tikz-3dplot and drawing curves and surfaces in 3D

Question

I used the tiikz and tikz-3dplot packages a bit to draw figures in 3D and 3D. I however never yet plotted a mathematical function, and am not sure how to do it.

Could someone please give me a minumal example? The examples I find on the internet are all very complicated plots and drawings.

In particular I would like to plot the curve: z(x) = 1-x^2, on the interval x\in(-1, 1). For the moment I would only need this in 2D, but ultimately it should be a curve in a 3D plot, where I also want to plot the surface z(x,y) = 1-x^2-y.

Thanks for help!

Answer 1

It is a bit hard to answer this question, but I'll try. Can one draw plots with tikz-3dplot? Certainly, this is why it has this name. You can always draw parametric plots, simply because TikZ can, and then the main purpose of tikz-3dplot is to install an orthographic view. The functions you mention can be plotted e.g. with

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{60}{110}

\begin{tikzpicture}[pics/3d axes/.style={code={%
 \draw[-latex] (0,0,0) -- (#1,0,0) node[pos=1.1]{$x$}; 
 \draw[-latex] (0,0,0) -- (0,#1,0) node[pos=1.1]{$y$}; 
 \draw[-latex] (0,0,0) -- (0,0,#1) node[pos=1.1]{$y$}; 
}}]
\begin{scope}[tdplot_main_coords]
 \pic{3d axes=2.5};
 \draw plot[variable=\x,domain=-1:1,samples=73,smooth] 
 (\x,0,{1-\x*\x});
\end{scope} 
\begin{scope}[tdplot_main_coords,xshift=5cm]
 \pic{3d axes=2.5}; 
 \foreach \Y in {-1,-0.8,...,1}
 {\draw plot[variable=\x,domain=-1:1,samples=73,smooth] 
 (\x,\Y,{1-\x*\x-\Y});}
 \foreach \X in {-1,-0.8,...,1}
 {\draw plot[variable=\y,domain=-1:1,samples=73,smooth] 
 (\X,\y,{1-\X*\X-\y});}
\end{scope} 
\end{tikzpicture}
\end{document}

These tricks will enable you to reproduce the plots of the arXiv paper you link to rather closely. You also see in the surface plot that it does not hide hidden surfaces. As long as you only draw lines, you will be fine, otherwise consider switching to pgfplots or asymptote.

ADDENDUM: Some arrows. Notice that, as usual with decorations, there might be dimension too large errors, which occur if the curvature is too large. In this case, they would appear if you reinstalled smooth in the plots in which the arrows are added. If you insist on smooth, you need to draw the path twice, once the full thing and once with the arrow up to the position where the arrow head should be.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{arrows.meta,bending,decorations.markings}
\tikzset{% 
    arc arrow/.style args={%
    to pos #1 with length #2}{
    decoration={
        markings,
         mark=at position 0 with {\pgfextra{%
         \pgfmathsetmacro{\tmpArrowTime}{#2/(\pgfdecoratedpathlength)}
         \xdef\tmpArrowTime{\tmpArrowTime}}},
        mark=at position {#1-\tmpArrowTime} with {\coordinate(@1);},
        mark=at position {#1-2*\tmpArrowTime/3} with {\coordinate(@2);},
        mark=at position {#1-\tmpArrowTime/3} with {\coordinate(@3);},
        mark=at position {#1} with {\coordinate(@4);
        \draw[-{Stealth[length=#2,bend]}]       
        (@1) .. controls (@2) and (@3) .. (@4);},
        },
     postaction=decorate,
     }
}

\begin{document}
\tdplotsetmaincoords{60}{110}

\begin{tikzpicture}[pics/3d axes/.style={code={%
 \draw[-latex] (0,0,0) -- (#1,0,0) node[pos=1.1]{$x$}; 
 \draw[-latex] (0,0,0) -- (0,#1,0) node[pos=1.1]{$y$}; 
 \draw[-latex] (0,0,0) -- (0,0,#1) node[pos=1.1]{$y$}; 
}}]
\begin{scope}[tdplot_main_coords]
 \pic{3d axes=2.5};
 \draw plot[variable=\x,domain=-1:1,samples=73,smooth] 
 (\x,0,{1-\x*\x});
\end{scope} 
\begin{scope}[tdplot_main_coords,xshift=5cm]
 \pic{3d axes=2.5}; 
 \foreach \Y in {-1,-0.8,...,1}
 {\draw[arc arrow=to pos {0.5-0.25*\Y} with length 2mm]  
 plot[variable=\x,domain=-1:1,samples=31] 
 (\x,\Y,{1-\x*\x-\Y});}
 \foreach \X in {-1,-0.8,...,1}
 {\draw
 plot[variable=\y,domain=-1:1,samples=31] 
 (\X,\y,{1-\X*\X-\y});}
\end{scope} 
\end{tikzpicture}
\end{document}