Implicit Function

• Solve implicit func.

To solve implicit functions

A functon which is not defined in a form of y=f(x) but written as
f(x)+g(y)=0 cannot be defined in gnuplot. To draw such an implcit
function, we have to use a parametric representation, or the function is
once defined as a 3-D surface, and find an intersection curve with z=0
plane.

If the function can be defined by using parameters, it is easy to
make a figure. For example, an equation for circle, x*x+y*y=r*r, is
re-defined by x=r*sin(t) and y=r*cos(t). In gnuplot:

gnuplot> set parametric

dummy variable is t for curves, u/v for surfaces
gnuplot> set size square
gnuplot> r=1
gnuplot> plot r*sin(t),r*cos(t)

When the function is so complicated, and use of parameter is
difficult, we define the function as a 3-D surface. The next example is
for x^3-2xy + y^3=0 .

gnuplot> f(x,y)= x**3 - 2*x*y + y**3

An intersection of f(x,y) and z=0 is the solution. To set the z=0 plane
at the bottom of 3-D plot, we use set ticslevel 0 .

gnuplot> set xrange [-2:2]
gnuplot> set yrange [-2:2]
gnuplot> set zrange [0:5]
gnuplot> set ticslevel 0
gnuplot> set view 40,340
gnuplot> set isosample 100,100
gnuplot> splot f(x,y)

The curve shown at the bottom is the solution of f(x,y)=0. The contour lines
give us the values of this curve, which are written on a file.

gnuplot> set xrange [-2:2]
gnuplot> set yrange [-2:2]
gnuplot> set zrange [*:*]
gnuplot> set cont base
gnuplot> set cntrparam levels discrete 0,0
gnuplot> unset surface
gnuplot> set surface
gnuplot> set term table
gnuplot> set output "cont.dat"
gnuplot> splot f(x,y)

Then, the data on the file are plotted, in a usual manner.

gnuplot> set xrange [-2:2]
gnuplot> set yrange [-2:2]
gnuplot> set size square
gnuplot> plot "cont.dat" w l

The following method does not use a temporary file. The contour lines
are shown at -0.4 with an increment of 0.1, and each line corresponds
to a solution of x^3-2xy + y^3=const.

gnuplot> set xrange [-2:2]
gnuplot> set yrange [-2:2]
gnuplot> set view 0,0
gnuplot> set isosample 100,100
gnuplot> set size square
gnuplot> set cont base
gnuplot> set cntrparam levels incre -0.4,0.1,0.4
gnuplot> unset surface
gnuplot> splot f(x,y)