10.4.3 Point/Procedural parametric (offset, evolute, etc. ) curve intersection Ro∩R=R(t)A≤t≤B In general there is no known and easily computable convex box decreasing arbitrarily with subdivision! An approximate solution method may involve minimization of F(t=R(t-R where tE [A, B]. This would involve Checking end points, ie. if F(A), F(B) are very small - Initial estimate for the possible minima, perhaps using linear approximation of r(t) to start the process However Convergence of the above minimization processes is not guaranteed in general There may exist more than one minima Convergence to local and not global minimum(where F(t)+0)is possible For certain classes of procedural curves such as offsets and evolutes of rational curves involving radicals of polynomials, it is possible to use the"auxiliary variable method"to reduce the point to curve intersection(or minimum distance) problem to a set of (a larger number of) nonlinear polynomial equations. Such systems can be solved robustly and efficiently using the nonlinear solver describe in the next section 1210.4.3 Point/Procedural parametric (offset, evolute, etc.) curve intersection R0 ∩ R = R(t) A ≤ t ≤ B • In general there is no known and easily computable convex box decreasing arbitrarily with subdivision! • An approximate solution method may involve minimization of F(t) = |R(t) − R| 2 where t ∈ [A, B]. This would involve – Checking end points, ie. if F(A), F(B) are very small. – Initial estimate for the possible minima, perhaps using linear approximation of R(t) to start the process. However, – Convergence of the above minimization processes is not guaranteed in general. – There may exist more than one minima. – Convergence to local and not global minimum (where F(t) 6= 0 ) is possible. For certain classes of procedural curves such as offsets and evolutes of rational curves involving radicals of polynomials, it is possible to use the “auxiliary variable method” to reduce the point to curve intersection (or minimum distance) problem to a set of (a larger number of) nonlinear polynomial equations. Such systems can be solved robustly and efficiently using the nonlinear solver describe in the next section. 12