This time I'm trying hard to be as specific as I can. Please don't close the thread, help me to make a specific question. Thanks.
The description of formula with formula itself:
The fft-based representation of a wave height ﬁeld expresses the wave height h(x, t) at the horizontal position x = (x, z) as the sum of sinusoids with complex, time-dependent amplitudes:
where t is the time and k is a two-dimensional vector with components k = (kx, kz), kx = 2πn/Lx, kz = 2πm/Lz, and n and m are integers with bounds −N/2 ≤ n < N/2 and −M/2 ≤ m < M/2. The fft process generates the height ﬁeld at discrete points x = (nLx/N,mLz/M). The value at other points can also be obtained by switching to a discrete fourier transform, but under many circumstances this is unnecessary and is not applied here. The height amplitude Fourier components, ˜ h(k, t), determine the structure of the surface.
Another vesrion of the same formula:
Here X is a horizontal position of a point whose height we are evaluating. The wave vector K is a vector pointing in the direction of travel of the given wave, with a magnitude k dependent on the length of the wave (l):
k =2p /lamda
And the value of h (K,t) is a complex number representing both amplitude and phase of wave K at time t. Because we are using discrete Fourier transformation, there are only a finite number of waves and positions that enters our equations. If s is dimension of the heightfield we animate, and r is the resolution of the grid, then we can write:
K = (kx,kz) = (2pn / s,2pm /s)
where n and m are integers in bounds –r/2 £ n,m < r/2. Note that for FFT, r must be power of two.
How many k do I need? Does it mean, that k takes all possible values (on the grid I mean)?
And what do I get as a result - complex number that shows height and (what?) in concrete point so I need make computations for all points on the grid or the whole grid itself?
How to apply fft to this formula?