Geometric Reasoning and Artificial Intelligence: Introduction to the Special Volume

Geometry plays an important role in our understanding of the world. It is therefore important for Artificial Intelligence (AI) systems to be able to represent and reason about geometry. This special volume presents a formal approach to the representation of geometric concepts in selected AI applications and describes some recent developments in algebraic methods for reasoning about geometry. Geometry is also the oldest subject in mathematics to be subjected to considerable rigor and formalism. The axiomatic treatment of geometry has been actively investigated for thousands of years, starting with Euclid. In the early 1950s and 1960s, when there was considerable excitement about theorem proving in AI, Gelernter and his associates [14, 15] initiated an effort in mechanical theorem proving in geometry. After some initial success, Gelernter's approach faced the stumbling block of combinatorial search space, even for relatively simple plane geometry theorems. Gelernter's approach towards mechanical geometry theorem proving was synthetic, i.e., it used properties of points, lines, triangles, etc. He employed the properties and definitions of these key geometry concepts in plane geometry to prove theorems using diagram and geometric construction. Earlier in the