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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.

  • 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](http://www.wolframalpha.com/input/?i=plot%204%20%281/4%29%20x%20y%20%28x%5E2%20-%20y%5E2%29%20=%20%28x%5E2%20%2b%20y%5E2%29%5E2%20%28sqrt%28x%5E2%20%2b%20y%5E2%29%20-%201%29) in Cartesian form.2012-11-01

2 Answers 2

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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.

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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.

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    @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?2012-10-30