Cours de robustesse a Nancy
46 Pages
English

Cours de robustesse a Nancy

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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)

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Language English
Robustness issues in arithmetics and the
Sylvain Pion
INRIA Sophia Antipolis
CGAL kernel
:
Links between geometry and
Floating point arithmetic
Exact arithmetic
Arithmetic filters
CGAL implementation
Plan
arithmetics
1
Intro
duction
2
Examples of geometric predicates
orientation(p, q r) = , sign((x(p)x(r))×(y(q)y(r))(x(q)x(r))×(y(p)y(r)))
Predicate of
Sylvain Pion
degree 2.
3
Sylvain
Pion
Examples
of
geometric
constructions
4
From geometry to arithmetic
Geometric algorithm
Sylvain Pion
Geometric operations (predicates and constructions)
Algebraic operations over coordinates/coefficient . . .)
Arithmetic operations (+, ,×,÷,
5
ArithmeticGeometry
Costof arithmeticTimecomplexityof geometric algorithms
Approximatethritimecassenortsubproblems of geometric algorithms
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6
The Real-RAM model
Real computer model with random access (RAM = Random access machine).
Theoretical model specifying the behavior of real arithmetic on computers.
All arithmeticoperationsover reals costO(1) time(and are exact).
All real variables takeO(1) memory space.
Complexity analyses of geometric algorithms are traditionnaly performed within this model.
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7
Relationship with the reality of computers ?
Two approaches :
Floating point arithmetic,approximate.
Exact arithmetic,slower.
For geometry : which approach is the best in practice ?
What is the precise cost of the exact approach ?
8
Floating
point
arithmetic
9
IEEE 754 Standard
Standardization of basic FP operations on computers (1985). Machine representation of(1)s×1.m×2e(for double precision, 64 bits):
s exponent 1 11
5 operations:+,,×,÷,
mantissa 52
4 rounding modes: to nearest (representable number), towards0, towards +, towards−∞.
Special values:+,−∞, denormals, NaNs.
supported by the industry (languages, compilers, processors).Relatively well
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Sylvain Pion
10