Converting Spherical to Cartesian

In summary: Then paste it below so that the summary is directly underneath the original problem.In summary,The problem is that you are missing a factor of cos(phi) in the numerator on the right side. To solve the equation, you need to rewrite everything in terms of sines and cosines, rearrange terms so the trig functions multiply rho, and identify what cartesian coordinates the various products are equal to.
  • #1
bglb212
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0
Problem solved!
 
Last edited:
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  • #2
Your equation is equivalent to [itex]\rho[/itex] = cos([itex]\phi[/itex])/(1 - cos2([itex]\phi[/itex]).

Your relevant equations show the converstion from spherical to Cartesian coordinates. Do you know the conversions going in the other direction?

One trick that will be helpful is to multiply both sides by rho. This potentially adds a point (rho = 0) that might not be a solution of your original equation, so you should check whether this is already a solution of your equation.
 
  • #3
Solved!
 
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  • #4
As an intermediate step, look at what I have in my previous post. Use trig identities to get to what I showed, then go from there.
 
  • #5
Look at the equations for x, y, and z. Note that they're all of the form rho times some combination of sines and cosines. So the first thing you should do is rewrite everything in terms of sines and cosines. Next, rearrange terms so the trig functions multiply rho. If needed, do as Mark suggested, and multiply both sides by rho. Then try to identify what cartesian coordinates the various products are equal to.
 
  • #6
Solved!
 
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  • #7
You're missing a factor of cos(phi) in the numerator on the right side.
 
  • #8
Solved!
 
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  • #9
Try bringing the (1+cos^2(phi)) to the other side.
 
  • #10
bglb212 said:
I don't think so, because doesn't pcos(phi)=z?
OK. I didn't catch what you were doing.

vela said:
Try bringing the (1+cos^2(phi)) to the other side.
Don't you mean (1 - cos^2(phi))?
 
  • #11
bglb212 said:
x^2+y^2+z^2=z/(1-cos(phi)^2
It's probably helpful not to change the rho^2 on the left side just yet.
 
  • #12
Solved!
 
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  • #13
Let us decide whether nothing good comes out of it. Show us what you tried.

You have rho^2 = z/(1 - cos^2(phi))
What do you get when you multiply both sides by 1 - cos^2(phi)?
 
  • #14
Solved!
 
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  • #15
Mark44 said:
Don't you mean (1 - cos^2(phi))?
Yes, I did. Thanks for catching that.
 
  • #16
bglb212 said:
got it, I like you mark. you're helpful. may your children be plentiful and live long
That's a very nice thing to say! Thank you!
 
  • #17
Solved!
 
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  • #18
csc(x) = 1/sin(x), cot(x) = cos(x)/sin(x)

Are those the ones you mean?
 
  • #19
Notice to Homework helpers. Please be sure to use the quote function to make a copy of the poster's problem.
 

1. What is the process of converting spherical coordinates to Cartesian coordinates?

The process of converting spherical coordinates to Cartesian coordinates involves using mathematical formulas to calculate the corresponding x, y, and z coordinates. The equations used depend on the given values of the spherical coordinates, which are typically expressed as r (radius), θ (polar angle), and φ (azimuthal angle).

2. Can you explain the difference between spherical and Cartesian coordinates?

Spherical coordinates use a radial distance, polar angle, and azimuthal angle to locate a point in three-dimensional space, while Cartesian coordinates use x, y, and z coordinates to represent a point. Spherical coordinates are often used in physical and engineering applications, while Cartesian coordinates are used more commonly in mathematics and computer graphics.

3. How do you convert a point from spherical coordinates to Cartesian coordinates?

To convert a point from spherical coordinates to Cartesian coordinates, the formulas r = √(x^2 + y^2 + z^2), θ = arccos(z/r), and φ = arctan(y/x) can be used. These equations can be rearranged to solve for the x, y, and z coordinates of the point.

4. What is the purpose of converting from spherical to Cartesian coordinates?

Converting from spherical to Cartesian coordinates allows for a point to be represented in a different coordinate system, which can be useful for various applications. For example, converting spherical coordinates to Cartesian coordinates can make it easier to plot points on a graph or to perform calculations involving distance and angles.

5. Are there any limitations to converting between spherical and Cartesian coordinates?

One limitation of converting between spherical and Cartesian coordinates is that it can be a complex process, especially for points with large absolute values. Additionally, some points may have multiple representations in spherical coordinates, making it difficult to find the corresponding Cartesian coordinates. It is important to carefully consider the given values and which formulas to use in order to accurately convert between the two coordinate systems.

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