  We have developed in the past several algorithms with intrinsic complexity
bounds for the problem of point finding in real algebraic varieties. Our aim
here is to give a comprehensive presentation of the geometrical tools which are
necessary to prove the correctness and complexity estimates of these
algorithms. Our results form also the geometrical main ingredients for the
computational treatment of singular hypersurfaces.
  In particular, we show the non--emptiness of suitable generic dual polar
varieties of (possibly singular) real varieties, show that generic polar
varieties may become singular at smooth points of the original variety and
exhibit a sufficient criterion when this is not the case. Further, we introduce
the new concept of meagerly generic polar varieties and give a degree estimate
for them in terms of the degrees of generic polar varieties.
  The statements are illustrated by examples and a computer experiment.
