# Second Maple lab for Holmes Math 275
# 
# Our first example is to graph an ellipsoid
> equ := x^2+2*y^2+3*z^2 = 10;

                             2      2      2
                     equ := x  + 2 y  + 3 z  = 10

> A:=solve(equ,z);  #this solution is a list

                  2      2      1/2         2      2      1/2
             (-3 x  - 6 y  + 30)       (-3 x  - 6 y  + 30)
        A := ----------------------, - ----------------------
                       3                         3

> B:=A[1]; C:= A[2];

                               2      2      1/2
                          (-3 x  - 6 y  + 30)
                     B := ----------------------
                                    3


                                2      2      1/2
                           (-3 x  - 6 y  + 30)
                    C := - ----------------------
                                     3

> plot3d(B,x=-5..5,y=-5..5,axes=box,scaling=constrained);

> plot3d(C,x=-5..5,y=-5..5,axes=box,scaling=constrained);

> plot3d({B,C},x=-5..5,y=-5..5,axes=box,scaling=constrained);

# I define a function which does this without all the typing
> equationplot:=(equ,a,b,c,d)->plot3d({solve(equ,z)},x=a..b,y=c..d,axes=
> box,scaling=constrained);

  equationplot := (equ, a, b, c, d) -> plot3d({solve(equ, z)},

        x = a .. b, y = c .. d, axes = box, scaling = constrained)

> equationplot(equ,-5,5,-5,5);

# Unfortunately, this doesn't work well here because the part of the
# surface between the two "branches" is too steep,
# and not enough points get plotted.  You can use the "numpoints"
# parameter in the plot3d command to improve this somewhat:
> plot3d({B,C},x=-5..5,y=-5..5,axes=box,scaling=constrained,numpoints=50
> 00);

# There are some other approaches.
> with(plots);
Warning, the name changecoords has been redefined


  [animate, animate3d, animatecurve, arrow, changecoords, complexplot,

        complexplot3d, conformal, conformal3d, contourplot,

        contourplot3d, coordplot, coordplot3d, cylinderplot,

        densityplot, display, display3d, fieldplot, fieldplot3d,

        gradplot, gradplot3d, graphplot3d, implicitplot,

        implicitplot3d, inequal, interactive, listcontplot,

        listcontplot3d, listdensityplot, listplot, listplot3d,

        loglogplot, logplot, matrixplot, odeplot, pareto, plotcompare,

        pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d,

        polyhedra_supported, polyhedraplot, replot, rootlocus,

        semilogplot, setoptions, setoptions3d, spacecurve,

        sparsematrixplot, sphereplot, surfdata, textplot, textplot3d,

        tubeplot]

> implicitplot3d(equ,x=-5..5,y=-5..5,z=-5..5,axes=box,scaling=constraine
> d);

# As you can see, the implicitplot command doesn't give very many points
# on the graph.
> implicitplot3d(equ,x=-5..5,y=-5..5,z=-5..5,axes=box,scaling=constraine
> d,numpoints=10000);

# This is better!  Another elegant solution is to use spherical or
# cylindrical coordinates.  Spherical is best for this example.
> equ;

                         2      2      2
                        x  + 2 y  + 3 z  = 10

> x:=r*cos(theta)*sin(phi); y:=r*sin(theta)*sin(phi); z:=r*cos(phi);

                      x := r cos(theta) sin(phi)


                      y := r sin(theta) sin(phi)


                           z := r cos(phi)

> equ;

   2           2         2      2           2         2
  r  cos(theta)  sin(phi)  + 2 r  sin(theta)  sin(phi)

              2         2
         + 3 r  cos(phi)  = 10

> F:=solve(equ,r);

                              10
  F := -------------------------------------------------,
                          2         2              2 1/2
       (30 + 10 sin(theta)  sin(phi)  - 20 sin(phi) )

                                 10
        - -------------------------------------------------
                             2         2              2 1/2
          (30 + 10 sin(theta)  sin(phi)  - 20 sin(phi) )

> G:=F[1];

                                    10
        G := -------------------------------------------------
                                2         2              2 1/2
             (30 + 10 sin(theta)  sin(phi)  - 20 sin(phi) )

> sphereplot(G,theta=0..2*Pi,phi=0..Pi,axes=box,scaling=constrained);

# To me, this seems by far the best picture, and it is much less
# computationally expensive than the implicit plot with 10000 points.
> x:=evaln(x); y:=evaln(y); z:=evaln(z); equ;  #this shows how to get
> rid of the dependence of x,y,z on r,theta,phi.

                                x := x


                                y := y


                                z := z


                         2      2      2
                        x  + 2 y  + 3 z  = 10

# Your lab assignment:
# 
# Exercise 1:  plot one figure of each of the kinds described in section
# 12.6.  For each figure, try plotting it with my equationplot
# command and with either cylindrical or spherical coordinates
# (cylindrical coordinates are appropriate for many of the figures,
# and there is a "cylinderplot" command which works just like
# sphereplot, except that it uses cylindrical coordinates (which can
# be set up and removed in the same way that I set  up and removed
# spherical coordinates above.
# 
# To help visualize things in section 13.1, read the brief introduction
# to parametric plotting below.  There is a second exercise at the end.
> plot3d([t,t^2,t^3],s=-2..2,t=-2..2,axes=box,scaling=constrained); 
> #Here is the twisted cubic in all its glory:  try moving it around.

> plot3d([cos(t),2*sin(t),t],s=-2..2,t=-10..10,axes=box,scaling=constrai
> ned);  #Here's an elliptical helix

> 
# The command above has two parameters because it is actually for
# plotting parametric _surfaces_:
> plot3d([2*s-3*t,t-s,5*s+7*t],s=-2..2,t=-2..2,axes=box,scaling=constrai
> ned);  #notice that this gives a plane.

# Exercise 2:  Try to write down a parametric plot which will give the
# same plane as the equation x+2y-2z=4.  Hint:  find
# two vectors that lie in the plane and a point in the plane.
> equationplot(x+2*y-2*z=4,-5,5,-5,5);

>