The implicitization problem asks for techniques to find the closed image of a rational map to projective space. This is an old problem in algebraic geometry, and while solved by Groebner bases, it has received renewed attention since the 90s. New approaches address the relation between the algebra of the "coordinates" of the map and the geometry of its base locus and image.

We will first present the two rather different main attempts — the method of moving surfaces and the approximation complex. We then introduce a family of matrices derived from the Rees algebra (or the normal cone) which surprisingly unify the latter two and give a good candidate for a tool for systematic study.

Throughout the talk we shall take a look at a few very concrete and rather interesting examples which highlight the geometric nature of the construction and its immunity to some of the other approaches' weaknesses.