Verifiable implementations of geometric algorithms using finite precision arithmetic

Connectivity in the regular polytope representation

“…The conventional approach to this issue is the use of topological encoding of regions and networks [7][8][9], combined with a form of normalisation to prevent situations arising where imprecise calculations can cause difficulties [10]. This normalisation is often combined with the validation process, but may in itself lead to breakdown of the spatial algebra.…”

Topologically exact evaluation of polyhedra defined in CSG with loose primitives

“…The result is the intersection of polylines that approximate the original line segments, but have the same topology. A similar approach is used in Milenkovic 30 for computing the intersections of polygons on two dimensions, but instead of vertex snapping, line coefficients are rounded-off to fixed precision. The idea of producing a consistent representation that is "close" to the correct one was also used by Fortune for robustly computing convex hulls and triangulations in two dimensions 31 .…”

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.…”