The ``ab initio'' approach to geometry led us in a natural way to coordinate geometry. Thus algebra (and arithmetic) play a very important role in geometry. Algebra is very useful since we can discuss curves and other objects with much greater ease using coordinates. This is not to say that everything in geometry cannot be discussed without bringing in algebra. One can discuss ``loci''; each locus is thought of as a geometric figure ``traced'' by a point moving according to some specific geometric constraint. We will discuss how conics can be described in this way. Though this description is beautiful, it also shows how limited we are if we ignore algebraic techniques. Thus we shall bring in some algebra to study curves and then surfaces. In particular, we shall discuss whether curves are really curved and the curvature of a surfaces is superficial!

Kapil H. Paranjape 2001-01-20