vehlwn / fourier-epicycles

Browser app to interpolate points with complex Fourier series

fourier-epicycles

Browser app to interpolate given set of points in the complex plane using complex Fourier series and to draw epicycles and deferents. Requiers JavaScript to work.

Textbox to the right contains lines with space separated points' coordinates. It can be edited either manually or by clicking on the canvas to the left.

LMB -- add point.

Alt+LMB -- remove point.

Start button calculates Fourier coefficients from input points and starts animation with varying time argument.

Clear button removes all points.

Build

$ npm install
$ npm run dev
$ python3 -m http.server --bind 127.0.0.1 5000 -d dist

Details

Put $N$ be the length of a sequence of complex numbers $\left\{ x_i \right\} _ {i=0} ^ {N-1}$.

Coefficients $c_k$ are calculated as discrete Fourier transform:

$$ c_k = \frac{1}{\sqrt N} \sum_{j=0}^{N-1} x_j \, e^{-2i \pi \frac{j}{N} k}, \; k = \overline{0..N-1}. $$

Value of an interpolant calculated as Fourier series:

$$ S(t) = \frac{1}{\sqrt N} \sum_{j=0}^{N-1} c_j \, e^{+2i \pi \frac{f_j}{N} t}, \; t \in [0,N). $$

Frequencies $f_j$ defined as:

$$ f_j = \begin{cases} j & \text{if} \; j \le \left\lfloor \frac{N}{2} \right\rfloor \\ j - N & \text{otherwise} \end{cases} $$

Examples

vim: textwidth=85