@adel-00 Assuming that N, delta, Delta, w are real (which seems to be the case), how is your two-case formulation of r any different from sqrt(abs(N*(N+1) - (delta-Delta+w)^2))?

@mmcdara You have implied that the resolution option can increase the number of points. Perhaps that is true for some weird example. But the purpose of the option, as stated on the help page, is to reduce the number of points.

There has been no official response about this matter, not even acknowledgement of the existence of a problem. As I understand it, no one can display worksheets now. Thus this new issue has superceded as well as overshadowed my Question, which hasn't been answered: Specifically, Why haven't I been able to post any worksheet for a year even at times when other people could? 

@adel-00 And what is d?

Note that r can be replaced by simply sqrt(abs(N*(N+1) - (delta-Delta+w)^2)).

@snowman You wrote:

• You mean that when numpoints is specified, increasing the gridrefine may make the needed numpoints surpass the specified numpoints, as a result, making the figure is rougher?  Only when the number of needed numpoints for gridrefine is less than the specified numpoints, can the figure is smooth?

The numpoints option just causes confusion, and it shouldn't be used at all. It's functionality is completely superceded by the grid option. It causes confusion because it has no relation to the numpoints option of the regular plot command, nor does it specify the number of computed points, nor even the initial number of computed points, in an implicitplot. All that it does is set the initial grid at N x N, where N is roughly sqrt(numpoints). I emphasize initial grid because the gridrefine option may increase the grid, but only within the rectangles that have a point of the curve.

• By the way, how do you figure out that For gridrefine=3 F(x,y) is calculated at 515 points, while 4119 points are used for gridrefine=6.

As long as the resolution option is not used, the number of computed points can be determined as the number of rows in the point-data matrix of the plot structure. This is op([1,1,2,1,2], P) where P is the plot structure (the return value of the implicitplot command). For example, using your F:

P||(0..6):= plots:-implicitplot~(F, 0..20, 0..10, gridrefine=~ [$0..6])[]:
map2(op, [1,1,2,1,2], [P||(0..6)]);
             [83, 165, 331, 659, 1317, 2631, 5263]

We see that the number of computed points approximately doubles (and that's a very close approximation) for each unit increase in gridrefine, which is exactly what I'd expect for a monotonic curve.

• There's a formula that The highest grid level for gridrefine=l will be [2^l*m,2^l*n] , which may not help me to achieve the value of 515, and 4119.

That formula is misleading because only grid rectangles that are already known to contain a point of the curve are subdivided. For a monotonic curve, that's relatively few of the rectangles. But suppose I change your F to a nearly space-filling curve:

F1:= (x,y)-> y-sin(99*x):
P||(0..6):= plots:-implicitplot~(F1, 0..20, -1..1, gridrefine=~ [$0..6])[]:
map2(op, [1,1,2,1,2], [P||(0..6)]);
       [759, 2647, 5935, 49501, 126235, 771225, 1892517]

• The results of  "showstat(`plots/implicitplot`) )" are too long, which confuses me a lot.

You won't be able to understand the roughness issue by reading any Maple code. All the Maple code does is compute some number of points (x,y) such that F(x,y) = 0 (and some other trivialities like defining the axes and colors). The end result of the Maple code can be examined by using ​​​​​​op(P) (note: no index numbers for this op) on the returned plot structure P. You'll see that it's trivial, (which is not to say that the computations done by implicitplot are trivial, just that in this case the end result of those computations is), and it's still fairly far from an actual visual plot. To actually display the plot on a physical device, the plot structure is passed to rendering code. This code is not written in Maple, and you can't see it. Obviously, the number of points (or pixels) actually plotted on your screen is determined by your screen's resolution (the number of pixels that it has).

@adel-00 The expression L1 contains a parameter r. It can't be numerically integrated without specifying a value for r.

@adel-00 I will check your code soon. However, again it seems like you're apologizing for using loops. There's no problem with using loops, and I never said to not use them! What I don't want you to use are integration techniques, whether they be Maple's packaged routines or your own ad hoc method.

Now perhaps you can't produce a symbolic expression for the integrand; that is often the case. In that case, can you write a procedure that numerically evaluates the integrand for any real numeric w? That would be sufficient. Note that I said integrand, not integral.

@Earl Quotes are very often the key to a successful Google search. I think that without quotes you get something close to the intersection of the separate searches for "pedal" and "curve". This intersection is of course much too large to be of any practical value.

@mmcdara The reason that I said that I was editing rather than correcting is that your usage of contributor wasn't incorrect, and indeed contributor can be synonymous with contributing factor (however, this isn"t the most-usual usage). However, in this particular context, that led to the ambiguity mentioned by the OP; indeed, I had noticed exactly that same ambiguity immediately upon initially reading your sentence. I thought that it was possible, although unlikely, that you were referring to a human co-author of the worksheet as a contributor​​​​​. 

Googling "pedal surface" (with the quotes) turns up several useful hits (after ignoring numerous articles about the surfaces of mechanical pedals, of course), including JSTOR articles from 1863 and 1881.

It's surprising that this obvious generalization isn't at all mentioned in either the Wikipedia or MathWorld articles "Pedal curve". By the way, that Wikipedia article is quite detailed.

@ecterrab I think that you've misinterpreted Preben''s comment. He's not objecting to the RootOf. Rather, he's pointing out that this expression cannot be evaluated at x=0, thus one can't solve for the constants of integration in the normal way.

Hopefully Preben will verify my guess about his intended meaning. 

@mmcdara I wasn't correcting you--I was only editing--although I have no compunction against correcting (or elaborating on subtle ambiguities in) our contributors' English usage when it seems worthwhile to do so. Anyway, perhaps you realize at this point that contributor usually refers to a person who contributes, whereas the most-common term for a nonperson entity that contributes is contributing factor (even in nontechnical contexts where factor has no special meaning).

@adel-00 You claim that you want to "integrate L1 w.r.t. w from -infinity to infinity numerically." Well, then show me an expression L1 that depends on w. The end result of the worksheet that you just posted is a sequence of numbers L1[-10] ... L1[10]; that's not an expression that depends on w. Those numbers may be approximations to various parts of the integral; I can't tell for sure. So, I'm telling you again, for the time being, forget about integrating this expression. Instead, give me a worksheet that does nothing more than construct the expression L1(w) which is to be integrated. If you need to use loops to define that expression, that's totally fine. If the expression is too long to display, that's fine.

@dnaviaux I'd guess that in mmcdara's final sentence, "contributor" could be replaced by "factor that contributes to the error".

Before we can integrate the expression, we need to have the expression in Maple. So, forget about integrating for the time being. Concentrate on simply having this rather complicated expression entered into Maple such that it will display the expression in prettyprinted form. After that, we can discuss integrating it.

As it stands, it's difficult for me to distinguish the parts of your code that are for integrating from the parts that are for simply defining the integrand.