Plotting in other coordinate systems Worksheet by Mike May, S.J.- maymk@slu.edu Revised by Russell Blyth - blythrd@slu.edu restart; We look at how we can plot with Maple in other coordinate systems. The trick is that Maple has a coords option in its plot commands.

We start with the standard Cartesian coordinate system. Maple assumes that we write y as a function of x. We can also plot curves parametrically using [x(t), y(t), t=trange]. plot((x-1)^2-2, x=-1..3); plot([t, (t-1)^2-2,t=-1..3]);

To use polar coordinates, note that Maple expects r to be a function of theta. If we are plotting parametrically, then Maple expects the form [r(t), theta(t), t=trange]. plot(1+2*cos(theta), theta=0..2*Pi, coords=polar); plot([1+2*cos(t), t, t=0..2*Pi], coords=polar);

We can put plots in different coordinate systems together with the display command. Note that we end the commands of the named plots with a colon rather than a semicolon (otherwise we get an ugly list of the postscript commands that make up the plot structures). To use display we first need to load it using the with(plots) command. with(plots): plota:= plot((x-1)^2-2, x=-1..3, color=red): plotb:= plot(1+2*cos(theta), theta=0..2*Pi, coords=polar, color=blue): display({plota, plotb});

1. Plot the graph of a 5 petal rose of radius 2 with a petal cut by the positive x-axis. (You should remember the formula for this from pre-calculus.) 2. Plot the cardioid defined by r=1-sin(theta) on the same axes as the graph of y=1+sin(Pi*(x+0.5)) to create a picture of a heart with a hat.

We start with surfaces that are the plots of functions in two variables with Cartesian coordinates. In a parallel fashion, Maple assumes that we write z as a function of x and y. If we parameterize a surface, it is assumed to be in the form ([x(u,v), y(u, v), z(u, v)], u=urange, v=vrange). plot3d(sin(x^2+y^2), x=-3..3, y=-3..3, style=patch, axes=boxed); plot3d([u, v, sin(u^2+v^2)], u=-3..3, v=-3..3, style=patch, axes=boxed);

In cylindrical coordinates Maple assumes that we express r as a function of theta and z. (By contrast, in class we typically express z as a function of r and theta.) This means that we have to use the parametric form if we want to graph the sombrero surface above. (The graph above has many r values corresponding to a single value of theta and z, so r is not a function of theta and z.) Interestingly, the parametric description in Maple arranges the variables in the more familiar (r, theta, z) pattern so the form is ([r(u,v), theta(u,v), z(u,v)], u=urange, v=vrange). plot3d([r, theta, sin(r^2)], r=0..4, theta=0..2*Pi, coords=cylindrical, style=patch, axes=boxed); The cylindrical form is useful for plotting surfaces obtained by rotating curves around the z axis. Cylinders are the easiest example of this. They are obtained by rotating vertical lines about the z-axis. We show such a cylinder together with another surface of rotation. plot3d({1, 3+sin(z)}, theta=0..2*Pi, z=-4..4, style=patch, axes=boxed, coords=cylindrical);

Maple assumes that surfaces described in spherical coordinates express the radial distance rho as a function of theta and phi. The easiest surface to graph is, as the name of the system suggests, a sphere centered at the origin. plot3d(2, theta=0..2*Pi, phi=0..Pi, coords=spherical, style=patch, axes=boxed); If we are graphing a more complicated surface we may want to use the parametric form. In that case Maple assumes that the form is ([rho(u,v), theta(u,v), phi(u,v)], u=urange, v=vrange). plot3d(4+5*cos(2*phi)+3*sin(3*theta), theta=0..2*Pi, phi=0..Pi, coords=spherical, style=patch, axes=boxed); plot3d([4+5*cos(2*phi)+3*sin(3*theta), theta, phi], theta=0..2*Pi, phi=0..Pi, coords=spherical, style=patch, axes=boxed);

Once again, a nice trick is to use display3d to put together surfaces that are easy to describe in different coordinate systems. The code below puts together the plane z=x+y (Cartesian coordinates), the cylinder r=2 (cylindrical coordinates), and the sphere rho = 3 (spherical coordinates). plotplane := plot3d(x+y, x=-4..4, y=-4..4, style=patch, color=red, axes=boxed): plotcyl := plot3d(2, theta=0..2*Pi, z=-8..8, style=patch, color=blue, axes=boxed, coords=cylindrical): plotsph := plot3d(3, theta=0..2*Pi, phi=0..Pi, style=patch, color=green, axes=boxed, coords=spherical): display3d({plotplane, plotcyl, plotsph});

3. Plot a sphere of radius 3 centered at the origin in each of the three coordinate systems we have discussed. Which is easiest? 4. Plot a cone with point at the origin, height 4 and radius 3 using the coordinate system of your choice.