Lectures 10-12 Intersection Problems, Nonlinear Solvers and robustness issues 10.1 Overview of intersection problems Intersections are fundamental in computational geometry, geometric modeling and design analysis and manufacturing applications. Examples of intersection problems include Shape interrogation(eg. visualization) through contouring (intersection with a series of parallel planes, coaxial cylinders, and cones etc. Numerical control machining (milling) involving intersection of offset surfaces with a series of parallel planes, to create machining paths for ball(spherical) cutters Representation of complex geometries in the "Boundary Representation"scheme; for example, the description of the internal geometry and of structural members cars airplanes, ships, etc, involves Intersections of free-form parametric surfaces with low order algebraic surfaces planes, quadrics, torii Intersections of low order algebraic surfaces in a process called boundary evaluation, in which the Boundary Representation is cre- ated by "evaluating " the Constructive Solid Geometry model of the object. during this process, intersections of the surfaces of primitives(see Figure 10.1)must be found during Boolean operations Boolean opertations on point sets A, B include(see Figure 10.2) Union:A∪B . Intersection: An B. and Difference:A-B 3Lectures 10 - 12 Intersection Problems, Nonlinear Solvers and Robustness Issues 10.1 Overview of intersection problems Intersections are fundamental in computational geometry, geometric modeling and design, analysis and manufacturing applications. Examples of intersection problems include: • Shape interrogation (eg. visualization) through contouring (intersection with a series of parallel planes, coaxial cylinders, and cones etc.) • Numerical control machining (milling) involving intersection of offset surfaces with a series of parallel planes, to create machining paths for ball (spherical) cutters. • Representation of complex geometries in the “Boundary Representation” scheme; for example, the description of the internal geometry and of structural members of cars, airplanes, ships, etc, involves – Intersections of free-form parametric surfaces with low order algebraic surfaces (planes, quadrics, torii). – Intersections of low order algebraic surfaces. in a process called boundary evaluation, in which the Boundary Representation is cre￾ated by “evaluating” the Constructive Solid Geometry model of the object. During this process, intersections of the surfaces of primitives (see Figure 10.1) must be found during Boolean operations. Boolean opertations on point sets A, B include (see Figure 10.2) • Union: A ∪ B, • Intersection: A ∩ B, and • Difference: A − B. 3