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
Figure 13.7: Offsets to a parabola r=[t, t(thick solid line)with offsets d=-03,-05,-08, dapted from [5. At d=-03 the tangent and normal vectors of the offset have the same sense that of the progenitor, while at d=-08 they flip directions By setting T=i+y and if r(t) is a rational polynomial curve, the computation of cusps can be reduced to system of two nonlinear polynomial equations that can be solved using the methods of Chapter 10 Examples(see Figures 13.7 and 13.8) Given a parabola=(t, t), the unit tangent and principal normal vectors are given by dr dr dt (1, 2t) V1+42' n=txe (2,-1 dt ds The curvature and its derivative are given ()=xE rP(1+412)2 (t) 24(1+42) (1+4t2)3 Since A(0)=0, k(t) reaches an extremum at t=0 and furthermore as K(0)<0, k(O)is a maximum with a curvature value K(0)=2. Therefore when d> t the offset is non- degenerate, while when d=-i,t=0 is an extraordinary point. Let us solve k(t)=-1/d t which yield 3√4d-1 We can easily see that if d>-1/2, there is no real root. This means that there singularity as long as radius of curvature is smaller than 2. If d=-1/2, there exists a single root t=0, while if d <-1 /2 there exist two symmetric values t1, t2 Self-intersections
Figure 13.7: Offsets to a parabola r = [t,t 2 ] (thick solid line) with offsets d=-0.3, -0.5, -0.8, adapted from [5]. At d = −0.3 the tangent and normal vectors of the offset have the same sense that of the progenitor, while at d = −0.8 they flip directions. By setting τ 2 = x˙ 2 + y˙ 2 and if r(t) is a rational polynomial curve, the computation of cusps can be reduced to system of two nonlinear polynomial equations that can be solved using the methods of Chapter 10. Examples (see Figures 13.7 and 13.8) Given a parabola r = (t,t 2 ), the unit tangent and principal normal vectors are given by t = dr ds = dr dt dt ds = (1, 2t) √ 1 + 4t 2 , n = t × ez = (2t, −1) √ 1 + 4t 2 The curvature and its derivative are given by κ(t) = (r˙ × ¨r) · ez |r˙| 3 = 2 (1 + 4t 2) 3 2 , κ˙(t) = −24t(1 + 4t 2 ) 1 2 (1 + 4t 2) 3 Since κ˙(0) = 0, κ(t) reaches an extremum at t = 0 and furthermore as κ¨(0) < 0, κ(0) is a maximum with a curvature value κ(0) = 2. Therefore when d > − 1 2 the offset is nondegenerate, while when d = − 1 2 , t = 0 is an extraordinary point. Let us solve κ(t) = −1/d for t which yields t = ± q 3 √ 4d 2 − 1 2 . We can easily see that if d > −1/2, there is no real root. This means that there is no singularity as long as radius of curvature is smaller than 2. If d = −1/2, there exists a single root t = 0, while if d < −1/2 there exist two symmetric values t1, t2. • Self-intersections 7
Self-intersections of an offset curve(see also Figures 13.7 and 13. 8)can be obtained by seeking pairs of distinct parameter values sf t such that r(s+dn(s=r(t)+dn(t) (13.11) Substitution of equation(13. 2)in(13. 11) yields the system [17] r(s)+、()+y(=()+-i(d i(sd √x(t)+y2(t) (t)d 以()()+(5=0)-√( (13.12) If r(t)is a rational polynomial curve, this system can be converted to a nonlinear poly- nomial system of four equations in four variables s, t, T and o where r2=i2(s)+2( 2=i2(t)+j2(t) (13.14) Such a system can be solved using the IPP algorithm, see also [17. However s are trivial solutions, and we must exclude them from the system, otherwise a Bernstein subdivision-based algorithm would attempt to solve for an infinite number of roots. In this case we have addressed the problem by dividing out the common factor by some Figure 13.8: Self-intersection of the offset curve of a parabola. Left: Interior offsets to the parabola r(t)=t, t] with d=-08 and cutter path; Right: Trimmed interior offsets to the parabola r(t)=[t, t2] with d=-0.8 and cutter path
Self-intersections of an offset curve (see also Figures 13.7 and 13.8) can be obtained by seeking pairs of distinct parameter values s 6= t such that r(s) + dn(s) = r(t) + dn(t). (13.11) Substitution of equation (13.2) in (13.11) yields the system [17] x(s) + y˙(s)d p x˙ 2(s) + y˙ 2(s) = x(t) + y˙(t)d p x˙ 2(t) + y˙ 2(t) y(s) − x˙(s)d p x˙ 2(s) + y˙ 2(s) = y(t) − x˙(t)d p x˙ 2(t) + y˙ 2(t) (13.12) If r(t) is a rational polynomial curve, this system can be converted to a nonlinear polynomial system of four equations in four variables s, t, τ and σ where τ 2 = x˙ 2 (s) + y˙ 2 (s) (13.13) σ 2 = x˙ 2 (t) + y˙ 2 (t). (13.14) Such a system can be solved using the IPP algorithm, see also [17]. However s = t are trivial solutions, and we must exclude them from the system, otherwise a Bernstein subdivision-based algorithm would attempt to solve for an infinite number of roots. In this case we have addressed the problem by dividing out the common factor by some algebraic manipulations [17]. Figure 13.8: Self-intersection of the offset curve of a parabola. Left: Interior offsets to the parabola r(t) = [t,t 2 ] with d = −0.8 and cutter path; Right: Trimmed interior offsets to the parabola r(t) = [t,t 2 ] with d = −0.8 and cutter path 8