Kitonum, this works pretty well.  It is missing the uppermost part of the graph where there is a smooth cap.

However, more important, of the two curves shown one is spurious.  I expected to see two because the equations for f and g  are degree four.  It looks like your method creates arrays full of data.  I want to weed out the spurious points.

I know another equation that the true data points must satisfy.  I paste it below.  How can I use this?

Thanks.

256 - 32 x^2 + x^4 - 32 y^2 + 2 x^2 y^2 + y^4 + 128 z +  8 x^2 z + 8 y^2 z + 48 z^2 + 2 x^2 z^2 + 2 y^2 z^2 + 

 8 z^3 + z^4 = 0, or maybe realistically, that the absolute value < 10^-12 or so.

Thank you Kitonum.  I will study this.

Thank you Kitonum.  From also graphing g (as you did f) I see that the Maple commands produce the part of the graph between t = 2.27 and t = 3.47.  It is reasonable and seems correct.

BUT THEN:  what do I need to do to get the entire graph plotted?  There is much more than this.  I tried a competing CAS and I can get much more plotted, indeed too much, as there is a spurious feature.  

Perhaps a Maple command that does not use ODEs?

@Markiyan Hirnyk  Thank you for the idea.  But then what??  You have defined f, g, and h as procedures, but what would the rest of the process be?  

@Preben Alsholm Do you mean like this:

ics:=solve(%,x=x0, y=y0, z=z0);    for the correct numeric values of x0, y0, z0 ?

@Kitonum Unfortunately the real problem of interest is far too complex to eliminate the t. 

@Preben Alsholm I wil try this thanks.  However, won't      eval(eqs2,t=0);

  find several initial points? (Maybe not on the trivial example I posted)  I already know the correct initial point.  I don't want a very messy graph with lots of spurious curves.

@one man I'm not sure what you mean by "solve this sytem for only one t".  The simple example has low degree polynomials that could be solved analytically.  My real problem is far too complex for that.

@acer Grrr.  I have no idea why it stoppoed working with Maple 2015.  For the next two days, I only have access to Maple 12.0.

The Maple 12 instructions given above work.  HOWEVER,

- the gif produced is rather small (apparently 512x512 pixels).  I found some documentation online about plotoptions to change width and height, but it doesn't work, i.e., plotsetup accepts the specifications for height and width, but the animation does not; the gif remains empty with no error message.

- Worse, the obeserver's perpective in the animation is not good.  How do I specify that in the animate command?

- Animation does one forward pass.  How do I make it at least do forward and back?

@rlewis Or rather, I can't save anymore to gif.  I went to the machine where I run Maple 2015 and saved wondeful animated gifs two days ago to tweak it and save again.  Nope.  It refuses to save.  I restarted Maple, did everything I can think of.   It creates a gif of size 0 and will not put anything into it.

I save by right-clicking on the anmation, choosing export -> gif.  

BTW, saving on the machine (Mac) with Maple 12.0 never worked, in spite of the advice posted abpve.

@acer I will try these commands on Maple 12.

@acer Expporting works fine on Maple 2015.  It failed on Maple 12.

Thanks to all who helped.  This discussion board seems to be much better than the one for Mathematica.

@rlewis On Maple12, I tried right-cliciking and a menu comes up, but it doesn't actually work.  Please explain how I can save the animation as an animated GIF.

@acer I didn't see the play controls at the top!   Thanks!

Wow, it works great, with my actual application!

I don't mind telling you, this is easier and better looking that anythng I could get from Sage or Mathematica.

@rlewis I don't see a movie.  I see some triangles in 3-space that I can grab and rotate, whcich is good, but I want to see a movie.