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There are several ways to specify a curve in the plane. A different plotting function is used for each of these specifications. Graphs of functions are plotted using Plot. Curves given parametrically are plotted using ParametricPlot. ImplicitPlot plots curves that are given implicitly as the solutions to equations.

Plots of curves given implicitly.

There are two methods ImplicitPlot can use to plot the solution to the given equations. The method that is used is determined by the form of the variable ranges given. One method uses Solve to find solutions to the equation at each point in the range. It carefully avoids dangerous points, plotting to within machine precision of those points, to generate an apparently smooth graph. This is the method used if you just give the range for . The second method treats the equation as a function in three-dimensional space, and generates a contour of the equation cutting through the plane where equals zero. This method is faster than the Solve method and handles a greater variety of cases, but may generate rougher graphs, especially around singularities or intersections of the curve. This method is used if you specify a range for both and .

This loads the package.

In[1]:= << Graphics`ImplicitPlot`

This plots an ellipse using the Solve method.

In[2]:= ImplicitPlot[x^2 + 2 y^2 == 3, {x, -2, 2}]


Because range specifications for both and are given, the ContourPlot method is used.

In[3]:= ImplicitPlot[Sin[2 x] + Cos[3 y] == 1,
{x, -2 Pi, 2 Pi},{y, -2 Pi, 2 Pi},


Both methods can accept standard graphics options; the Solve method accepts the options of Plot, while the ContourPlot method accepts ContourPlot options.

Here multiple curves are displayed.

In[4]:= ImplicitPlot[{(x^2 + y^2)^2 == (x^2 - y^2),
(x^2 + y^2)^2 == 2 x y}, {x,-2,2},


You can find other interesting examples of using a contour plot to do implicit plotting in the book Exploring Mathematics with Mathematica, by Theodore Gray and Jerry Glynn (Addison-Wesley, 1991).