How to find cylindrical co-ordinates of point of intersection of r=a and \(\rho=a\)?

[Win_odd Dhamnekar]

Let \(P=(a,\theta,\phi)\) be a point in spherical co-ordinates,with a>0, and \(0 <\phi <\pi\). Then P lies on the sphere \(\rho=a\). Since \(0 <\phi <\pi\), the line segment drawn from the origin to P can be extended to intersect the cylinder given by r=a.(in cylindrical co-ordinates).

Now, how to find the cylindrical co-ordinates of that point of intersection?

I don't have any hint to answer this question till now. Answer provided to me is \((a, \theta, a\cot{\phi})\). I am working on this question to find out step by step solution.

Meanwhile, if any member of free math help forum, knows the correct answer may reply with correct answer.

 

[Dr.Peterson]

Your title asks a different question than your content. Do you want to intersect the sphere [MATH]\rho = a[/MATH], or the line [MATH]\theta = \theta_1, \phi = \phi_1[/MATH], with the cylinder [MATH]r = a[/MATH]? The former, of course, is easy, so I assume the title is wrong.

I might either write the line in cylindrical coordinates, or the cylinder in spherical coordinates, and then solve. Or I might just imagine a vertical plane through P and the origin, and think about triangles.

 

[Win_odd Dhamnekar]

Your title asks a different question than your content. Do you want to intersect the sphere [MATH]\rho = a[/MATH], or the line [MATH]\theta = \theta_1, \phi = \phi_1[/MATH], with the cylinder [MATH]r = a[/MATH]? The former, of course, is easy, so I assume the title is wrong.

I might either write the line in cylindrical coordinates, or the cylinder in spherical coordinates, and then solve. Or I might just imagine a vertical plane through P and the origin, and think about triangles.

Hello,
\(\rho=a\) is an equation of sphere. The parametric spherical equation of the line segment drawn from the origin to P having a length \(\rho\) is \(\rho=a, \theta=\theta, \phi=\phi\) since \(P=(a,\theta,\phi)\) is given as spherical co-ordinates of P.

But if you think putting \(\theta =\theta_1, \phi=\phi_1\), it is easy to answer this question, you may do so.

 

[Win_odd Dhamnekar]

Hello,
After carefully thinking, i got the answer to this question \( (a, \theta, a*\cot{\phi})\) as cylindrical co-ordinates of point of intersection between cylinder given by r=a and line given by spherical parametric equation \((\rho=a, \theta=\theta_1,\phi=\phi_1)\).

But i can't show it mathematically here. Would it show it mathematically here ?

 

[Dr.Peterson]

Hello,
\(\rho=a\) is an equation of sphere. The parametric spherical equation of the line segment drawn from the origin to P having a length \(\rho\) is \(\rho=a, \theta=\theta, \phi=\phi\) since \(P=(a,\theta,\phi)\) is given as spherical co-ordinates of P.

But if you think putting \(\theta =\theta_1, \phi=\phi_1\), it is easy to answer this question, you may do so.

In your equation of a sphere, [MATH]a[/MATH] is a constant, right? But in your parametric equation of a line, [MATH]a[/MATH] is a parameter while [MATH]\theta[/MATH] and [MATH]\phi[/MATH] are constants [EDIT]. This is rather confusing! It's essential to distinguish constants from variables, so that your [MATH]\theta=\theta[/MATH] is more than a tautology. Failure to do this (by temporarily renaming the constants) may be the cause of your difficulty.

I still don't see what the sphere has to do with your question, though.

Hello,
After carefully thinking, i got the answer to this question \( (a, \theta, a*\cot{\phi})\) as cylindrical co-ordinates of point of intersection between cylinder given by r=a and line given by spherical parametric equation \((\rho=a, \theta=\theta_1,\phi=\phi_1)\).

But i can't show it mathematically here. Would it show it mathematically here ?

Are you saying something goes wrong when you try to attach your work? Or are you asking me (not "it") to show my work?

The line, in cylindrical coordinates, is given parametrically by [MATH]r = t, \theta = \theta_1, z = t\cot(\phi_1)[/MATH], which leads immediately to the answer. The question for you, I guess, is how to determine the conversion between cylindrical and spherical coordinates, [MATH]z = r\cot(\phi)[/MATH]. To help you with that, I have to see what you know about each coordinate system. If you have been taught a conversion directly between cylindrical and spherical, then this is just the conversion formula [MATH]r = z\tan(\phi)[/MATH].