 not so Frequently Asked Questions  update 2004/9/15


not so FAQ 
About Parametric Functionsuse of parametersIn the usual 2dimensional plot of gnuplot, the Y coordinate is expressed by y=f(x), however you can also use a parametric expression which uses the parameter t, x = f(t) y = g(t) With this expression, more complicated functions can be plotted with gnuplot. Note that the 3dim. plot with two parameters u,v is given in the spherical harmonics section. First of all, you need to use the command set parametric to tell gnuplot that the function is defined by a parameter. Then, the plot command followed by a function f(t) which is the Xcoordinate and a function g(t) for Ycoordinate, is give like, plot f(t),g(t) . to draw a vertical lineThe most simple but it is impossible to express by the y=f(x) form is a vertical line which is x=const. This function can be written as: x=const y=t with the parameter t, when t is varied. The range of t is controlled by the command set trange . gnuplot> set parametric dummy variable is t for curves, u/v for surfaces gnuplot> const=3 gnuplot> set trange [1:4] gnuplot> set xrange [0:5] gnuplot> set yrange [0:5] gnuplot> plot const,t In this case the vertical line is draw at x=3. Since we used set trange [1:4] , the range of this truncated line is from 1 to 4. If trange not set, the vertical line is drawn from the bottom to top border lines. to draw a circle, polygonsThe parametric expression of a circle is x=sin(t) y=cos(t) and the circle can be drawn if one changes the t parameter from 0 to 2pi. The graph is "squared" here, and the t range is given by an option of plot command. gnuplot> set parametric dummy variable is t for curves, u/v for surfaces gnuplot> set size square gnuplot> set xrange [1:1] gnuplot> set yrange [1:1] gnuplot> plot [0:2*pi] sin(t),cos(t) The parameter t is not changing continuously, and actually this is controlled by the value which is set by the set samples command. The default value is 100. In the case of set samples 8 , gnuplot generates eight t values from zero to 2*pi, and the graph becomes a regular heptagon. If you need a regular Ngon, just type set samples N+1. The 2dim. parametric representation is convenient to draw a function which is in a polar coordinate. The 2dim. polar coordinate has two variables which are radius r and angle theta. The gnuplot parameter t is for the theta, and the radius r is expressed by a function of angle, namely r(t). A (x,y) coordinate is given by x=r(t)*cos(t) y=r(t)*sin(t) The circle is a special case of which r(t)=const. When the radius of circle is proportional to t, you get a spiral. gnuplot> set xrange [10*pi:10*pi] gnuplot> set yrange [10*pi:10*pi] gnuplot> plot [0:10*pi] t*sin(t),t*cos(t) The following example shows r(t)=const*(1+cos(t)), which is called Cardioid. gnuplot> set parametric dummy variable is t for curves, u/v for surfaces gnuplot> r(t) = 1+cos(t) gnuplot> plot [0:2*pi] r(t)*cos(t),r(t)*sin(t) exchange X and YaxesFunctions are normaly expressed by y=f(x), but the parametric expression allows us to make a graph of x=f(y). The y values are the same as t, and the x values are calculated with a function of f(t). gnuplot> set parametric dummy variable is t for curves, u/v for surfaces gnuplot> c=2*pi gnuplot> set size square gnuplot> set trange [c:c] gnuplot> set xrange [c:c] gnuplot> set yrange [c:c] gnuplot> plot c*sin(t),t with lines, t,c*cos(t) with impulses Two functions are shown, one (green stripe) is y=2pi*cos(x), and the other (red solid line) is x=f(t)=2pi*sin(y). The option with impulse draws a vertical line from the Y=0 axis. If you use with impulses for the red curve which is x=2pi*sin(y), you still get a vertical stripe, not a horizontal one. 