Surfaces in Space

and some level curves and traces

>	with(plots):

Warning, the name changecoords has been redefined

Here is the surface described by .

>	surface1:=plot3d(16*x^2+9*y^2,x=-3..3,y=-(4/3)*sqrt(9-x^2)..(4/3)*sqrt(9-x^2),axes=boxed,labels=[x,y,z],orientation=[45,45]):

>	display(surface1);

Next I will add the level curve corresponding to z = 81 in blue.  To draw the level curve I need to parameterize .

>	levelcurve1:=spacecurve([(9/4)*sin(t),3*cos(t),81],t=0..2*Pi,thickness=4,color=blue):

>	display(surface1,levelcurve1);

Here I add the trace in the yz-coordinate plane (x = 0) in green.  Thus I needed to parameterize .

>	trace1:=spacecurve([0,t,9*t^2],t=-4..4,thickness=4,color=green):

>	display(surface1,levelcurve1,trace1);

Next I will add the trace in the plane x = 2 also in green.  Thus I needed to parameterize .

>	trace2:=spacecurve([2,t,9*t^2+64],t=-3..3,thickness=4,color=green):

>	display(surface1,levelcurve1,trace1,trace2);

Finally, I will add the trace in the xz-coordinate plane (y = 0) in red.  I had to parameterize .

>	trace3:=spacecurve([t,0,16*t^2],t=-3..3,thickness=4,color=red):

>	display(surface1,levelcurve1,trace1,trace2,trace3);

Here is the surface described by .

>	surface2:=plot3d(16*x^2-9*y^2,x=-10..10,y=-(4/3)*sqrt(x^2-9)..(4/3)*sqrt(x^2-9),axes=boxed,labels=[x,y,z],orientation=[45,45]):

>	display(surface2);

Here is the same surface described using an implicitplot3d command.

>	surface2a:=implicitplot3d(z=16*x^2-9*y^2,x=-10..10,y=-10..10,z=-1600..1600,axes=boxed,labels=[x,y,z],orientation=[45,65]):

>	display(surface2a);

Here is the level curve corresponding to z = 900 in blue.  I had to parameterize .  You can't get a good picture if you try to do it all at once.

>	levelcurve2a:=spacecurve([(7.5)*sec(t),10*tan(t),900],t=0..Pi/4,thickness=4,color=blue):

>	levelcurve2b:=spacecurve([(7.5)*sec(t),10*tan(t),900],t=-Pi/4..0,thickness=4,color=blue):

>	levelcurve2c:=spacecurve([(7.5)*sec(t),10*tan(t),900],t=Pi..5*Pi/4,thickness=4,color=blue):

>	levelcurve2d:=spacecurve([(7.5)*sec(t),10*tan(t),900],t=3*Pi/4..Pi,thickness=4,color=blue):

>	display(surface2a,levelcurve2a,levelcurve2b,levelcurve2c,levelcurve2d);

This time I will add in the plane z = 900 in red.

>	plane2:=implicitplot3d(z=900,x=-10..10,y=-10..10,z=-1600..1600,axes=boxed,labels=[x,y,z],orientation=[45,65],color=red):

>	display(surface2a,levelcurve2a,levelcurve2b,levelcurve2c,levelcurve2d,plane2);

Here is the trace in the yz-coordinate plane (x = 0) in red.

>	traceyz:=spacecurve([0,t,-9*t^2],t=-10..10,thickness=4,color=red):

>	display(surface2a,levelcurve2a,levelcurve2b,levelcurve2c,levelcurve2d,traceyz);

The picture below has a different viewing orientation.

>	surface2aBelow:=implicitplot3d(z=16*x^2-9*y^2,x=-10..10,y=-10..10,z=-1600..1600,axes=boxed,labels=[x,y,z],orientation=[45,125]):

>	display(surface2aBelow,levelcurve2a,levelcurve2b,levelcurve2c,levelcurve2d,traceyz);

I'll put in the plane x = 0 in green.

>	plane3:=implicitplot3d(x=0,x=-10..10,y=-10..10,z=-1600..1600,axes=boxed,labels=[x,y,z],orientation=[45,65],color=green):

>	display(surface2aBelow,levelcurve2a,levelcurve2b,levelcurve2c,levelcurve2d,traceyz,plane3);

Here is an animation showing the surface along with 30 planes paralled to the xy-coordinate plane illustrating 30 level curves.

>	surface3:=implicitplot3d(z=16*x^2-9*y^2,x=-20..20,y=-20..20,z=-1600..1600,axes=boxed,labels=[x,y,z],orientation=[45,65],numpoints=2500):

>	planes:=animate3d(t,x=-20..20,y=-20..20,t=-1500..1500,color=red,frames=30):

>	display(surface3,planes);

The next surface is the "cylinder" described by .

>	implicitplot3d(y=2*x^2,x=-3..3,y=0..18,z=-5..5,axes=boxed,numpoints=900);

Here is the cylinder described by .

>	plot3d(3*sin(x),x=-2*Pi..2*Pi,y=-5..5,axes=boxed,labels=[x,y,z],numpoints=2500);

Here we have the surface formed by revolving the graph of  about the x-axis using an x-interval of [-1.5,3].  The equation of the surface is

>

.

>	implicitplot3d(y^2+z^2=(x^3-2*x^2-3*x+10)^2,x=-1.5..3,y=-20..20,z=-20..20,axes=boxed,numpoints=2500);

Here are a couple vriations on how you display your plot.  Other variations can be found in the plot options.

>	implicitplot3d(y^2+z^2=(x^3-2*x^2-3*x+10)^2,x=-1.5..3,y=-20..20,z=-20..20,axes=boxed,numpoints=2500,glossiness=1,light=[45,45,0.8,0.8,0.2],style=patchnogrid);

>