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

polar coordinates 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 (analogous to t ...

r^n = a^n \cos(n \theta)\,

where is a nonzero constant and is a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

other than 0. With a rotation about the origin, this can also be written :

r^n = a^n \sin(n \theta).\,

The term "spiral" is a misnomer, because they are not actually

spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Rectangular hyperbola () *

Line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

() *

Parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...

() *

Tschirnhausen cubic In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation :r = a\sec^3 \left(\frac\right) where is the secant function. History The curve was studied by v ...

() * Cayley's sextet () *

Cardioid In geometry, 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 sinusoida ...

() *

Circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

() *

Lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. ...

() The curves were first studied by

Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bein ...

Equations

Differentiating :

r^n = a^n \cos(n \theta)\,

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

\frac\cos n\theta + r\sin n\theta =0

. Then :

\left(\frac,\ r\frac\right)\cos n\theta \frac
= \left(-r\sin n\theta ,\ r \cos n\theta \right)
= r\left(-\sin n\theta ,\ \cos n\theta \right)

which implies that the polar

tangential angle 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 -axis. (Some authors define the angle as the deviation from the direction of th ...

is :

\psi = n\theta \pm \pi/2

and so the tangential angle is :

\varphi = (n+1)\theta \pm \pi/2

. (The sign here is positive if ''r'' and cos ''n''θ have the same sign and negative otherwise.) The unit tangent vector, :

\left(\frac,\ r\frac\right)

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

\frac = r \cos^ n\theta = a \cos^ n\theta

. In particular, the length of a single loop when

n>0

is: :

a\int_^ \cos^ n\theta\ d\theta

The

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

is given by :

\frac = (n+1)\frac = \frac \cos^ n\theta

Properties

The

inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...

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 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. Examples: # The orthoptic of a parabola is its directrix (proof: see below), # The orthoptic of an ellipse \tfrac + ...

pedal A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control p ...

and negative pedal of a sinusoidal spiral are different sinusoidal spirals. One path of a particle moving according to a

central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...

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 In mathematics, a polynomial lemniscate or ''polynomial level curve'' is a plane algebraic curve of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''. For any such polynomial ''p'' and positive real number ' ...

References

*Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Spiral" p. 213–214
"Sinusoidal spiral" at www.2dcurves.com
*{{MathWorld , title=Sinusoidal Spiral , urlname=SinusoidalSpiral Plane curves Algebraic curves