f(R)=0 R q(R)=0 Figure 10.5: Curves meet at small angle {f(R)=9(R)=0} When curves f=0,g=0 meet at a small angle (h tg= 1), then the condition d61 19l<e and 82=fo<8 (where If1, 191, 81, 82 are evaluated with R= Ro and 6, 8<1) are not enough to guarantee proximity of Ro to the intersection of f, g, see Figure 10.5 vf 83 vg Figure 10.6: Approximate curves with straight lines Using a linear approximation, and letting vf v9 V升| be the angle of intersection as in Figure 10.6 near the intersection point, a better criterion for evaluating if Ro is near the intersection of f and 63=o <6<1. R0 g(R) = 0 f(R) = 0 Figure 10.5: Curves meet at small angle. • Example B: R0 ∩ {f(R) = g(R) = 0} When curves f = 0, g = 0 meet at a small angle ( 5f |5f| · 5g |5g| ∼= 1), then the condition |f| <  and δ1 = |f| |5f| < δ |g| <  and δ2 = |g| |5g| < δ (where |f|, |g|, δ1, δ2 are evaluated with R = R0 and , δ  1) are not enough to guarantee proximity of R0 to the intersection of f, g, see Figure 10.5. δ δ δ 1 2 3 φ g f Figure 10.6: Approximate curves with straight lines. Using a linear approximation, and letting φ = cos−1 | 5f | 5 f| · 5g | 5 g| | be the angle of intersection as in Figure 10.6 near the intersection point, a better criterion for evaluating if R0 is near the intersection of f and g is δ3 = φ −1 { |f| | 5 f| + |g| | 5 g| } < δ  1 9