Towards implementing robust geometric computations

Validation and Automatic Repair of Planar Partitions Using a Constrained Triangulation

“…Additionally, robustness issues are an important issue in polygon intersection tests [Hoffmann et al, 1988].…”

The intrinsic spread of a configuration in 𝑅^{𝑑}

“…A few years ago these observations led Bernard Chazelle to pose the problem of how large a grid was needed to accommodate all simple planar npoint configurations up to order type [4]. An answer to Chazelle's question is relevant to the computational problem of accurately representing configurations of points and arrangements of lines [6] in an environment of finite precision arithmetic; see also [5,11,14,18,20], in which the problem of finding robust geometric algorithms in such an environment is addressed. In this paper we solve Chazelle's problem by proving In §2 we establish the lower bound by first constructing a "rigid" configuration that is very spread out in the intuitive sense, then modify it via a recent construction of [15] to a configuration of points in general position which achieves at least the same spread in every realization.…”

Epsilon geometry: building robust algorithms from imprecise computations

“…Note that if the algorithm cannot decide between any two cases, it will simply merge the corresponding uncertainty intervals. 3 Approximate Point Inclusion…”

“…Our approach is more similar to those of Milenkovic [S] and of Hoffmann, Hopcroft, and Karasick [3]. These methods compute an exact result for a perturbed version of the input data, but they assume a perturbation bounded by a constant chosen a priori.…”