A planar offset curve r(t) with signed offset distance d to the progenitor r(t) is defined by r(t=r(t)+dn(t) (13.5) where d>0 corresponds to positive("exterior")and d<0 corresponds to negative The unit tangent vector of the offset curve(see Figure 13.7 for illustration) r 1+kd i1+ The unit normal vector of the offset curve(see Figure 13.7 for illustration) n=t xez 1+d (137) . Curvature of the offset curve 13.2.2 Singularities of parametric offset curves There are two kinds of singularities on the offset curves, irregular points and self-intersections Irregular points Isolated points: This point occurs when the progenitor curve with radius R is a circle and the offset is d Cusps: This point occurs at a point t where the tangent vector vanishes d (13.9) A cusp at t=te can be further subdivided into 7 1. Ordinary cusps when A(tc)#0 2. Extraordinary points when i(tc)=0 and k(tc)#0 Note that(1+Kd)/1+nd in equations(13.6)and(13.7) changes abruptly from-1 to 1 Equation(13.9)for r(t)=a(t), y(t)) can be reduced to Cusp, while at extraordinary when the parameter t passes through t= tc at an ordinary points(1+nd)/1+ nd does not change its value, see Fi e13.7. 1(0(6-y/2(+2()(2()+0(=0• A planar offset curve ˆr(t) with signed offset distance d to the progenitor r(t) is defined by ˆr(t) = r(t) + dn(t) (13.5) where d > 0 corresponds to positive (“exterior”) and d < 0 corresponds to negative (“interior”) offsets. • The unit tangent vector of the offset curve (see Figure 13.7 for illustration) ˆt = ˙ˆr | ˙ˆr| = 1 + κd |1 + κd| t (13.6) • The unit normal vector of the offset curve (see Figure 13.7 for illustration) nˆ = ˆt × ez = 1 + κd |1 + κd| n (13.7) • Curvature of the offset curve κˆ = κ |1 + κd| (13.8) 13.2.2 Singularities of parametric offset curves There are two kinds of singularities on the offset curves, irregular points and self-intersections. • Irregular points Isolated points: This point occurs when the progenitor curve with radius R is a circle and the offset is d = −R. Cusps: This point occurs at a point t where the tangent vector vanishes. κ(t) = − 1 d (13.9) A cusp at t = tc can be further subdivided into [7]: 1. Ordinary cusps when κ˙(tc) 6= 0 2. Extraordinary points when κ˙(tc) = 0 and κ¨(tc) 6= 0. Note that (1 + κd)/|1 + κd| in equations (13.6) and (13.7) changes abruptly from -1 to 1 when the parameter t passes through t = tc at an ordinary cusp, while at extraordinary points (1 + κd)/|1 + κd| does not change its value, see Figure 13.7. Equation (13.9) for r(t) = {x(t), y(t)} can be reduced to d [x¨(t)y˙(t) − x˙(t)y¨(t)] − q x˙ 2(t) + y˙ 2(t) h x˙ 2 (t) + y˙ 2 (t) i = 0 (13.10) 6