I am truly bemused now. How do you make the shortest path possible, via an indirect arc? The only way to direct the finger straight into the keybed is either to allow the whole arm to fall through each depression- or to extend the finger slightly. Otherwise, the geometry is literally impossible. There is no direct path via an arc- only an indirect one.
Who says the "shortest possible path" is the most efficient path in human terms?
I think you've confused yourself by trying to force the human hand to fit the "letter of physical laws".
Since the strongest most natural movement the finger makes is to be flexed in its natural arc, THIS is its most efficient movement and path, eventhough by physics standards it may not be.
P2U posted a short film of Rachmaninoff moving his index finger in this manner, in essence, illustrating this simple principal. (again NO CURLING!)
So, if you hold your fingers straight in the air from a straight upward pointing forearm and upward pointing hand, and then flex the finger starting at 12 O'Clock down to 9 O'Clock, this is about the full range of motion of the finger without curling.
Now to play a piano key, with this same configuration held horizontally resting atop the piano keyboard, the pull or flex would only require a distance -- using the clock again but now "laying on its side" -- for the finger to move from 12 O'Clock to about 11 O'Clock.
This is a very short distance (if even that) for the finger to go in an almost invisible arc since it takes such a small distance to send the finger into the key to the keybed.
So how much distance do you save by doing your extension movement? One millimeter... two...
And for saving that very small distance, extending the finger fails to capitalize on the natural strength and speed inherent in the natural flexing movement in the natural finger arc.
A straight line being the shortest distance between two points, is not necessarily efficient in piano terms.
You mention working on the Mephisto Waltz... Ok, here's an excellent example... you're of course familiar with the famous "jumps" toward the end of the piece... you know... A up to A Octave, A# up to A# Octave... playing the theme.
Many pianists play the lower notes with the thumb, jumping up to play the octave believing that this is the shortest and most efficient path.
But this is not so. If you play the lower note, A, A#, D# etc with the INDEX FINGER, this make your hand and arm move in a slightly elliptical pattern to play the upper octave and then returning completing a somewhat circular movement to play the next single melody note again with the index finger.
This way, you are making slightly rounded movements, highly flexible movements, instead of the stiffer co-contraction encountered in the stopping and starting movements you encounter when using the thumb instead.
Again, not the most direct path in physics terms, but much easier to execute in human/piano terms.