An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.

Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in O(n log n) time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time. 1

The domain-theoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is computable if it is the effective limit of a sequence of finitary partial objects of the same type as the original object. We are also provided with two different quantitative measures for approximation using the Hausdorff metric and the Lebesgue measure. In this thesis, we introduce a new data type to capture imprecise data or approximate points on the plane, given in the shape of compact convex polygons. This data type in particular includes rectangular approximation and is invariant under linear transformations of coordinate system. Based on the new data type, we define the notion of a number of partial geometric operations, including partial perpendicular bisector and partial disc and we show that these operations and the convex hull, Delaunay triangulation and Voronoi diagram are Hausdorff and Scott continuous and nestedly Hausdorff and Lebesgue computable. We develop algorithms to obtain the partial convex hull, partial Delaunay triangulation and partial Voronoi diagram. We prove that the complexity of the partial convex hull is N log N in 2D and 3D, whereas the partial Delaunay triangulation and partial Voronoi diagram algorithms for non-degenerate data have the same complexity as their classical counterparts. 2