Paraboloid Equations: Coordinates & Relationships

[as390]

Hi all,
I just joined this forum and I desperately need the coordinates of a paraboloid in any orthogonal curvilinear coordinates. Like a sphere is easy to present in spherical coordinates and vice versa. In the same way, I will be very thankful if someone can relate any point on a paraboloid where the paraboloid rotates from the x-axis. So then what will be the coordinates of any point in spherical coordinates and its relation to cartesian coordinates on the surface of such paraboloid.

 

[HallsofIvy]

It's not clear to me what you want. You say you want the equation of a paraboloid in any orthogonal curvilinear coordinates? Clearly there is no "general" formula. It would be possible, for example, to set up your coordinate system so that the paraboloid itself is one of the coordinate surfaces (say, a "u", "v" surface) with the third coordinates, w, say, measured along the curves orthogonal to the paraboloids. In that case, the equation would be simply w= constant, just as a sphere is particularly easy in spherical coordinates. In other coordinate systems, the equation of a paraboloid might be very complicated. Again, there is no general formula.

But then, at the end, you seem to be asking for the equation of a paraboloid in spherical coordinates only. That's relatively easy: write the equation of the paraboloid in Cartesian coordinates, then replace x with $\rho cos(\theta) sin(\phi)$, y with $\rho sin(\theta) sin(\phi)$, and z with $\rho cos(\phi)$.

For example, the "basic" paraboloid, z= x²+ y² becomes
$$\rho cos(\phi)= \rho^2 cos^2(\theta) sin^2(\phi)+ \rho^2 sin^2(\theta) sin^2(\phi)= \rho^2 sin^2(\phi)$$
or simply
[tex]cos(\phi)= \rho sin^2(\phi)[/itex]<br /> in spherical coordinates.<br /> <br /> I don't understand what you mean by "rotates away from the x-axis".[/tex]

 

[Defennder]

I think he was referring to this:
http://en.wikipedia.org/wiki/Paraboloidal_coordinates

Unfortunately I can't vouch for the accuracy of Wikipedia.

 

[as390]

I may explain my point through an example of a cone.

Le X*,Y*,Z* be the usual cartesian coordinates where X* passes through the centre of the cone and that the cone rotates from the X*-axis.Let \phi be the half angle of the cone.
The origin is at the vertex of the cone.Let x is the distance from the origin along the cone side to a point where the perpendicular distance from the X*-axis is say r*. Then
clearly r*=x sin(\phi).Also let z is the normal to the cone surface at point where r=x sin(\phi)
let \theta be the angle through which the cone rotates.Then it is already established that the cartesian coordinates can be related in terms of x,\theta,z as under

X*=x cos(\phi)-z sin(\phi)

Y*=(x sin(\phi)+z cos(\phi)) sin(\theta)

Z*=(x sin(\phi)+z cos(\phi))cos(\theta)

I want similar relations for a paraboloid. We can consider one angle of revolution say \theta and the other the slope angle at the point p on the surface of paraboloid say.
Also note that the coordinates in the above cone becomes simply cylindrical when we replace \phi by right angle.

I will be thankful if anyone takes interest in my problem.

 

[as390]

Adding further to my problem, if you see at any point of the cone surface we constructed an orthogonal system of coordinates which is x,\theta,z. And these coordinates are well defined because they can be defined in terms of X*,Y*,Z*.
Whichever, orthogonal coordinates can one set for an arbitrary point on the paraboloid surface, are acceptible in my problem but those must be well defined.
Any one who help me in sorting out this,must be referenced in my postgrad final thesis.
I am waiting impatiently to hear it from anyone in the forum. I tried it quite for long but goes in vain.

 

[as390]

It seems like I can figure out my proplem in prolate spheriodal coordinates.
However,I would be very happy if some one can figure out it for me in the common two systems of coordinates like spherical,cylindrical because the rectangular system ofcourse cannot make a physical orthogonal system at the surface point of my chosen body.