Robustness issues in CGAL :arithmetics and the kernelSylvain PionINRIA Sophia AntipolisPlan Links between geometry and arithmetics Floating point arithmetic Exact arithmetic Arithmetic lters CGAL implementation1Introduction2Examples of geometric predicatespositiveC1orientation pqC2 C’1r p’C’2p negativeorientationx(p) x(p’) xorientation(p,q,r) =sign((x(p) x(r))(y(q) y(r)) (x(q) x(r))(y(p) y(r)))Predicate of degree 2.Sylvain Pion 3Examples of geometric constructionsyp C1py(p)C2c rqO x(p) xSylvain Pion 4From geometry to arithmeticGeometric algorithm⇒ Geometric operations (predicates and constructions)⇒ Algebraic operations over coordinates/coe cients√⇒ Arithmetic operations (+, ,,, ...)Sylvain Pion 5Arithmetic ⇒ GeometryCost of arithmetic⇒ Time complexity of geometric algorithmsApproximate arithmetic⇒ robustness problems of geometric algorithmsSylvain Pion 6The Real-RAM modelReal computer model with random access (RAM = Random access machine).Theoretical model specifying the behavior of real arithmetic on computers. All arithmetic operations over reals cost O(1) time (and are exact). All real variables take O(1) memory space.Complexity analyses of geometric algorithms are traditionnaly performed withinthis model.Sylvain Pion 7Relationship with the reality of computers ?Two approaches : Floating point arithmetic, approximate. Exact arithmetic, slower.For geometry : which approach is the best in ...