Computational synthetic geometry deals with methods for realizing abstract geometric objects in
concrete vector spaces. This research monograph considers a large class of problems from
convexity and discrete geometry including constructing convex polytopes from simplicial
complexes vector geometries from incidence structures and hyperplane arrangements from
oriented matroids. It turns out that algorithms for these constructions exist if and only if
arbitrary polynomial equations are decidable with respect to the underlying field. Besides such
complexity theorems a variety of symbolic algorithms are discussed and the methods are applied
to obtain new mathematical results on convex polytopes projective configurations and the
combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and
oriented matroids are introduced providing a new basis for applying computer algebra methods in
this field. The necessary background knowledge is reviewed briefly. The text is accessible to
students with graduate level background in mathematics and will serve professional geometers
and computer scientists as an introduction and motivation for further research.
