How to determine parabolic object.

[Thundagere]

This is more of an mathematics application question than anything, but.
Let's say I'm building a satellite or some sort of focusing device. I obviously need a parabola. If I have an object that resembles a parabola—for example, a pot of some sort—how can I determine that's it's in reality a parabola, and not just a random shape?
Also, what would I need to do to find the area inside of that pot?
Thanks.

 

[flyingpig]

You mean a paraboloid?

It should have the following equation

$$f(x,y) = a(x - x_0)^2 + b(y - y_0)^2 + k$$

 

[Thundagere]

Yep.
So I would measure out it's parameters, and say, mark a point as the origin? Then (let's assume inches instead of cm), mark another point on the edge and find it's coordinates? Would I keep doing that until I had an equation, then plug in values and make sure it fit roughly?

 

[Office_Shredder]

You're saying you actually have an aluminum object in real life whose paraboloidness you wish to determine? Assuming it's reflective you can shoot lasers parallel to the rotational axis and they all should pass through a point... so you could try pointing a bunch simultaneously around the edges, hold up a sheet in the middle and see what kind of spread you get as you move it back and forth. This would also just tell you how close to a focusing device you've actually constructed.

To get the area I would just find the density of the material, measure the thickness (assuming it's uniform) and weigh the thing. No need to be fancy if you don't have to be

Trying to measure a set of points and get relative coordinates, followed by finding a best fit equation and margin of error seems like a pretty difficult and error prone method especially if you're doing it by hand