This report introduces a geometrical constraint kernel for handling the location in space and time of polymorphic k-dimensional objects subject to various geometrical and time constraints. The constraint kernel is generic in the sense that one of its parameters is a set of constraints on subsets of the objects. These constraints are handled globally by the kernel. We first illustrate how to model several placement problems with the constraint kernel. We then explain how new constraints can be introduced and plugged into the kernel. Based on these interfaces, we develop a generic k-dimensional lexicographic sweep algorithm for filtering the attributes of an object (i.e., its shape and the coordinates of its origin as well as its start, duration and end in time) according to all constraints where the object occurs. Experiments involving up to hundreds of thousands of objects and 1 million integer variables are provided in 2, 3 and 4 dimensions, both for simple shapes (i.e., rectangles, parallelepipeds) and for more complex shapes.