Polynomial/Rational Approximation of Minkowski Sum Boundary Curves

Anomaly Detection Based on Zone Partition for Security Protection of Industrial Cyber-Physical Systems

“…where '⊕' denotes the minkowski sum [36] which means that each set in the expression is regarded as a mutual independent element, such as c s , {e 1 s }, {s} and so on, and the state of the node s can be described by any element in d s . The detectableset for the basic structure in Fig.…”

Rational surfaces with linear normals and their convolutions with rational surfaces

“…Applications include motion planning for polygonal objects in the presence of polygonal obstacles. Later, these concepts have been generalized to arbitrary shapes in the plane and in space, see (Lee et al, 1998a(Lee et al, , 1998bKaul and Farouki, 1995;Mühlthaler and Pottmann, 2003;Peternell and Manhart, 2003), where the notion of the convolution of two (not necessarily convex) objects has been introduced. 1 Now we consider two regular surfaces A and B in three-dimensional space, which are given by parametric representations a(u, v) and b(s, t) with parameter domains (u, v) ∈ Ω A ⊆ R 2 and (s, t) ∈ Ω B ⊆ R 2 , respectively.…”

“…In the curve case, various algorithms for computing Minkowski sums exist (Kaul and Farouki, 1995;Kohler and Spreng, 1995;Lee et al, 1998aLee et al, , 1998bRamkumar, 1996;Farouki, 2003). The main issue is to trim away those parts of the convolution curve that do not contribute to the outer boundary of the Minkowski sum.…”

Contributing vertices-based Minkowski sum computation of convex polyhedra

“…Minkowski sum implementation is of a particular interest and used in a variety of domains such as computer-aided design and manufacturing [2], computer animation and morphing [3], morphological image analysis [4,5], similarity measures for convex polyhedra [6], penetration depth computation and dynamic simulation [7], robot motion planning [8], and solid modeling. (facets), one-dimensional faces (edges), and zero-dimensional facets (vertices).…”