08-24-2021, 08:54 AM
I explain now the plugin Fourier3. (The other one, "Fourier3 with control of symmetry" will be in turn later.)
If you run the plugin Fourier3 with the default values you get the figure on the left:
That curve was not my invention. I took it from a writing by Frank A. Farris:
https://scholarcommons.scu.edu/cgi/viewc...th_compsci
which you may wish to glance at (it is mathematics but there are some pretty pictures). The idea in the plugin is pure mathematics but when you try to get some understanding of what the plugin does it is better to think as follows:
Look at the drawing on the right above. Or look at Figure 1 in Farris's paper; it is clearer. Now imagine the following: We have three wheels. The largest has its center fixed and it revolves around the center with some constant rate. The second wheel has its center attached to some point on the perimeter of the first wheel, so it follows along when the first wheel revolves. At the same time the second wheel revolves at some constant rate around its own center. And the third wheel is attached to some point on the second wheel's perimeter, and it too revolves at some constant rate. And then there is a pen attached to the third wheel's perimeter. That pen draws some curve (the dashed curve above) when the wheels revolve.
Can you imagine that? Can you see the wheels going round and the curve to be drawn? It is difficult, and I wish I could make an animation of it. But you can look at the first half minute of the video
https://www.youtube.com/watch?v=r6sGWTCMz2k
There there are many more wheels than 3, but it gives the idea.
So that is the best way to think about it, but if you know some mathematics you can think of it as the first 3 terms of a complex Fourier series. That explains the name "Fourier3" of the plugin.
You may remember that the spirograph has two wheels going round. Here we have three. Indeed, what the spirograph draws can be drawn with this Fourier3 plugin too, though the connection is not obvious or easy. And even the rhodoneas (rose curves) can be drawn with Fourier3. In this sense, Fourier3 generalizes both of my Spiropath and Rhodonea plugins. But those two may be easier to use.
The main inputs to Fourier3 are:
(Just for completeness sake: The coefficients are usually floats, but you may also input complex numbers. The complex parts cause phase shifts to the wheels. Never mind about that now. You can just use floats; then all resulting curves will have vertical reflection symmetry.)
When one starts to use Fourier3, the main impression is that it is rather frustrating. It can draw pretty curves, certainly. But the inputs are complicated (3 integers and 3 floats or complex numbers). And if you have in your mind some particular kind of a curve, in practice there is no way to know from beforehand what would be good inputs. The only way to use the plugin is to keep trying with different inputs, and if you happen to get some pleasing picture, grab it and save the inputs for later use. This is something I cannot help. I have no more insight to this problem.
To save the inputs you need not write them down: If you run the plugin with the last input "Display messages" set to "Yes", there will be messages in the error console, including suitable inputs to make that curve (possibly a little different from what you used). You can copy those to a file.
Varying coefficients
In these three figures I used the same frequencies (2,-3,-18) but I made little changes in the coefficients:
The path on the left I got with with the default coefficients (1, 0.5, 0.25). The second figure was made with otherwise the same inputs except that I changed the third coefficient from 0.25 to 0.125. That little change made the curve rather different. But that I couldn't guess from beforehand! There is no way but to try different inputs and to see what comes. The third figure was made by the same frequencies but with coefficients (1, 0.25, 0.125), a little change in the second coefficient.
About rotation symmetry
Why the default figure (first picture) has 7-fold rotation symmetry and the other three (second picture) have 5-fold symmetry? It can be explained. It is solved in the nice theorem by Farris (in the site mentioned above); look there if you want to see the rule precisely. For anybody curios, I say now just this: In these examples, in the first picture the frequencies (2,-5,-19) are all of the form 7q+2 for some integer q. In the second picture the frequencies (2,-3,-18) are all of the form 5q+2. Note the 7 and the 5.
So, it is the frequencies that dictate the rotation symmetry.
Varying frequencies
Three examples with the default coefficients (1, 0.5, 0.25) but with different frequencies:
The frequences are, respectively, (1, -8, -35), (1, -10, -32), and (1, 4, -11), hence the symmetries are 9-fold, 11-fold, and 3-fold (Farris!).
The next plugin "Fourier3 with control of symmetry" makes it easier to obtain some desired symmetry. That also makes experimenting a little more efficient. I explain that in my next post.
