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udvat

point inside paraboloid

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The equation of paraboloid is x^2/a^2 + y^2/b^2 = z/c the radius r and height h is given. How can I determine whether a point p is inside/outside the paraboloid? Would anyone please help me? Thanks a lot in advance.

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If you want an exact solution, I'm not sure I can help, but I'll give it a shot.

Is this for an algorithm?

If it is, translate the paraboloid to the origin. Translate the point by the same amount.

If it is assumed that the parabaloid will always be in the positive or negative y direction(up and down), check to see if the distance of a 2D vector consisting of Px and Pz(from the point p) is less than the given radius. If it is greater, return false.

Set up an equation that rearranges the given equation to solve for y.

Then plug in the x and z components of the point p, call them Px and Pz.

This will give you the y value(height if you're using y is up coordinate system) at the x,z location. Just compare Py to the answer to see if it is inside our outside.

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Your quadratic equation does not involve r and h. Did you mean that your object is bounded by z <= h?

For the unbounded paraboloid, and assuming c > 0, (x,y,z) is strictly inside the paraboloid when x^2/a^2 + y^2/b^2 < z/c and is strictly outside the paraboloid when x^2/a^2 + y^2/b^2 > z/c.

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