Hi Let z=ρsinθ where ρ is the length of the radius vector for a point P on the surface of the paraboloid and θ is the angle of the vector between it and the x-y plane.
x=-ρcosθsinϕ and y=ρcosθcosϕ so x²+y²=ρ²cos²θ, z=x²+y² becomes ρsinθ=ρ²cos²θ.
Therefore, ρ=secθtanθ in polar coords for θ in (-π/2,π/2) and all ϕ in [0,2π).
Point F(x,y,z) is shown as a corner of a cuboid such that OF is the vector for point F, which can be related to ρ, θ and ϕ. For right-handed orthogonal axes we have to write x as a negative quantity. The dashed lines represent the positive part of the x, y and z axes. The dotted lines are edges that would be hidden if the cuboid were solid.