If you run the plugin Fourier3 with the default values you get the figure on the left:
That curve was not my invention. I took it from a writing by Frank A. Farris:
https://scholarcommons.scu.edu/cgi/viewc...th_compsci
which you may wish to glance at (it is mathematics but there are some pretty pictures). The idea in the plugin is pure mathematics but when you try to get some understanding of what the plugin does it is better to think as follows:
Look at the drawing on the right above. Or look at Figure 1 in Farris's paper; it is clearer. Now imagine the following: We have three wheels. The largest has its center fixed and it revolves around the center with some constant rate. The second wheel has its center attached to some point on the perimeter of the first wheel, so it follows along when the first wheel revolves. At the same time the second wheel revolves at some constant rate around its own center. And the third wheel is attached to some point on the second wheel's perimeter, and it too revolves at some constant rate. And then there is a pen attached to the third wheel's perimeter. That pen draws some curve (the dashed curve above) when the wheels revolve.
Can you imagine that? Can you see the wheels going round and the curve to be drawn? It is difficult, and I wish I could make an animation of it. But you can look at the first half minute of the video
https://www.youtube.com/watch?v=r6sGWTCMz2k
There there are many more wheels than 3, but it gives the idea.
So that is the best way to think about it, but if you know some mathematics you can think of it as the first 3 terms of a complex Fourier series. That explains the name "Fourier3" of the plugin.
You may remember that the spirograph has two wheels going round. Here we have three. Indeed, what the spirograph draws can be drawn with this Fourier3 plugin too, though the connection is not obvious or easy. And even the rhodoneas (rose curves) can be drawn with Fourier3. In this sense, Fourier3 generalizes both of my Spiropath and Rhodonea plugins. But those two may be easier to use.
The main inputs to Fourier3 are:
- three "frequencies"
- three "coefficients".
(Just for completeness sake: The coefficients are usually floats, but you may also input complex numbers. The complex parts cause phase shifts to the wheels. Never mind about that now. You can just use floats; then all resulting curves will have vertical reflection symmetry.)
When one starts to use Fourier3, the main impression is that it is rather frustrating. It can draw pretty curves, certainly. But the inputs are complicated (3 integers and 3 floats or complex numbers). And if you have in your mind some particular kind of a curve, in practice there is no way to know from beforehand what would be good inputs. The only way to use the plugin is to keep trying with different inputs, and if you happen to get some pleasing picture, grab it and save the inputs for later use. This is something I cannot help. I have no more insight to this problem.
To save the inputs you need not write them down: If you run the plugin with the last input "Display messages" set to "Yes", there will be messages in the error console, including suitable inputs to make that curve (possibly a little different from what you used). You can copy those to a file.
Varying coefficients
In these three figures I used the same frequencies (2,-3,-18) but I made little changes in the coefficients:
The path on the left I got with with the default coefficients (1, 0.5, 0.25). The second figure was made with otherwise the same inputs except that I changed the third coefficient from 0.25 to 0.125. That little change made the curve rather different. But that I couldn't guess from beforehand! There is no way but to try different inputs and to see what comes. The third figure was made by the same frequencies but with coefficients (1, 0.25, 0.125), a little change in the second coefficient.
About rotation symmetry
Why the default figure (first picture) has 7-fold rotation symmetry and the other three (second picture) have 5-fold symmetry? It can be explained. It is solved in the nice theorem by Farris (in the site mentioned above); look there if you want to see the rule precisely. For anybody curios, I say now just this: In these examples, in the first picture the frequencies (2,-5,-19) are all of the form 7q+2 for some integer q. In the second picture the frequencies (2,-3,-18) are all of the form 5q+2. Note the 7 and the 5.
So, it is the frequencies that dictate the rotation symmetry.
Varying frequencies
Three examples with the default coefficients (1, 0.5, 0.25) but with different frequencies:
The frequences are, respectively, (1, -8, -35), (1, -10, -32), and (1, 4, -11), hence the symmetries are 9-fold, 11-fold, and 3-fold (Farris!).
The next plugin "Fourier3 with control of symmetry" makes it easier to obtain some desired symmetry. That also makes experimenting a little more efficient. I explain that in my next post.