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

 

[CellCoree]

To determine if an object is truly parabolic, you would need to measure its shape and compare it to the mathematical definition of a parabola. A parabola is a symmetrical curve with a single focus point and a directrix line that is perpendicular to the axis of symmetry. You can use a measuring tool, such as a ruler or caliper, to measure the distance from the focus point to various points on the curve and compare them to the distance from those same points to the directrix line. If they are equal, then the object is a parabola. Additionally, you can use a mathematical equation to plot the shape of the object and see if it matches the expected parabolic curve.

To find the area inside of the pot, you can use the formula for the area of a parabolic segment, which is (2/3)πr^3, where r is the radius of the pot. Alternatively, if the pot is not a perfect parabola, you can use the more general formula for the area under a curve, which involves integration. This would require knowledge of calculus and the specific equation that represents the shape of the pot.