4

Consider a sine wave having $4$ cycles wrapped around a circle of radius 1 unit.

$$ y = \sin(4x) $$

To find the equation of the sine wave with circle acting, one approach is to consider the sine wave along a rotated line. But it doesn't suffice for the circular path.

RAM
  • 225
  • Let $P(x,y)$ be a point on your desired wave, and let $M$ be its midpoint. Write the desired function of the distance $|PM|$. – Bart Michels Oct 30 '12 at 17:11
  • Do you mean something like the curve given by the polar equation $r=1+a \sin \theta$? – Matthew Conroy Oct 30 '12 at 18:30
  • @Matthew: assuming we're interpreting the quedtion correctly, wouldn't it be $\sin 4\theta$? – Javier Oct 30 '12 at 18:41
  • @JavierBadia Oops. Yeah, $r=1+a \sin 4 \theta$. Thanks for catching that. – Matthew Conroy Oct 31 '12 at 01:03
  • @Neeraj: From what I can tell, people have understood your question correctly, but it is you who have not understood their comments and answers. Instead of asking them to re-read your question, I think you would do better to mention what it is you find lacking in their responses. –  Oct 31 '12 at 05:31
  • 1
    @NeerajTuteja: Try a smaller $a$, such as $a=0.25$: http://www.wolframalpha.com/input/?i=polar+plot+r+%3D+1+%2B+0.25+sin%284t%29 – Blue Oct 31 '12 at 07:24
  • 1
    I don't see the distinction you're making; polar and Cartesian coordinates are simply different ways of expressing the same graph. For example, you can substitute $r = \sqrt{x^2+y^2}$ and $\sin 4\theta=4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta=(4x^3y-4xy^3)/r^4$ into the polar equation to get $4axy(x^2-y^2)=(x^2+y^2)^2(\sqrt{x^2+y^2}-1)$, which is the same graph in Cartesian form. –  Nov 01 '12 at 20:53

2 Answers2

7

it should be, in cartesian coordinates

x = (R + a · sin(n·θ)) · cos(θ) + xc

y = (R + a · sin(n·θ)) · sin(θ) + yc

where

R is circle's radius

a is sinusoid amplitude

θ is the parameter (angle), from 0 to 2π

xc,yc is circle's center point

n is number of sinusoids on circle

you can also get a pure cartesian equation (non-parametric) on x/y, but just for half circle, solving second for sin(θ) and replacing it on first one.

Max
  • 86
3

Do it first for the circle centered at the origin in polar coordinates.

Then switch do Cartesian coordinates, then shift to the actual center of the circle.

Phira
  • 20,860
  • 2
    @NeerajTuteja Your comment is not telling me where your are stuck, is it? Maybe you are not familiar with polar coordinates, maybe you do not really know what you mean by "wrapping around the circle", maybe you have trouble switching to Cartesian coordinates, how should I know when you just tell me my answer must be wrong? – Phira Oct 30 '12 at 22:00