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Difference between revisions of "Sinusoidal spiral"

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$$\rho^m=a^m\cos m\phi.$$
 
$$\rho^m=a^m\cos m\phi.$$
  
When $m$ is rational, this is an [[Algebraic curve|algebraic curve]]. In particular, when $m=1$ it is a circle, when $m=-1$ — an equilateral hyperbola, when $m=1/2$ — a [[Cardioid|cardioid]], and when $m=-1/2$ — a parabola.
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When $m$ is rational, this is an [[algebraic curve]]. In particular, when $m=1$ it is a circle, when $m=-1$ — an equilateral hyperbola, when $m=1/2$ — a [[cardioid]], and when $m=-1/2$ — a parabola.
  
 
For general $m>0$ the sinusoidal spiral passes through the pole, and is entirely contained within a circle of radius $a$. When $m$ is negative, the radius vector of the curve can take arbitrarily large values and the curve does not pass through the pole. The sinusoidal spiral is symmetric with respect to the polar axis, and when $m=p/q$ is rational (where $p$ and $q$ are relatively prime numbers), it has $p$ axes of symmetry passing through the pole. When $m$ is a positive integer, the radius vector of the curve is a periodic function of period $2\pi/m$. As $\phi$ goes from 0 to $2\pi$, the curve consists of $m$ branches, each contained in an angle of $\pi/m$. In this case the pole is a multiple point (see Fig.). When $m=p/q$ is a positive rational number, the curve consists of $p$ intersecting branches. When $m$ is a negative integer, the curve consists of $|m|$ infinite branches, which can be obtained by inverting the spiral with $m'=-m$.
 
For general $m>0$ the sinusoidal spiral passes through the pole, and is entirely contained within a circle of radius $a$. When $m$ is negative, the radius vector of the curve can take arbitrarily large values and the curve does not pass through the pole. The sinusoidal spiral is symmetric with respect to the polar axis, and when $m=p/q$ is rational (where $p$ and $q$ are relatively prime numbers), it has $p$ axes of symmetry passing through the pole. When $m$ is a positive integer, the radius vector of the curve is a periodic function of period $2\pi/m$. As $\phi$ goes from 0 to $2\pi$, the curve consists of $m$ branches, each contained in an angle of $\pi/m$. In this case the pole is a multiple point (see Fig.). When $m=p/q$ is a positive rational number, the curve consists of $p$ intersecting branches. When $m$ is a negative integer, the curve consists of $|m|$ infinite branches, which can be obtained by inverting the spiral with $m'=-m$.
 
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{{:Sinusoidal spiral/Fig1}}
<center><asy>
 
import graph;
 
size (200);
 
 
 
real r = 2.3;
 
real m = 4;
 
 
 
real eps=10.^(-10);
 
for  (int k=0; k<m; ++k) {
 
  draw ( polargraph(  new real(real x) {return cos(m*x)^(1/m);}, -(pi/2m)+eps+k*2pi/m, (pi/2m)-eps+k*2pi/m ),
 
defaultpen+1.5 );
 
  draw ( -r*expi(-pi/2m+k*2pi/m)..r*expi(-pi/2m+k*2pi/m), dashed );
 
  draw ( -r*expi(pi/2m+k*2pi/m)..r*expi(pi/2m+k*2pi/m), dashed );
 
}
 
label( "$m=4$", (0.58,0.02), fontsize(7pt) );
 
 
 
real eps=10.^(-2);
 
for  (int k=0; k<m; ++k) {
 
  draw ( polargraph(  new real(real x) {return cos(m*x)^(-1/m);}, -(pi/2m)+eps+k*2pi/m, (pi/2m)-eps+k*2pi/m ),
 
defaultpen+1.5 );
 
}
 
label( "$m=-4$", (1.55,0.02), fontsize(7pt) );
 
 
 
label( "sinusoidal spiral: $a=1$", (0,2.3) );
 
draw ( unitcircle, dashed );
 
</asy></center>
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov,   "Planar curves" , Moscow (1960) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves", Moscow (1960) (In Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.D. Lawrence, "A catalog of special plane curves", Dover (1972) {{ISBN|0-486-60288-5}} {{ZBL|0257.50002}}</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> E.H. Lockwood, "A book of curves" , Cambridge Univ. Press (1967)</TD></TR>
 +
</table>
  
  
 
+
{{OldImage}}
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Lockwood,  "A book of curves" , Cambridge Univ. Press  (1967)</TD></TR></table>
 

Latest revision as of 06:36, 24 April 2023

A plane curve whose equation in polar coordinates has the form

$$\rho^m=a^m\cos m\phi.$$

When $m$ is rational, this is an algebraic curve. In particular, when $m=1$ it is a circle, when $m=-1$ — an equilateral hyperbola, when $m=1/2$ — a cardioid, and when $m=-1/2$ — a parabola.

For general $m>0$ the sinusoidal spiral passes through the pole, and is entirely contained within a circle of radius $a$. When $m$ is negative, the radius vector of the curve can take arbitrarily large values and the curve does not pass through the pole. The sinusoidal spiral is symmetric with respect to the polar axis, and when $m=p/q$ is rational (where $p$ and $q$ are relatively prime numbers), it has $p$ axes of symmetry passing through the pole. When $m$ is a positive integer, the radius vector of the curve is a periodic function of period $2\pi/m$. As $\phi$ goes from 0 to $2\pi$, the curve consists of $m$ branches, each contained in an angle of $\pi/m$. In this case the pole is a multiple point (see Fig.). When $m=p/q$ is a positive rational number, the curve consists of $p$ intersecting branches. When $m$ is a negative integer, the curve consists of $|m|$ infinite branches, which can be obtained by inverting the spiral with $m'=-m$.

References

[1] A.A. Savelov, "Planar curves", Moscow (1960) (In Russian)
[2] J.D. Lawrence, "A catalog of special plane curves", Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002
[a1] E.H. Lockwood, "A book of curves" , Cambridge Univ. Press (1967)



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How to Cite This Entry:
Sinusoidal spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sinusoidal_spiral&oldid=35331
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article