Are you assuming the four corner points are on the surface of the cone? If yes, then to determine the surface of the cone, you need four parameters: (x,y,z) of the tip, and the slop k

Thinking out loud here...

If you have a point on the surface (x1, y1, z1) and the peak (xp, yp, zp), doesn't that give you the slope k of the cone?

I don't think k and z are independent, though it may be easier to find one or the other.