The procedure works well for the few combinations of p,q I tried except the fsolve command produces an error for p,q = 2,3 (Coxeters loxodromic sequence of tangent circles).

A change of plottools:-circle to plottools:-disk displays neither circles nor disks.

@dharr In the next while I will work on understanding the logic of your answer and ask you to clarify anything that puzzles me.

You have given me much to enjoy and probably some new insights to geometry. Thank you!

@dharr I thank you for your reply, however Maple apparently cannot convert even the first few lines of the Mathematica code in the website to which you provided the link.

I am completely unfamiliar with this conversion so it is likely that I am doing it incorrectly using the command MmaToMaple.

Website https://en.wikipedia.org/wiki/Coxeter%27s_loxodromic_sequence_of_tangent_circles contains some specifications for the Coxeter version of the Doyle spiral but nothing describing the logarithmic spirals on which the tangent circles have their centrers.

@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.