Fourier Transform of the Heart

This video illustrates the way in which a Fourier series approximates an arbitrary function. The heart outline defines a mapping from the real interval [0,2pi] to the complex plane. In the Fourier series, each term is represented by a link rotating at a set speed. The first two links do one rotation during the video, in opposite directions. The next two links do two rotations, and so on. The lengths and starting angles of the links are the coefficients in the Fourier Series. The coefficients for terms from -4 to 4 (so the constant term is the centre of the list) are: - - - - - - Those were obtained by numerical fitting to the functions x = 16 .* sin(t).^3 ; y = 13.*cos(t) - 5.*cos(2.*t) -2.*cos(3.*t) - cos(4.*t) ; The central constant term may as well be taken as zero.