Parabolic Coordinates 3d

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\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{math}
%\usetikzlibrary{patterns}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{1pt}
%
% File name: Parabolic-Coordinates-3D.tex
% Description: 
% A geometric representation of the parabolic coordinates in 3D-space is shown.
% 
% Date of creation: March,12th, 2023.
% Date of last modification: March,12th, 2023.
% Author: Efraín Soto Apolinar.
% https://www.aprendematematicas.org.mx/author/efrain-soto-apolinar/instructing-courses/
% Source: to appear in the second edition of the book titled:
% Illustrated Glossary for School Mathematics.
% https://tinyurl.com/yayyxrbz
%
% Terms of use:
% According to TikZ.net
% https://creativecommons.org/licenses/by-nc-sa/4.0/
% Your commitment to the terms of use is greatly appreciated.
%
\begin{document}
%
\tdplotsetmaincoords{80}{120}
\begin{tikzpicture}[tdplot_main_coords]
    \tikzmath{function u(\x,\y) {return abs(sqrt(sqrt(\x * \x + \y * \y) - \y)) ;};}
    \tikzmath{function v(\x,\y) {return sqrt(sqrt(\x * \x + \y * \y) + \y) ;};}
    \tikzmath{function p(\x,\y) {return atan(\y / \x) ;};}
    %
%   \tikzmath{function x(\u,\v,\p) {return \u * \v * cos(\p r);};}
%   \tikzmath{function y(\u,\v,\p) {return \u * \v * sin(\p r);};}
%   \tikzmath{function z(\u,\v,\p) {return 0.5 * (\u * \u - \v * \v);};}
    %
    \pgfmathsetmacro{\px}{1.0}
    \pgfmathsetmacro{\py}{3.0}
    \pgfmathsetmacro{\pz}{2.0}
 
 
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