View source for 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>