Spherical coordinates equation

[yitriana]

Homework Statement

Identify surface whose equation in spherical coordinates is given
p = sin(theta)*sin(fi)

The Attempt at a Solution

I know that y = r*sin(theta)*sin(fi). and thus, y = rp.

This yields y = (x² + y²)^0.5*(x² + y²+z²)^0.5

However, this is rather ugly. The answer is supposed to be "a sphere with radius 0.5, center (0. 0.5, 0)" however, I don't see how that results from this expression.

 

[Office_Shredder]

What is p supposed to be?

 

[yitriana]

p is ro. p = (x^2 + y^2 + z^2)^0.5

 

[Fenn]

I'm guessing that the expression should read something like

$$r = \sin\theta\sin\phi$$

where $$\theta$$ and $$\phi$$ are your polar angles. First, look in the x-y plane, where $$\theta=90^\circ$$. This simplifies to

$$r = \sin\phi.$$

Now, apply $$r$$ to the equations linking polar and Cartesian coordinates, for $$\theta=90^\circ$$, as

$$x = r\cos\phi$$

$$y = r\sin\phi.$$

You'll see they come out to be

$$x = \sin\phi\cos\phi$$

$$y = \sin^2\phi.$$

Next, recall the double-angle formulas that $$\sin(2x) = 2\sin x\cos x$$ and $$\cos(2x) = 1 - 2 \sin^2 x$$. I'll leave it as an exercise to substitute these back into the expressions for $$x$$ and $$y$$. What you should notice is that your values should now look like

$$x = A \sin(2\phi)$$

and

$$y = A \cos(2\phi) + B$$

where $$A$$ and $$B$$ are numbers. You should recognize this as the parametric representation of a circle, at the coordinate $$(0,B)$$.

Now, this doesn't exactly answer your question, but it should hopefully get you to visualize how the surface should be a sphere. Particularly, if you repeat this exercise in the y-z plane, where $$\phi=90^\circ$$, you should find find another case where the result is a circle, offset from the origin by some distance $$B$$.

I think that, once you've identified what $$B$$ is, you can look into the coordinate transformation of

$$x' = x, \quad y' = y + B, \quad z' = z$$

and then determine $$r' = \sqrt{x'^2 + y'^2 + z'^2}$$ which should be a constant. If this is the case, it shows that this surface is a sphere centered at $$(0,B,0)$$.