11.5 Cylinders and cylindrical coordinates

Transcription

1 11.5 Cylinders and cylindrical coordinates Definition [Cylinder and generating curve] A cylinder is a surface composed of the lines that pass through a given plane curve parallel to a given line in space. The curve is a generating curve for the cylinder. Definition [Cylindrical coordinates] Cylindrical coordinates represent a point in P space by ordered triples ( r, θ, z) in which r and θ are polar coordinates for the vertical projection of P on the xy-plane, and z is the rectangular vertical coordinate. Equations relating rectangular and cylindrical coordinates x=r cos( θ ), y=r sin( θ ), z=z r 2 = x 2 + y 2 tan( θ) = y x Plotting in other coordinate systems Worksheet by Mike May, S.J.- maymk@slu.edu > restart: It is worthwhile to 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. Functions of one variable Cartesian Coordinates We start with the standard cartesian coordinate system. Maple assumes that we will write y as a function of x. We can also plot 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]);

2 Polar Coordinates If we want to use polar coordinates we should note that Maple expects r to be a function of theta. If we are plotting parametrically Maple expects [r(t), theta(t), t=trange].

3 > plot(1+2*cos(theta), theta=0..2*pi, coords=polar); plot([1+2*cos(t), t, t=0..2*pi], coords=polar); Mixed Coordinates One of the useful things we can do is to put plots done with different coordinate

4 systems together with the display command. Note that we want to 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 with the 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}); Warning, the name changecoords has been redefined [ animate, animate3d, animatecurve, changecoords, complexplot, complexplot3d, conformal, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, display3d, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d, inequal, listcontplot, listcontplot3d, listdensityplot, listplot, listplot3d, loglogplot, logplot, matrixplot, odeplot, pareto, pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d, polyhedra_supported, polyhedraplot, replot, rootlocus, semilogplot, setoptions, setoptions3d, spacecurve, sparsematrixplot, sphereplot, surfdata, textplot, textplot3d, tubeplot] Exercises: 1) Plot the graph of a 5 petal rose of radius 2 with a petal cut by the positive x-axis.

5 (You should remember the formula for this from pre-calculus. > 2) Plot the cartioid defined by r=1-sin(theta) on the same axes as the graph of y=2+sin(pi*x) to create a picture of a heart with a hat. > Functions of two variables Cartesian Coordinates We start surfaces that are the plots of functions in two variables with cartesian coordiantes. In a parallel fashion, Maple assumes that we will write z as a function of x and y. If we are going to parameterize the 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, grid=[60,60]); plot3d([u, v, sin(u^2+v^2)], u=-3..3, v=-3..3, style=patch, axes=boxed, grid=[60,60]);

6 Cylindrical Coordinates The next coordinate system we want to look at is the cylindrical coordinates. Maple assumes that we will express r as a function of theta and z. (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 do 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 form 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, grid=[100,100]);

7 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 lines of the form y=c about the x-axis. > plot3d({1, 3+sin(z)}, theta=0..2*pi, z=-4..4, style=patch, axes=boxed, coords=cylindrical, grid=[60,60]);

8 > Spherical coordiantes Maple assumes that spherical coordinates will express 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, grid=[60,60],scaling=constrained);

9 If we are doing a more complicated surface we may want to use the parametric form. In that Case Maple asumes that the form will be ([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, grid=[60,60]); plot3d([4+5*cos(2*phi)+3*sin(3*theta), theta, phi], theta=0..2*pi, phi=0..pi, coords=spherical, style=patchnogrid, axes=boxed, grid=[40,40], lightmodel=light1);

10 Mixed coordinates 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 putes togeter 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=yellow, axes=boxed, grid=[60,60]): plotcyl := plot3d(2, theta=0..2*pi, z=-8..8, style=patch, color=pink, axes=boxed,

11 coords=cylindrical, grid=[60,60]): plotsph := plot3d(3, theta=0..2*pi, phi=0..pi, style=patch, color=green, axes=boxed, coords=spherical, grid=[60,60]): display3d({plotplane, plotcyl, plotsph}); Exercises: 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 your favorite coordinate system. >