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Question: Shortest curve on a surface connecting 2 points

Question:Shortest curve on a surface connecting 2 points

one man 1330 Maple
shortest-distance
July 23 2024
4

I liked the recent question from user goebeld and especially the answer from Rouben Rostamian.
I admit, I didn’t even realize that Maple had VariationalCalculus procedures.
But what if the red and green  points are on the surface x1^4 + x2^4 + x3^4 -1 = 0 ? 
Points coordinates (-0.759835685700000, -0.759835685700000, 0.759835685700000) and
 (0.759835685700000, 0.759835685700000, -0.759835685700000).

Where will the shortest distance between these points on a given surface be? Taking into account symmetry, of course.
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