O17-088
> restart;
> 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]

> T:=[(3+2*cos(u))*cos(t),(3+2*cos(u))*sin(t),2*sin(u)];

    T := [(3 + 2 cos(u)) cos(t), (3 + 2 cos(u)) sin(t), 2 sin(u)]

> plot3d(T, t=-Pi..Pi, u=-Pi..Pi);
> 

> A:=8*Pi*int(3+2*cos(u),u=0..Pi);

                                       2
                             A := 24 Pi

> plot3d(sqrt(6*sqrt(x^2+y^2)-x^2-y^2-5),x=-5..5,y=-5..5);

> surf:=36*(x^2+y^2)-(x^2+y^2+z^2+5)^2;

                         2       2     2    2    2     2
             surf := 36 x  + 36 y  - (x  + y  + z  + 5)

> plantang:=(b,c)->(3-b)*y-c*z+3*b-5;

           plantang := (b, c) -> (3 - b) y - c z + 3 b - 5

> c:=2*sqrt(5)/3;

                                     1/2
                                  2 5
                             c := ------
                                    3

> solve({surf=0,x=0,z=c});

                 1/2                           1/2
              2 5                           2 5
  {x = 0, z = ------, y = 5/3}, {x = 0, z = ------, y = -5/3},
                3                             3

                       1/2                            1/2
                    2 5                            2 5
        {x = 0, z = ------, y = 13/3}, {x = 0, z = ------, y = -13/3}
                      3                              3

> b:=5/3;

                               b := 5/3

> theplan:=plantang(b,c);

                                          1/2
                                 4 y   2 5    z
                      theplan := --- - --------
                                  3       3

> villarceau:=simplify(subs(y=sqrt(5)*z/2,surf));

                       2         2    4        2  2   81  4
     villarceau := 26 x  + 45/2 z  - x  - 9/2 x  z  - -- z  - 25
                                                      16

> factor(villarceau);

              2                  2      2                  2
          (4 x  - 16 x - 20 + 9 z ) (4 x  + 16 x - 20 + 9 z )
        - ---------------------------------------------------
                                  16

>