Transform of the Heart

Visualization of the Fourier series decomposition of a heart-shaped parametric curve

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Curve Parametrization‾\textbf{\underline{Curve Parametrization}}
x(t)=17 sin⁡3(t)x(t) = 17\,\sin^3(t)
y(t)=15 cos⁡(t)−7 cos⁡(2t)−2 cos⁡(3t)−cos⁡(4t)y(t) = 15\,\cos(t) - 7\,\cos(2t) - 2\,\cos(3t) - \cos(4t)
Fourier Series Decomposition‾\textbf{\underline{Fourier Series Decomposition}}
f(t)=∑n=0N−1An ei(ωnt+ϕn)f(t) = \sum_{n=0}^{N-1} A_n\,e^{i(\omega_n t + \phi_n)}
An= amplitude A_n = \text{ amplitude }
ωn= the angular frequency\omega_n = \text{ the angular frequency}
ϕn= the phase of the nth term\phi_n = \text{ the phase of the } n^\text{th} \text{ term}

Each Fourier component represents an epicycle.

The sum reconstructs the heart shape via Fourier synthesis.