Trajectory Explorer

This interactive visualization shows a parametric curve defined by:

\[ \begin{align}\begin{aligned}x(t) = x_0 + r_1\cos(t+\varphi_1) + r_2\cos(2t+\varphi_2) + r_3\cos(3t+\varphi_3)\\y(t) = y_0 + r_1\sin(t+\varphi_1) + r_2\sin(2t+\varphi_2) + r_3\sin(3t+\varphi_3)\end{aligned}\end{align} \]

Use the sliders below to adjust the parameters and see how the trajectory changes!

Interactive Trajectory Explorer

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About this visualization

The left plot shows the actual trajectory in the x-y plane. The right plot shows how the polar angle θ (measured from the origin) changes as we trace along the curve.

Parameters:

  • r₁, r₂, r₃: The radii of the Fourier components (harmonics)

  • x₀, y₀: The center offset of the trajectory

  • φ₁, φ₂, φ₃: Phase shifts for each harmonic

This type of curve appears in problems involving winding numbers and complex analysis. Try adjusting the parameters to see:

  • How different combinations of r₁, r₂, r₃ create different shapes

  • When the curve winds around the origin multiple times

  • How the center position (x₀, y₀) affects whether the curve encloses the origin