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 pathSelf-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 poly￾nomial 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