@mehdi jafari The OP is not using e as the familiar constant but rather as a substitute for the epsilon that is used in the paper. One problem (caused by 2D input) is that it is being interpretted as a function rather than as a coefficient. 

What is the significance of a base-dimension unit (like meters) raised to a fractional power like 1/6?

If you remove the units, you'll get an answer immediately from solve. 

@gdoug I see what you mean. Here's a corrected version.

HeadToTail:= proc(L::list(Vector(2)))
local L0:= [<0,0>, L[..-2][]]; print(L0, L -~ L0);
     plots:-arrow(zip(`[]`, L0, L -~ L0), args[2..]);
end proc:

Please let me know how that works for you.

@Markiyan Hirnyk 

parse will work for the example list that I showed, but I don't know if the OP's actual list contains all strings that simple. I chose not to use parse because the OP's list might contain something like "a b".

@tomleslie I think that your criticism is too harsh:

1. The OP did upload a worksheet using the big green arrow. The amount of clipping on the right is browser dependent and the OP has no control over that. Note that there is a horizontal scroll bar so that you can read the right sides of the three lines (in Google Chrome) that got clipped.
2. No one has to retype anything or do any cutting and pasting whatsoever because the OP included a link to the uploaded worksheet.
3. I for one have no trouble understanding what the OP is asking. Your interpretation is in fact correct.

I will accept Kitonum's implied answer to my question above. You're asking for a cheap way to do it. By cheap, do you mean fast? Faster than fsolve? Do you have a large number of points that you want to project onto a specific curve? If so, a Newton iteration could be constructed to do it for that curve. The iteration can be automatically constructed and optimized for each given curve.

@Kitonum I'm guessing that the phenomenon that I described is an anomaly of Maple 16 (and maybe 17 and 15).

Download plus_vs_add.mw

I've been able the duplicate the above on different computers. But in Maple 2015, I get that add is faster than `+`.

What does the projection of a point onto a curve mean? Can you give me a formula or reference for it?

@Alejandro Jakubi I wonder if add(U) is equivalent to `+`(U[]). Of course they produce the same result. But I think that `+`(U[]) is faster than add(u, u= U). That is, when the list of summands already exists (and only when it already exists) the prefix `+` form is faster than the older form of add.

You'll need to post the complete code so that we can duplicate your problem. You can post a worksheet by using the green uparrow tool, which is the last item on the second row of the toolbar in the MaplePrimes editor.

@acer 

Okay, I made the tables explicitly constructed as tables, which takes care of all your objections except the last. Regarding your last objection, I find it rather distasteful for code to use more variable names than are necessary. For one thing, it interferes with the garbage collection of the tables. A better solution is to not use names that reveal the underlying data structure.

@tomleslie Yes, your worksheet works in Maple 16 and gives a very reasonable answer. When N is increased to 100, it gets the answer to 4 decimal places.

I don't understand the significance of the ScatterPlot3D and the lowess surface.

kegj: If you want me to check your code, you'll have to upload the most recent version. The version that you most recently uploaded obviously doesn't work because you've mixed up lowercase and uppercase letters.

This extra space is a long-standing and oft-reported bug. AFAIK, there is no known workaround.

@kegj In Maple, the familiar constant Pi is spelled with a capital P; when you spell it pi, it just becomes a regular variable name.

@tomleslie The different behavior of Pi in floating-point expressions is intentional and documented.