### 7.5 Geometry interpolation

depends on the slice geometry variables , , and at the horizon
position. In
practice these variables are usually only known on the (3-dimensional) numerical grid of the underlying numerical-relativity
simulation,
so they must be interpolated to the horizon position and, more generally, to the position of each
intermediate-iterate trial shape the apparent horizon finding algorithm tries in the process of (hopefully)
converging to the horizon position.
Moreover, usually the underlying simulation gives only and , so must be numerically
differentiated to obtain . As discussed by Thornburg [156, Section 6.1], it is somewhat more efficient to
combine the numerical differentiation and interpolation operations, essentially doing the differentiation inside the
interpolator.

Thornburg [156, Section 6.1] argues that for an elliptic-PDE algorithm (Section 8.5), for best
convergence of the nonlinear elliptic solver the interpolated geometry variables should be smooth
(differentiable) functions of the trial horizon surface position. He argues that that the usual
Lagrange polynomial interpolation does not suffice here (in some cases his Newton’s-method
iteration failed to converge) because this interpolation gives results which are only piecewise
differentiable.
He uses Hermite polynomial interpolation to avoid this problem. Cook and Abrahams [51], and
Pfeiffer et al. [124] use bicubic spline interpolation; most other researchers either do not describe their
interpolation scheme or use Lagrange polynomial interpolation.