Isometric Embedding and Visualization of the Apparent Horizons in Binary Black Hole Simulations

We present an efficient algorithm for embedding topological two-spheres with specified 2-metrics into three-dimensional Euclidean space.  We show that the algorithm converges for a large class of test surfaces with known embeddings, even for those with negative Gaussian curvature, independent of the initial guess. The algorithm is then applied to visualize the apparent horizons in numerical binary black hole simulations using the Spectral Einstein Code (SpEC). This algorithm also allows one to test various definitions for Quasi-local mass in general relativity and study the energy dynamics during black hole mergers.