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By Su Bu-Qing, Liu Ding-Yuan, Chang Geng-Zhe
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Additional info for Computational Geometry. Curve and Surface Modeling
Example text
It has been proved in practice that this method for fairing curves is simple and efficient. 13) for convexity preserving of interpolatory cubic spline functions is due to Wang Rishuang (1979). The results have been extended to splines in tension and to cases of large deflection by Liu Dingyuan ([2], 1979). For interpolatory quadratic spline functions, note that elements on the main diagonal of the coefficient matrix of the continuity equations are all 3. +1) (i = 0 , l , . . , / i ) . 14) For equally spaced knots, A, = μ,,· = | .
The first part of the above inequalities always holds, as we have (3-2«)(m + n - 2 m n ) - 3 m ( l - « ) 2 = « { l - m n + 2 ( l - m ) ( l - « ) } > 0 . But the second part does not hold, as 4m(l - n)2- (3 -2n)(m + n - 2 m « ) = m - n -2n{\ - n) < 0 . Hence m and n do not exist in this case. This completes the proof. 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.
11) of inequalities holds. 11) hold. For pi = 0 (i; = 1, 2 , . . , n), T(x) reduces to the ordinary interpolatory cubic spline function. 12) in which λ 0 = μ„ = 1. Set which equals the second divided difference of the interpolated data >,. 12) can be written as F f > | ( M / F * _ 1 + AiF*fl) (i = 0, 1, n). 13) This is a sufficient condition for convexity preserving of the interpolatory cubic spline function, which can be used as a criterion for fairing data points in mathematical lofting and in geometric design.