@acer  OK, do you know how to add a color bar  like the figure below.

@acer Thanks a lot! Yes, I need a color bar to indicate the value and export it to a single image. By the way, what do you mean by "if not r::numeric then return 'procname'(args); end if;"   what does procname (args) mean ?

@acer  sorry to let you misunderstand, the first coordinate is "r", and the second coordinate is "f(r)", which is independent of theta, so the figure is independent of the angle.  I do not have the analytical expression for f(r) but a list of data. what I want is to plot with color according to the value of f(r).

@Carl Love Great! your answer cover from practices to theory, thanks a lot

@Carl Love  Thank you very much,  you really let me understand this question, thanks again, and Merry Christmas!

@acer  Sorry, It's not a toy model, this is the function I meet during  the deduction, I have some other function which is more complex to implicitplot, and I choose this simplest one to ask this question.

@mmcdara Thank you and Merry Christmas, but I still have some problems, 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?

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.  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.

Besides, how do you know "(Note these numbers are twice the numbers I obtained by examining the output of showstat(`plots/implicitplot`) )." the results of  "showstat(`plots/implicitplot`) )" are too long, which confuses me a lot. 

You also mention the "resolution", from the "HELP", I know that " The option resolution = r applies resolution-based smoothing of the output curve to reduce the size of the output data structure, but retain the visual features of the curve up to a resolution of r x r pixels", does this mean that command resolution increases the absolute error of implicitplot indeed? 

@vv Thanks a lot!

@Kitonum Thanks a lot, but what I want is a 2d figure, not a 3d. Nevertheless, I still benefit from your answer.

@vv Thanks a lot. However, when I copy your code into my maple, the figure is not very smooth. I have increased the Digits to 100 , but it doesn't work at all.