Geometry, Duality and Robust Computation in Engineering VACLAV SKALA Department of Computer Science and Engineering Faculty of Applied Sciences, University of West Bohemia Univerzitni 8, CZ 306 14 Plzen Czech Republic skala@kiv.zcu.cz http://www.VaclavSkala.eu Abstract: - Robustness of computations in engineering is one of key issues as it is necessary to solve technical problems leading to ill conditioned solutions. Therefore the robustness and numerical stability is becoming a key issue more important that the computational time. In this paper we will show selected computational issues in numerical precision, well known cases of failures in computations. The Euclidean representation is used in today’s computations, however the projective space (an extension of the Euclidean space) representation leads to more compact and robust formulations and to matrix-vector operations supported in hardware, e.g. by GPU. Key-Words: - Euclidean space, projective space, homogeneous coordinates, duality, intersections, barycentric coordinates, planes intersection, Plucker coordinates, numerical precision. 1 Introduction Data processing is one of the main fields in computer science. Data processing itself can be split to two main areas: • processing of textual data • processing of numerical data Nowadays, computers use binary system for information and data representation. We use octal or hexadecimal representation for data representation. If we would be direct descendants of tetrapods we would have a great advantage as they had 8 fingers on a hand, see Fig.1. However, we have 5 fingers at a hand and use a decimal numeral system and for computation we use numbers with a decimal point, rational and irrational ones. The above mentioned main two areas are quite different, but have many common algorithms, e.g. hashing. In the case of textual data we have “unlimited” dimensionality (”unlimited” length of a string) but limited interval of values (usually given by a number of symbols in the given alphabet). On the contrary in the case of numerical or geometrical data we have a limited dimensionality (usually 2 or 3 in the case of E 2 or E 3 ) but “unlimited” interval of values (usually (-∞, ∞)). In the case of hashing techniques it lead us to a “unified” approach of hashing, but different construction and specification of the hash function used [8], [36], [37]. Name Base Digits E min E max BINARY B 16 Half 2 10+1 −14 15 B 32 Single 2 23+1 −126 127 B 64 Double 2 52+1 −1022 1023 B 128 Quad 2 112+1 −16382 16383 DECIMAL D 32 10 7 −95 96 D 64 10 16 −383 384 D 128 10 34 −6143 6144 IEEE 758-2008 standard Table 1 Figure 1 WSEAS Trans.on Computers, Vol.11, No.9, ISSN 1109-2742, E-ISSN 2224-2872, pp.275-291, 2012