Consequently we need to compute integrals over primitives(which can be evaluated analyti cally) and integrals involving intersections of primitives AoB fdV which can be approximated using a ray casting, ray classification and integral approximation method the boundary representation, which is the most generally used representation today, represents object in terms of their boundary elements (e.g. vertices, edges, faces). For evaluating the eorems from vector calculus are useful 7 1. Green's Theorem If C is a piecewise smooth, simple closed curve that bounds a region R, and if P(a, y) and Q(, y are continuous functions which have continuous partial derivatives along C and throughout R. then (Pd.r+Qdy dQ aP da (196 2. Divergence Theorem(also called Gauss'Theorem The flux of vector field F Howing outward through a closed surface S equals the integral of the divergence of F over the region R bounded by S ∥ F. ndA VDv where n is the outward unit normal vector and V·F k 19 where i,j, k are the unit coordinate vectors Ba In the sequel we will apply these theorems to compute the integral properties of geometric dels represented by the B-rep method (in 1-3 dimensions) 19.6 Integral properties of curves 1 9.6. 1 Planar curves Let a planar curve be defined by r=(x(t),y(t),to≤t≤t1 (19.9) ngth L dt (19.10) V=(0)+(t 15Consequently we need to compute integrals over primitives (which can be evaluated analyti￾cally) and integrals involving intersections of primitives R A∩B fdV which can be approximated using a ray casting, ray classification and integral approximation method. Boundary representation, which is the most generally used representation today, represents the object in terms of their boundary elements (e.g. vertices, edges, faces). For evaluating the integral properties for B-rep solids, the following theorems from vector calculus are useful [7]: 1. Green’s Theorem If C is a piecewise smooth, simple closed curve that bounds a region R, and if P(x, y) and Q(x, y) are continuous functions which have continuous partial derivatives along C and throughout R, then I C (Pdx + Qdy) = ZZ R ∂Q ∂x − ∂P ∂y ! dA (19.6) 2. Divergence Theorem (also called Gauss’ Theorem) The flux of vector field F flowing outward through a closed surface S equals the integral of the divergence of F over the region R bounded by S; ZZ S F · ndA = ZZZ R ∇FdV (19.7) where n is the outward unit normal vector and ∇ · F = ∂F ∂x · i + ∂F ∂y · j + ∂F ∂z · k. (19.8) where i, j, k are the unit coordinate vectors. In the sequel we will apply these theorems to compute the integral properties of geometric models represented by the B-rep method (in 1-3 dimensions). 19.6 Integral properties of curves 19.6.1 Planar curves Let a planar curve be defined by r = (x(t), y(t)), to ≤ t ≤ t1 (19.9) • Length L = Z s1 s0 ds = Z t1 t0 q r˙(t) · r˙(t)dt (19.10) = Z t1 t0 q x˙ 2(t) + y˙ 2(t)dt 15