Each example takes a physical or mathematical problem and shows how an elliptic integral or Jacobi function gives its exact solution — formula, derivation, and interactive figure together, running entirely in the browser. Topics span celestial mechanics and orbital geometry, sub-Riemannian geodesics and optimal control, nonlinear pendulum dynamics, GPU-scale Keplerian propagation, cnoidal waves, and Weierstrass modular forms.
Celestial mechanics & orbital dynamics
Derive the arc length of an ellipse via $E(\varphi|m)$, trace Keplerian orbits in 3D, solve Kepler's equation, and watch the Carlson duplication algorithm converge step-by-step.
Real JPL orbital elements propagated on GPU. Throughput benchmarked across NumPy, PyTorch CUDA, and JAX. Position accuracy validated against JPL Horizons.
Sub-Riemannian geometry & optimal control
Optimal contour completion in V1 is a shortest-path problem on SE(2).
The solution curves are Euler's elastica; their curvature is $\kappa(s) = 2k\,\mathrm{sn}(s\mid k^2)$
and their spatial period is $4K(k^2)$. Motivation for the original elliptic package.
Semi-analytical stability index $S = 1 - k^2$ for each asteroid computed from osculating elements alone. The libration period $T = 4K(k^2)/\omega_0$ diverges at the separatrix — same $K(k^2)$ as the elastica problem.
Jacobi elliptic functions
Exact period via $K(k^2)$, live $\mathrm{sn}(t|m)$ animation, phase portrait with separatrix, and period divergence as $\theta_0 \to \pi$.
Torque-free rotation of an asymmetric top: $\omega_1 = \omega_0\,\mathrm{cn}(t|m)$, polhode animation, and Euler angle integration.
Nonlinear waves & signal processing
Exact KdV cnoidal wave solution $u(x,t) = A\,\mathrm{cn}^2(\kappa(x-vt)|m)$ animated across the soliton limit $m \to 1$.
Weierstrass & modular forms
Domain-colouring of $\wp(z)$ over the complex plane, lattice geometry from invariants $g_2, g_3$, and the group law on $y^2 = 4x^3 - g_2\,x - g_3$.