(a) Algebraic(polynomial) f(R)=g(R)=0 space curve 2=0, f(,y)=0 planar curves (b)Procedural offsets(eg. non-constant distance offsets involving convolution, see Pottmann 1997) 3. Surfaces Parametric R= R(u, v) where u, v vary in some finite domain, the parametric space (a)(Rational) Polynomial(eg: NURBS, Bezier, rational Bezier etc. (b)Procedural offsets blends generalIzed cylinders Implicit: Algebraic f(R)=0 where f is a polynomial 10.2.3 Classification by number system In our discussion of intersection problems, we will refer to various classes of numbers · integer numbers rational numbers, m/n, n+0, where m, n are integers floating point numbers in a computer(which are a subset of the rational numbers) radicals of rational numbers, eg. Vm/n,n+0, where m, n are integers algebraic numbers (roots of polynomials with integer coefficients transcendental (e, T, trigonometric, etc. ) numbers ●re imber interval numbers,(a, b], where a, b are real numbers; rounded interval numbers, [c, d], where c, d are floating point numbers Issues relating to floating point and interval numbers affecting the robustness of intersection Igorithms are addressed in the next section on nonlinear solvers(a) Algebraics (polynomial) f(R) = g(R) = 0 space curves z = 0, f(x, y) = 0 planar curves (b) Procedural offsets (eg. non-constant distance offsets involving convolution, see Pottmann 1997) 3. Surfaces • Parametric R = R(u, v) where u, v vary in some finite domain, the parametric space. (a) (Rational) Polynomial (eg: NURBS, B´ezier, rational B´ezier etc.) (b) Procedural – offsets – blends – generalized cylinders • Implicit: Algebraics f(R) = 0 where f is a polynomial. 10.2.3 Classification by number system In our discussion of intersection problems, we will refer to various classes of numbers: • integer numbers; • rational numbers, m/n, n 6= 0, where m, n are integers; • floating point numbers in a computer (which are a subset of the rational numbers); • radicals of rational numbers, eg. q m/n, n 6= 0, where m, n are integers; • algebraic numbers (roots of polynomials with integer coefficients); • transcendental (e, π, trigonometric, etc.) numbers. • real numbers; • interval numbers, [a, b], where a, b are real numbers; • rounded interval numbers, [c, d], where c, d are floating point numbers. Issues relating to floating point and interval numbers affecting the robustness of intersection algorithms are addressed in the next section on nonlinear solvers. 6