The numerics in computational geometry are used for making combinatorial decisions (do three points make a left or a right turn in the plane; is point p inside

The numerics in computational geometry are used for making combinatorial decisions (do three points make a left or a right turn in the plane; is point p inside or outside of the circumsphere defined by these three points; etc). These are called predicates in the CG literature. What makes this difficult is that multiple different predicates interact (e.g., take the three points in different order), and the answers they give need to be consistent. If you want some nice examples of how very simple things can go wrong, see L. Kettner, K. Mehlhorn, S. Pion, S. Schirra, and C. Yap, “Classroom examples of robustness problems in geometric computations,” Comput. Geom., vol. 40, no. 1, pp. 61–78, 2008. Most predicates boil down to a decision of whether some quantity is less than, equal, or greater than zero. CGAL implements filtered predicates — a work of art in my opinion — where they use ordinary computation and if the result is far enough away from zero, return its sign. If not, they switch to higher precision or interval arithmetic. A good explanation of how this works (what "far enough from zero" actually means) is in O. Devillers and S. Pion, “Efficient Exact Geometric Predicates for Delaunay Triangulations,” in Proceedings of the 5th Workshop on Algorithm Engineering and Experiments, pp. 37–44, 2003. The larger problem of symbolically perturbing input, so that it's in general position (in a sense, what to do if your predicate returns 0) was a very active research topic in the 90s, and both the problem and much of the work is explained really well in R. Seidel, “The Nature and Meaning of Perturbations in Geometric Computing,” Discrete Comput. Geom., vol. 19, no. 1, pp. 1–17, 1998.