@vv I will examine the answer in the link you have provided. Hopefully it will give me some insight into the Doyle spiral's math.

Here is a geodesic on a lumpy sphere.

Geodesic_on_a_lumpy_sphere.mw

I couldn't figure out how to display the geodesic path using standard plot commands.

@Rouben Rostamian  Thank you taking the time to explain Wikipedia's "in some sense". 

@Rouben Rostamian  Thanks for sending me this lovely picture, but shouldn't a geodesic be the shortest distance between the two points shown? If that is true then can your earlier statement about an infinity of geodesics between two points be true?

I probably won't be able to try to find geodesics on other surfaces until past this weekend.

@Rouben Rostamian  This is wonderfully simpler than your previous answer and more understandable at my level of math capability.

It is fascinating that a different geodesic is produced merely by stating the same end point in a different way!

I would try to choose this reply as a second best answer but I'm afraid that might screw up this website.

I will try this method for different surfaces, and if you do so please send me the results if they are interesting.

@Rouben Rostamian  Thanks for doing this lookup. I'll see what I can find (and understand).

Is this paper of any value?

https://www.pearson.com/content/dam/one-dot-com/one-dot-com/us/en/files/Jay%20Villanueva%20-%20geodesics9.pdf

@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.