli r n a n cos n 8 displaystyle r n a n cos n theta where a is a nonzero constant and n is a rational number other than 0 With a rotation about the origin this can also be written r n a n sin n 8 displaystyle r n a n sin n theta The term spiral is a misnomer because they are not actually spirals and often have a flower like shape Many well known curves are sinusoidal spirals including Rectangular hyperbola n 2 Line n 1 Parabola n 1 2 Tschirnhausen cubic n 1 3 Cayley s sextet n 1 3 Cardioid n 1 2 Circle n 1 Lemniscate of Bernoulli n 2 The curves were first studied by Colin Maclaurin Equations editDifferentiating r n a n cos n 8 displaystyle r n a n cos n theta nbsp and eliminating a produces a differential equation for r and 8 d r d 8 cos n 8 r sin n 8 0 displaystyle frac dr d theta cos n theta r sin n theta 0 nbsp Then d r d s r d 8 d s cos n 8 d s d 8 r sin n 8 r cos n 8 r sin n 8 cos n 8 displaystyle left frac dr ds r frac d theta ds right cos n theta frac ds d theta left r sin n theta r cos n theta right r left sin n theta cos n theta right nbsp which implies that the polar tangential angle is ps n 8 p 2 displaystyle psi n theta pm pi 2 nbsp and so the tangential angle is f n 1 8 p 2 displaystyle varphi n 1 theta pm pi 2 nbsp The sign here is positive if r and cos n8 have the same sign and negative otherwise The unit tangent vector d r d s r d 8 d s displaystyle left frac dr ds r frac d theta ds right nbsp has length one so comparing the magnitude of the vectors on each side of the above equation gives d s d 8 r cos 1 n 8 a cos 1 1 n n 8 displaystyle frac ds d theta r cos 1 n theta a cos 1 tfrac 1 n n theta nbsp In particular the length of a single loop when n gt 0 displaystyle n gt 0 nbsp is a p 2 n p 2 n cos 1 1 n n 8 d 8 displaystyle a int tfrac pi 2n tfrac pi 2n cos 1 tfrac 1 n n theta d theta nbsp The curvature is given by d f d s n 1 d 8 d s n 1 a cos 1 1 n n 8 displaystyle frac d varphi ds n 1 frac d theta ds frac n 1 a cos 1 tfrac 1 n n theta nbsp Properties editThe inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve s value of n For example the inverse of the lemniscate of Bernoulli is a rectangular hyperbola The isoptic pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral When n is an integer and n points are arranged regularly on a circle of radius a then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral In this case the sinusoidal spiral is a polynomial lemniscate nbsp Wikimedia Commons has media related to Sinusoidal spiral References editYates R C A Handbook on Curves and Their Properties J W Edwards 1952 Spiral p 213 214 Sinusoidal spiral at www 2dcurves com Sinusoidal Spirals at The MacTutor History of Mathematics Weisstein Eric W Sinusoidal Spiral MathWorld Retrieved from https en wikipedia org w index php title Sinusoidal spiral amp oldid 1125632246, wikipedia, wiki, book, books, library,

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