@Carl Love My searches will benefit from your advice.

@Rouben Rostamian  You have given me two gifts! First, I will carefully study your answer and try to absorb this lovely technique.

Secondly, you have opened up a new field of exploration for me as I attempt to apply this to a variety of other interesting surfaces.

You likely already are aware of the almost endless variety of these on https://mathcurve.com/surfaces.gb/surfaces.shtml 

@dharr Thanks for the complement. Maple's VectorCalculus has a TangentPlane command and I was aware of the verbal description of a pedal surface, but my math skills are not up to using this information to form a general definition. Hopefully I'll know more after examining Rouben Rostamian's answer above.

@Carl Love Thank you for this information. I posed this question of the forum after searching the internet as you did (without the quotes) and finding nothing beyond Wikipedia's "Pedal curve"

@Kitonum My wife found the oloid hard to visualize. Your animation will help her.

I'm intrigued. What other 3D shapes would you describe as "outlandish"?

@Rouben Rostamian  I greatly appreciate the educational aspect of this and many of your previous replies. In future, I will look for alternate and simpler parametrizations of surfaces. Thank you.

@acer I tried the option grid = [100,100] but it didn't affect the plot's smoothness. It seems I used the wrong grid values.

@vv I thank you for your  excellent reply.

Is there any way in which the "distorted" grid indicates the complex expression in F(x,y) or can this only be found by trial and error?

@vv Thank you for your reference. I will pursue it and its sub-references and hope to expand my knowledge.

@dharr I hope in future to have additional questions sufficiently intriguing to draw your interest.

@vv I have tried to "parse" your reply but I do not have the background knowledge of transformations to understand how f(x,y) maps the unit square to a quadrilateral.

Can you refer me to a source which explains transformations such as this at an undergrad level, which is where my math education stopped many years ago?

@dharr Your reply produces the full pencil of conics within and bordering the quadrilateral. I am deeply grateful for your insights into this problem's logic and Maple coding techniques.

@vv Your reply reveals a new approach of mapping a simple shape to a more complex one. Thank you for this food for thought.

@dharr Unfortunately your tips do not stop solve from running endlessly. I hope you will be able to return to this question with more time available.

In the meantime I am working on implementing in Maple the GeoGebra picture of an ellipse in a quadrilateral at https://www.geogebra.org/m/d4k6SjFX

@dharr Your reply is wonderful and gives me a base for attempting to display the other pencils in the book