If you want to use only second order surfaces to make a continuous shape that looks like it was turned on a lathe, here is a description, in broad outlines, of how you can do it.

Suppose the user has chosen some set of key points, where each key point is specified by a value along some preferred axis (say the z axis) and some radius in the other two axes (say the x and y axes). Your job is to create a set of sections of second order surfaces passing through these key points, such that, when pieced together, the surfaces appear to form one continuous surface of revolution.

The user adjusts the radius at a key point (z,r), as shown by the big black dots in the applet below. The trick is to compute the square of that radius, giving (z,r2). Then you can fit a smooth curve consisting of two parabolas in the interval between successive key points.

Each parabola will describe a section of a paraboloid of revolution of the form:

A r2 + B = z

for some constants A and B. When you take the square root in the radial direction, you get a surface equation of the form:

A r2 + B = z2

Substituting x2+y2 for r2 gives the quadratic surface equation:

A x2 + A y2 - z2 + B = 0

Geometrically, this means that each paraboloid of revolution section is converted into either a section of an ellipsoid (when A is negative) or a section of a hyperboloid (when A is positive and B is not zero), or a section of a cone (when A is positive and B is zero).