Integral properties of geometric models 19.5 Introduction One of the important advantages of using a CAD model for representing and designing an object is that we can easily compute the integral properties of such models such as edge curves ies include length volume. These are very useful in preliminary design. For example, surface area affects drag, volume affects the carrying capacity of a vehicle, centroids are useful in hydrostatic balance moments of inertia are used in dynamics and in hydrostatic stability calculation(for ships Computation of the integral properties of curves, surface patches and solids involves eval uation of single, double and triple integrals of the form f(P)dL ∫(P)dS, solid ∫(P)dV(19.2 where o is the required property, P is a point and f is a real-valued function, which depends on the type of property required. We have studied three classes of solid representation methods in the previous chapters, namely Decomposition methods, Constructive Solid Geometry(CSG methods, and Boundary Representation(B-rep) methods For decomposition methods, the integral over the solid reduces to a sum of integrals fdV=∑ Cell fav 193 solid where celli is the i-th cell which is either full or partially full. For the case of exhaustive spatial enumeration the cells are constant-sized cubes and for the octree decompositions they are variable-sized cubes [10, 13. For high resolution models it is enough to consider all the celli to be full entirely and for those cases the resulting integrals are elementary and can be computed using simple analytic As we have studied in the previous chapters, CSG is a tree whose nodes represent the Boolean operators and the leaves are the primitive solids. Therefore the computation of integral properties of CSG solids consists of applying the following formula recursively [10 fdv+/fdv dv f dv 19 14Integral properties of geometric models 19.5 Introduction One of the important advantages of using a CAD model for representing and designing an object is that we can easily compute the integral properties of such models such as edge curves, faces and volumes. Integral properties include length, area, centroid, moment of inertia, and volume. These are very useful in preliminary design. For example, surface area affects drag, volume affects the carrying capacity of a vehicle, centroids are useful in hydrostatic balance, moments of inertia are used in dynamics and in hydrostatic stability calculation (for ships). Computation of the integral properties of curves, surface patches and solids involves eval￾uation of single, double and triple integrals of the form φcurve = Z curve f(P)dL, φsurface = Z surface f(P)dS, φsolid = Z solid f(P)dV (19.2) where φ is the required property, P is a point and f is a real-valued function, which depends on the type of property required. We have studied three classes of solid representation methods in the previous chapters, namely Decomposition methods, Constructive Solid Geometry (CSG) methods, and Boundary Representation (B-rep) methods. For decomposition methods, the integral over the solid reduces to a sum of integrals Z solid fdV = X i Z celli fdV (19.3) where celli is the i − th cell which is either full or partially full. For the case of exhaustive spatial enumeration the cells are constant-sized cubes and for the octree decompositions they are variable-sized cubes [10, 13]. For high resolution models it is enough to consider all the celli to be full entirely and for those cases the resulting integrals are elementary and can be computed using simple analytic forms. As we have studied in the previous chapters, CSG is a tree whose nodes represent the Boolean operators and the leaves are the primitive solids. Therefore the computation of integral properties of CSG solids consists of applying the following formula recursively [10]: Z A∪B fdV = Z A fdV + Z B fdV − Z A∩B fdV (19.4) Z A−B fdV = Z A fdV − Z A∩B fdV. (19.5) 14