Easiest way to plot a function with PGF/TikZ

Question

I am new to TikZ and/or PGF (whatever the difference is :/), as you can see. Anyway, I need to draw the graph of the equation x^(2/3) + y^(2/3) = a^(2/3), where a = 2.
Any suggestions? Here is what I have so far.

\documentclass{article}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}
Some text goes here.
\begin{center}
\begin{tikzpicture}[>=latex']
    % Draw x-axis
    \draw[very thick,->] (-3,0) -- (3.5,0)
            node[right] {$x$}; 
    % Draw y-axis
    \draw[very thick, ->] (0,-3) -- (0,3.5) 
        node[above] {$y$};
    % Draw graph of equation
        \draw[smooth, color=blue, domain=-3:3, ultra thick, line cap=butt, samples=400]
            plot (\x,{ (2^{2/3} - (\x)^{2/3} )^{3/2} });
\end{tikzpicture}
\end{center}
\end{document}

Answer 1

While tikz can do basic graphs, it is more of drawing package than a graphing package. For graphs, I would recommend you use the pgfplots package, which internally uses tikz to do the actual drawing. To use pgfplots you invoke the axis environment.

The following graph in blue is the function you had in your MWE:

Based on the math explained in ガベージコレクタ's solution, I have parmaterically graphed the solution to the original equation:

Notes:

• Using pgfmathdeclarefunction isn't required, but makes it easier to read.

• This solution in the MWE has a discontinuity at 0, so if you use unbounded coords=discard, you will see messages of the sort:

  coordinate (2Y7.559544e-3],3Y0.0e0]) has been dropped because it is unbounded (in y)

  These go away if you use unbounded coords=jump.

• There have been a few cases where I needed to rewrite the expressions such as this as

  \pgfmathdeclarefunction{Function}{1}{%
      \pgfmathparse{exp((3/2)*ln( 2^(2/3) - exp((1/3)*ln(#1)) ))}}
  

  but it does not appear to be necessary in this case.

  Thought I would point it out in case you run into problems with other similar expressions.

• Also, it should be noted that pgf trigonometric functions expect the parameter to be in degrees, hence the use of the deg() function in the parametric graph.

References:

• There are a few cases where you need to use the math engine in gnuplot to generate the points, in which case it is useful to be able to Consistently specify function to be graphed with or without gnuplot.
• If you like the method of defining a separate function using \pgfmathdeclarefunction, and you want to be able to use this new function for computations, you should refer to Consistently specify a Function and use it for computation and plotting.

Code:

\documentclass{article}

\usepackage{pgfplots}% This uses tikz
\pgfplotsset{compat=newest}% use newest version

\newcommand*{\A}{2}
\pgfmathdeclarefunction{Function}{1}{% as per original MWE
  \pgfmathparse{(2^(\A) - (#1)^(1/3) )^(3/2)}%
}

\pgfmathdeclarefunction{SolutionX}{1}{%
    \pgfmathparse{\A*(cos(deg(\t)))^3}%
}

\pgfmathdeclarefunction{SolutionY}{1}{%
    \pgfmathparse{\A*(sin(deg(\t)))^3}%
}

\tikzset{My Line Style/.style={smooth, ultra thick, samples=400}}

\begin{document}
\begin{tikzpicture}
\begin{axis}[unbounded coords=discard]
    \addplot[My Line Style, color=blue, domain=0:3] (\x,{Function(\x)});
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[axis lines=middle, xmin=-2.5, xmax = 2.5, ymin=-2.5, ymax = 2.5]
    \addplot[My Line Style, color=red,  variable=\t, domain=-2*pi:0] 
        ({SolutionX(\t)},{SolutionY(\t)});
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\psset{algebraic,plotpoints=300}

\def\a{2}
\def\x(#1){\a*cos(#1)^3}
\def\y(#1){\a*sin(#1)^3}

\begin{document}
\begin{pspicture}[showgrid=true](-2,-2)(2,2)
    \psparametricplot[linecolor=red]{0}{\psPiTwo}{\x(t)|\y(t)}
\end{pspicture}
\end{document}

With PSTricks the solution is as follows,

Animated version:

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\psset{algebraic,plotpoints=300}
\usepackage[nomessages]{fp}
\FPeval\Delta{round(2*pi/30:2)}

\def\a{2}
\def\x(#1){\a*cos(#1)^3}
\def\y(#1){\a*sin(#1)^3}

\begin{document}
\multido{\n=0.00+\Delta}{31}{%
\begin{pspicture}[showgrid=true](-2,-2)(2,2)
    \psparametricplot[linecolor=red]{0}{\n}{\x(t)|\y(t)}
\end{pspicture}}
\end{document}

With implicit function plot:

\documentclass[pstricks,border=3pt]{standalone}
\usepackage{pst-func}
\begin{document}
\begin{pspicture*}(-2,-2)(2,2)
  \psplotImp[linecolor=red,stepFactor=0.05,algebraic](-2.1,-2.1)(2.1,2.1){(x^2)^(1/3)+(y^2)^(1/3)-4^(1/3)}
\end{pspicture*}
\end{document}

Answer 3

While I have been using TikZ a little bit over the last couple of years, I am brand new to pgf, per se, and tikzmath. I have a number of questions, but for now, out of simple curiosity, since Nathan was new to "tikz and/or pgf" why didn't you just tell him to use

\draw [ scale=0.5, domain=-3.142:3.142, smooth, variable=\t ]
plot ( {2*cos(\t r)^3}, {2*sin(\t r)^3} );  

or:

\begin{tikzpicture}[declare function={xx(\t) = 2*cos(\t)^3; yy(\t) = 2*sin(\t)^3;}]
    \draw [scale=0.5] plot [ domain=0:360, samples=73, smooth ] ( {xx(\x)}, {yy(\x)} );