  We examine implicit representations of parametric or point cloud models,
based on interpolation matrices, which are not sensitive to base points. We
show how interpolation matrices can be used for ray shooting of a parametric
ray with a surface patch, including the case of high-multiplicity
intersections. Most matrix operations are executed during pre-processing since
they solely depend on the surface. For a given ray, the bottleneck is equation
solving. Our Maple code handles bicubic patches in $\le 1$~sec, though
numerical issues might arise. Our second contribution is to extend the method
to parametric space curves and, generally, to codimension $> 1$, by computing
the equations of (hyper)surfaces intersecting precisely at the given object. By
means of Chow forms, we propose a new, practical, randomized algorithm that
always produces correct output but possibly with a non-minimal number of
surfaces. For space curves, we obtain 3 surfaces whose polynomials are of
near-optimal degree; in this case, computation reduces to a Sylvester
resultant. We illustrate our algorithm through a series of examples and compare
our Maple prototype with other methods implemented in Maple i.e., Groebner
basis and implicit matrix representations. Our Maple prototype is not faster
but yields fewer equations and seems more robust than Maple's {\tt
implicitize}; it is also comparable with the other methods for degrees up to 6.
