Fourier ShapeΒΆ

  • Animate a fourier series of different shapes

  • Highlights: integrate, sample, Trail

  • Inspiration: 3Blue1Brown

A Fourier Series of n terms is an approximation of a given function using sine waves of frequencies upto n. We can represent shapes (curves) using complex functions. We find the amplitudes of the frequencies forming the function by finding its Fourier transform

Given a (complex) function \(u(t) : [a, b] \to C\) The amplitude of a frequency f as a function of f is given by the fourier transform \(\widehat{u}(f)\) :

\[\widehat{u}(f) = \int_{a}^{b} u(t) e^{- 2 \pi i f t} dt\]

We compute this by using the Reimann sum of the integral instead: the function `math::integrate`

To find the function for an arbitrary shape

  1. trace over the shape in Blender

  2. export as a .obj file

  3. use obj_to_pts to import and parse the obj

  4. use ShapeFnFromPts to make a function which interpolates between the points and returns a complex number.