08-24-2021, 04:06 PM
I explain now the plugin "Fourier3 with control of symmetry". In it I implemented Farris's theorem (see the previous post) to give the user easy control of the rotation symmetry. You can make a curve with m-fold symmetry by inputting m in the plugin.
(And it would be very nice if other features than rotation symmetry could be controlled so easily. Alas, I don't know any counterparts to Farris's theorem concerning other features. And that theorem I happened to find quite by accident when I was looking for something else.)
The GUI is similar to Fourier3 but there are three more inputs: m, k, and a boolean input.
If you put, say, m=5, the plugin takes your inputs for frequencies and coefficients, and modifies the frequencies so that the curve will have 5-fold rotation symmetry.
That input k can be used to get some variation: different values may give even drastically different shapes (though sometimes changing k will change nothing). But the real variation comes from your input frequencies and coefficients. (The reason for such k is that it appears in Farris's theorem, and I didn't find any better way to get a value for it but to let the user to input it. You can just let it be k=1 if you like.)
The boolean input asks if you want to retain any symmetry that may be inherent in your input frequencies. Example: Suppose that you write in the GUI the frequencies (2,-5,-19). Those would give rise to 7-fold symmetry in Fourier3 (Farris's theorem!). Suppose that you now input m=5. If the boolean input is "No", the plugin demolishes that 7-fold symmetry and you are given a curve with 5-fold symmetry. But if the boolean input is "Yes", the 7-fold symmetry is preserved and you get 35-fold symmetry. Naturally, that is what you get when you have both 7-fold and 5-fold symmetries in the same figure.
(More precisely, the rule goes like this: Let M be the symmetry contained in your frequencies (above M=7). If the boolean input is "Yes", the plugin gives Mm/gcd(M,m)-fold symmetry (where gcd means the greatest common divisor). Namely, Mm/gcd(M,m) is the smallest positive integer divisible by both M and m.)
I think that this plugin is better for experimentation than Fourier3 since you can determine the rotation symmetry. On the other hand, if you for some strange reason have some 3 frequencies and you want to see what kind of curves you get with them, then use Fourier3; that plugin keeps faithfully your frequencies.
When you find inputs that make a nice curve and you want to be able to re-create the curve at some later time, do the following: Re-run the plugin with "Display messages" set to "Yes". Then in the error console you find (among other things) values for Frequencies and Coefficients which, when inserted in Fourier3, produce that curve. You can copy those values to some file and save.
As an example on varying k, below are the curves I got with default frequencies and coefficients but setting m=5 and the boolean input to "No", and tried different values for k. Here are those that pleased the eye: k=1,2,3,4 and k=-1,-2,-3,-4. Note the 5-fold symmetry everywhere.
(And please do not ask how m and k work precisely. The rules I coded into the plugin are ad hoc, to say the least. They may change if ever get some better ideas.)
The same exercise with frequencies (1,4,7), m=5, and k=1,2,3,4:
(And it would be very nice if other features than rotation symmetry could be controlled so easily. Alas, I don't know any counterparts to Farris's theorem concerning other features. And that theorem I happened to find quite by accident when I was looking for something else.)
The GUI is similar to Fourier3 but there are three more inputs: m, k, and a boolean input.
If you put, say, m=5, the plugin takes your inputs for frequencies and coefficients, and modifies the frequencies so that the curve will have 5-fold rotation symmetry.
That input k can be used to get some variation: different values may give even drastically different shapes (though sometimes changing k will change nothing). But the real variation comes from your input frequencies and coefficients. (The reason for such k is that it appears in Farris's theorem, and I didn't find any better way to get a value for it but to let the user to input it. You can just let it be k=1 if you like.)
The boolean input asks if you want to retain any symmetry that may be inherent in your input frequencies. Example: Suppose that you write in the GUI the frequencies (2,-5,-19). Those would give rise to 7-fold symmetry in Fourier3 (Farris's theorem!). Suppose that you now input m=5. If the boolean input is "No", the plugin demolishes that 7-fold symmetry and you are given a curve with 5-fold symmetry. But if the boolean input is "Yes", the 7-fold symmetry is preserved and you get 35-fold symmetry. Naturally, that is what you get when you have both 7-fold and 5-fold symmetries in the same figure.
(More precisely, the rule goes like this: Let M be the symmetry contained in your frequencies (above M=7). If the boolean input is "Yes", the plugin gives Mm/gcd(M,m)-fold symmetry (where gcd means the greatest common divisor). Namely, Mm/gcd(M,m) is the smallest positive integer divisible by both M and m.)
I think that this plugin is better for experimentation than Fourier3 since you can determine the rotation symmetry. On the other hand, if you for some strange reason have some 3 frequencies and you want to see what kind of curves you get with them, then use Fourier3; that plugin keeps faithfully your frequencies.
When you find inputs that make a nice curve and you want to be able to re-create the curve at some later time, do the following: Re-run the plugin with "Display messages" set to "Yes". Then in the error console you find (among other things) values for Frequencies and Coefficients which, when inserted in Fourier3, produce that curve. You can copy those values to some file and save.
As an example on varying k, below are the curves I got with default frequencies and coefficients but setting m=5 and the boolean input to "No", and tried different values for k. Here are those that pleased the eye: k=1,2,3,4 and k=-1,-2,-3,-4. Note the 5-fold symmetry everywhere.
(And please do not ask how m and k work precisely. The rules I coded into the plugin are ad hoc, to say the least. They may change if ever get some better ideas.)
The same exercise with frequencies (1,4,7), m=5, and k=1,2,3,4: