What is equation of paraboloid?

The equation of a paraboloid with axis of symmetry along the z-axis can be written as:

z = a(x^2 + y^2) + bz + c

where a, b, and c are constants. This equation describes a surface that opens upwards or downwards, depending on the sign of a. If a is positive, the paraboloid opens upwards, and if a is negative, it opens downwards.

The parameter b determines the position of the vertex of the paraboloid along the z-axis. If b is positive, the vertex is above the xy-plane, and if b is negative, it is below.

The constant c represents a translation of the paraboloid in the z-direction. If c is positive, the paraboloid is shifted upwards from the origin, and if it is negative, it is shifted downwards.

There are also other forms of the equation of a paraboloid, such as using x or y as the axis of symmetry. However, the general idea is the same: the equation describes a surface that is symmetric with respect to a certain axis and has a parabolic cross-section.