dnguyend/EmbeddedGeometry

Embedded Geometry

This is the supporting repository to our paper Geometry in global coordinates with applications in optimal transport and mechanics.

The derivation of the Christoffel function for the reflector antenna cost on an optimal transport problem on fixed-rank semidefinite matrices is given here.

The main formulas involves derivatives of a number of operators, including projection $\Pi$, metric $\mathsf{g}$, and Christoffel function $\Gamma$. We use Jax for numerical derivative, in particular, jvp is the main tool for directional derivative. The jax functions grad and jacfwd are also useful.

Examples verifies properties of the Christoffel function, the curvature formulas and Bianchi identities.

In JaxRigidBodyDynamics.ipynb we show examples for rigid body mechanics.

In SemidefiniteOptimalTransportAntennaCost.ipynb we show numerical verifications of the Christoffel function and the cross curvature for an optimal transport problem with reflector antenna cost.

In KMCTests.ipynb we show a few other numerical experiments for the Kim-McCann metric.