Parametric Surfaces

Graphing a surface defined by x=f(u,v), y=g(u,v), z=h(u,v). This page describes plotting of surfaces defined by the parametric equations of two variables, x=f(u,v), y=g(u,v), and z=h(u,v), for which the ranges of u and v are explicitly defined. The basic draw command for this function utilizes either the uncompiled function or compiled function format and uses the surface command to specify the three functions for the x, y, and z components of the surface. The general format for uncompiled functions is:

        draw(surface(f(u,v),g(u,v),h(u,v)), u=a..b, v=c..d)

where a..b and c..d are segments defining the intervals [a,b] and [c,d] over which the parameters u and v span. In this case the functions are not compiled until the draw command is executed. Here is an example of a surface plotted using the parabolic cylindrical coordinate system option:

In the case of compiled functions, the functions are named and compiled independentlyh. This is useful if you intend to use the functions often, or if the functions are long and complex. The following lines show functions whose parameters are of the type SmallFloat. The functions are compiled and stored by Axiom when entered.

Once the function is compiled the draw command only needs the names of the functions to execute. Here is a compiled functions example plotted using the toroidal coordinate system option:

Note that the parameter ranges do not take the variable names as in the case of uncompiled functions. The variables are entered in the order in which they are defined in the function specification. In this case the first range specifies the u-variable and the second range specifies the v-variable.