@Rouben Rostamian Rouben, your worksheet shows me the solution to my questions 1 and 2 and I overlooked a simple answer to question 3.
I regularly scan Maple Primes to find new Maple programming techniques and, occasionally, new math techniques and your worksheet has amazed me in both regards.
I had no knowledge of barycentric coordinates but now I see they have great power in solving otherwise tough math problems and they define a vector (V) which provides all the coordinates within the triangle as lambda1 and lambda2 cycle through their ranges.
Try as I might, there are several statements in your worksheet that baffle me.
I can't find a clear explanation or example in Maple 15 help pages of the syntax %T.
Please explain why the correct ranges for plotting the planar triangle are the ones you specified in your plot3d command.
Why does the statement V / sqrt(V^%T . V); not evaluate i.e. acts as an inert command?
Please also explain why plotting the unnormalized V gives a planar triangle while plotting the normalized V plots the same triangle fitted to the surface of the unit sphere.
Please explain why the ranges of function U within the spacecurves of the normalized V provide the edges of the spherical triangle.
Your worksheet is an education in itself and I greatly appreciate your attention.