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By Su Bu-Qing, Liu Ding-Yuan, Chang Geng-Zhe

ISBN-10: 0126756104

ISBN-13: 9780126756104

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These inequalities can be used as necessary but not sufficient conditions for the occurrence of two real inflection points. Put Ax = ax+\a2T + la,T\ Bx = bMb2T + lb,T\ Then VI becomes A2X + B2X = 1. + B\)(A\+ B\) 2* (A,A 2 + BXB2)2, the inequalities al+b]^(\pT-q)2, (1) (2) a22 + b22^(lpT2-r)2, (3) a2x + b2x^T2{lqT-\r)2, are obtained by different choices of A2 and B2. If at least one of the above three inequalities does not hold, then we conclude that two real inflection points do not occur on the curve segment simultaneously.

First of all, an affine coordinate system {O, A, μ} is established in the plane, as shown by Fig. 3-9. In the figure, the plane is divided into several regions by the A-axis, the μ-axis, the straight lines A = 3 and μ = 3, the hyperbola (A - 4 ) ( μ - 4 ) - 4 = 0, U—the segment of the parabola μ 2 - 3 μ + Α = 0 limited by the second quadrant— and l4—the segment of the parabola Α 2 - 3 Α + μ = 0 limited by the fourth quadrant. For example, N, represents the region in Fig. 3-9 characterized by 0 < A < 3 and 0 < μ < 3 .

5 Extending (λ, μ) to the Whole Plane Now we extend the results obtained in the previous section to the whole plane (Liu Dingyuan [4], 1981). 1) can be written in vector form P(t) = iP3t3 + ±P2t2 + Plt + P0. 2) q = [P,Pl], r = [PiP2l where the brackets [ ] denote the determinant formed by the components of the two vectors appearing in the brackets. Set G = q2-2pr. 3) Suppose that two end-points M 0 , M, and the tangent vectors A0, Ax at M 0 , Mu respectively, are given. 4) where L = M0MX. Hence P'(0) = A o , P"(0) = 6 L - 4 A o - 2 A i ; Ρ'(1) = Λ,, P"(l) = -6L + 2A0 + 4Al.

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Computational Geometry. Curve and Surface Modeling by Su Bu-Qing, Liu Ding-Yuan, Chang Geng-Zhe


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