Chebyshev net in the Sphere and Catenary

Question

Hello I have experience writing texts in latex, but I am a beginner in drawing in latex, I have tried to make these graphics with great effort but I have not succeeded.

\documentclass[crop,tikz]{standalone}
\usepackage{pgfplots}
\usepackage{pgfplotstable}
\usepackage{tikz-3dplot}
\usepackage{amsmath}
\usepackage{amssymb}

\usepgfplotslibrary{fillbetween}

\def \plotwidth {510.0pt}

\definecolor{color0}{RGB}{0,153,216}

\pgfplotsset{compat=1.12}
\pgfplotsset{ticks=none}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    view/h=45,
    axis equal,
    axis lines=center,
]
    \addplot3[surf, opacity=0.5, samples=30, domain=-1:1, y domain=0:2*pi, z buffer=sort] ({sqrt(1-x^2) * cos(deg(y))}, {sqrt( 1-x^2 ) * sin(deg(y))}, x);
\end{axis}
\end{tikzpicture}
\end{document}

output:

I'm sorry for all the inconvenience, I'm just a beginner. I try to make those graphics in RGB={0,153,216}. Thank you very much for all kinds of help, on the other hand if someone knows a book in Portuguese or Spanish to learn how to graph in latex I would appreciate it very much.

Answer 1

This looks like the sphere you gave. But I am pretty sure that this is not a net with equal side lengths.

Perhaps you can point us to a reference with a proper parametrization. The ones I found using google are all in French.

\documentclass[border=9,tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}
    \makeatletter
    \def\mdef{\pgfmathsetmacro}
    \def\r{10} % radius
    \begin{axis}[height=20cm, view/h=45, axis equal, axis lines=center]
        \addplot3[
            samples=20, domain=-60:60, unbounded coords=jump,
            surf, opacity=.7, z buffer=sort,
            pre filter/.code={
                \edef\u{\pgfplots@current@point@x@unfiltered} % parametric plot's #1
                \edef\v{\pgfplots@current@point@y@unfiltered} % parametric plot's #2
                \mdef\theta{\u + \v} % azimuth
                \mdef\phi{\u - \v} % elevation
                \mdef\TooCloseToPole{ \phi > 85 ? 1 : (\phi < -85) ? 1 : 0 }
                \if1\TooCloseToPole
                    \mdef\pgfplots@current@point@z{nan}
                \else
                    \mdef\cosphi{cos(\phi)}
                    \mdef\pgfplots@current@point@x{\r * \cosphi * cos(\theta)}
                    \mdef\pgfplots@current@point@y{\r * \cosphi * sin(\theta)}
                    \mdef\pgfplots@current@point@z{\r * sin(\phi)}
                \fi
            }
        ]
        (\x, \y, 0);
\end{axis}
\end{tikzpicture}
\end{document}