Recalling derivative coordinate functions
When I developed the equation for genetic fractals (paper) , I first defined a “path function”, i.e. an equation that could represent arbitrary curves. This is given as:
The notation is slightly different with respect to the paper. is a derivative of the coordinate function and is a derivative of the coordinate function .
To put this equation in words: the argument in the exponential form is the function which determines the direction in which the path evolves. The speed at which it evolves is determined by . So don’t worry about the integrals, we just need to define the derivative coordinate functions and and the integrals will keep track of the curve.
The reason we use the derivatives of the coordinate function is that they allow us to define the local behavior at the last point of the curve.
So much for the introduction, you can find more details in the paper.
Infinite parametrizations of a curve
One of the things I have been wanting to study concerns transformations between different coordinate systems using these derivative coordinates. Initially I focused on the transformation between these derivative coordinates and a Cartesian coordinate system. I made some attempts at this but I quickly ran into unsolvable integrals. I will pursue this some day but right now I came across another interesting idea.
If my thesis is correct, then there are infinitely many ways in which a single curve, which is a point set, can be represented using derivative coordinate functions. If this is so, then if I can find a transformation between this forms then we could transform the parametrization of the curve to suit our purpose. Note the two ifs.
Today I did some experimentation with an ellipse. I chose this geometric shape for a good reason. If indeed I can choose the parametrization of the ellipse at will, then I may be able to calculate the circumference (arc length) of the ellipse in a novel way. You may (or not) know that there exists no simple way of calculating the circumference of an ellipse. It can only done by approximation.
Two alternative parametrizations of an ellipse
There are two interesting parametrizations of an ellipse. Well, I can think of a few more, but for my purpose I focus on a parametrization using derivative coordinate functions for two special cases:
is constant, i.e. we let the curve evolve at constant speed and control the shape by varying the directional coordinate
is constant, i.e. we change the direction of the curve uniformly (“constant rotation”) and change the speed as needed.
Using excel, some guessing and a good deal of patience I obtained the results below. You can see that the curve generated using the derivative coordinate functions is superimposed on the ellipse we are modelling with axes a=1, b=2. You can also see that the parametrization is approximate, but good enough for our purposes.
By approximation, the two parametrizations work well and the plots above give a good indication of what the derivative coordinate functions look like when we fix the other coordinate.
If you have followed this so far, you will agree that there are many alternative parametrizations that will give the same ellipse, i.e the same point set. If we remove the restriction of keeping one of the derivative coordinate functions constant, then we can think of many alternative combinations of the two derivative coordinate functions.
As mentioned before, the parametrization with as a constant is interesting because we can calculate the circumference by: where S is the period of revolution. The theoretical circumference of this ellipse is 9.69. Even though the fit isn’t perfect we calculate 0.0485 x 200 = 9.70. That’s not bad. But neither is it meaningful. It’s just excel.
Interesting as this is – and I post it so that I can come back to this – this is just experimental maths. Now I need to figure out the transformation. Hopefully I won’t prove to ugly.