A conformal plot of a complex function F(z) from a+bi to c+di maps a two-dimensional grid $a\le x\le c$, $b\le y\le d$  from the plane into a second (curved) grid determined by the images of the original gridlines under F.  The result is a set of curves in the plane, which has the property that they also intersect at right angles at the points where F is analytic.

The conformal command produces a conformal plot of a complex function F, where F can be an expression or a procedure.  The first range, r1, defines the gridlines in the plane that are to be conformally mapped via the complex function F. The second range, r2, is optional and defines the view of the plot.  The default view includes the full range of the conformal lines.

The conformal3d command works in the same way as the conformal command, except that it plots F on the Riemann sphere, and it accepts only the first range parameter.

Remaining arguments are interpreted as options which are specified as equations of the form option = value.

To change the number of gridlines displayed, use the grid=[m, n] option, with m and n integers. This option specifies the number of gridlines in both x and y directions that are to be mapped conformally.  The default is 11 lines in either direction, making an 11 by 11 grid.

To change the number of points sampled, use the numxy=[m, n] option, with m and n integers.  This option specifies the number of points that are to be plotted in each gridline, with m points in the x direction and n points in the y direction. The default is 21 points in each direction.

To specify the color of the gridlines, use the color=c option, where c is a valid color as described on the plot/color help page.  The value c can also be a list of two colors; in this case, each color is used for gridlines in a single direction. With the conformal3d command, the spherecolor=c option can also be used to specify the color of the sphere.

To map a grid defined in a different coordinate system, use the coords=t option, where t is one of the coordinate systems listed in the plot/coords and plot3d/coords help pages.  With this option, the gridlines in the default Cartesian coordinate system are first mapped to the new coordinate system and then the conformal mapping is applied.

There are a number of other standard 2-D and 3-D plot options that are available with the conformal and conformal3d commands.  These include specifications for style, and the number of horizontal and vertical tickmarks.  For more details, see plot/options and plot3d/options.

$\mathrm{with}\left(\mathrm{plots}\right)\:$

$\mathrm{conformal}\left(z^2\,z=0..2+2\mathrm{I}\right)$

$\mathrm{conformal}\left(\frac{1}{z}\,z=-1-\mathrm{I}..1+\mathrm{I}\,-6-6\mathrm{I}..6+6\mathrm{I}\,\mathrm{color}=\mathrm{magenta}\,\mathrm{numxy}=\left[80\,80\right]\right)$

$\mathrm{conformal}\left(\cos \left(z\right)\,z=0..2\mathrm{\pi }+\mathrm{\pi }\mathrm{I}\,\mathrm{grid}=\left[8\,8\right]\,\mathrm{numxy}=\left[50\,50\right]\right)$

$\mathrm{conformal}\left(z^3\,z=0..2+2\mathrm{I}\,\mathrm{tickmarks}=\left[3\,6\right]\right)$

$\mathrm{conformal}\left(\frac{2z-1}{2-z}\,z=-2-2\mathrm{I}..2+2\mathrm{I}\,-2-2\mathrm{I}..2+2\mathrm{I}\right)$

$\mathrm{conformal}\left(z+\frac{1}{z}\,z=-3-3\mathrm{I}..3+3\mathrm{I}\,-3-3\mathrm{I}..3+3\mathrm{I}\right)$

$\mathrm{conformal}\left(\frac{z-\mathrm{I}}{z+\mathrm{I}}\,z=-3-3\mathrm{I}..3+3\mathrm{I}\,-4-4\mathrm{I}..4+4\mathrm{I}\,\mathrm{grid}=\left[30\,30\right]\,\mathrm{style}=\mathrm{line}\right)$

$\mathrm{conformal}\left(\mathrm{sqrt}\left(z\right)\,z=-\frac{\mathrm{\pi }\mathrm{I}}{2}..1+\frac{\mathrm{\pi }\mathrm{I}}{2}\,\mathrm{coords}=\mathrm{polar}\right)$

$\mathrm{conformal3d}\left(\cos \left(z\right)\,z=0..2\mathrm{\pi }+\mathrm{I}\mathrm{\pi }\right)$

$\mathrm{conformal}\left(z^3\,z=-1-\mathrm{I}..1+\mathrm{I}\right)$

$\mathrm{conformal3d}\left(\cos \left(z\right)\,z=0..2\mathrm{\pi }+\mathrm{I}\mathrm{\pi }\,\mathrm{color}=\left[''DeepPink''\,''Yellow''\right]\,\mathrm{spherecolor}=''black''\right)$