@Rouben Rostamian  How would your Maple code be changed to enable display of the geodesic on the torus which passes through (or joins) two selected points on the torus's surface?

@Rouben Rostamian  Your reply has a wealth of information.

For me to understand it and use it in a variety of situations (other surfaces, other initial directions of motion of the insect) I will need to delve into aspects of Differential Geometry, a subject of which I am dimly aware. I would like to do this and hope I have the required talent. Thanks to your pedagocical patience I stand a chance of this!

I thank you for another great lesson in mathematics.

In the worksheet below I have modified your excellent reply to enable displaying the patch anywhere on the globe i.e. on the unit sphere.

Color_a_spherical_patch2.mw

@Rouben Rostamian  Thank you for your quick response.

I will study this and get back to you if I have any questions

@Preben Alsholm A nice addition to Rouben Rostamian's reply. Thank you.

@Rouben Rostamian  I'll carefully compare your working implementation  to mine and see where I went wrong.

Your replies continue to serve as a wonderful personal tutor. Thank you.

@Mac Dude Your advice gives me plenty to consider.

@tomleslie I am thankful to you for even an incomplete answer. It is disappointing that Maple staff have not replied to a few basic "how to" questions.

@Carl Love I've been away from my computer for a few days so haven't replied to you till now.

Your additional attention to my original question is much appreciated. It will be a couple more days before I can study it in depth and add to this reply.

I've spent some time studying your latest replay. It's difficult for me to understand so I'll return to it later.

@acer Thank you. Implicitplot now shows the correct image.

@Carl Love Thanks.Your solution is pleasantly simple.

However in my Maple 2020 the implicitplot command produces an empty plot.

Here is a modification of your solution which displays a poor version of your plot.

SolY := solve(eval(W, params), y):  # too long to display
NopsSolY:=nops([SolY]);
display(seq(plot(SolY[i], x = -1.2000 .. 1.2000), i = 1 .. NopsSolY), scaling = constrained);

@Kitonum Complete plots of the three Watt's curves are displayed in the revised, uploaded worksheet.

WattsCurve.mw

@Kitonum It seems there is no way to display the complete curves

@Rouben Rostamian

This worksheet, like all your previous replies to my Maple Primes questions, gives me valuable new insights which I greatly appreciate.  

@dharr Thank you for your enlightening answer. I will study it more thoroughly for additional insights.