# On the approximation of curves by line segments using dynamic programming

ON THE APPROXIMATION OF CURVES BY LINE SEGMENTS USING DYNAMIC PROGRAMMING The R A N D Corp., S a n t a M o n i c a , C J i f o r n i a RICHARD BELLMAN A c o m p u t a t i o n a l solution along these lines requires a few seconds per stage, where N is the n u m b e r of stages; see [3]. Extensions I t requires v e r y little a d d i t i o n a l effort to a p p r o x i m a t e to g ( x ) b y q u a d r a t i c polynomials, or b y p o l y n o m i a l s of a n y fixed degree. Similarly, we can c o m p u t e the m i n i m u m of N-{-I_ f u j Introduction I n a recent p a p e r [l], Stone considered the p r o b l e m of d e t e r m i n i n g the 2 N A- 2 c o n s t a n t s , a~:, b~, i = l, 2, ¢ ¢ ¢ , N -4- 1 a n d the N p o i n t s of subdivision u~, u2, ." ¢ , ux so as to m i n i m i z e the f u n c t i o n F ( a l , a2 , . . . , aN+l ; bl , b2 , "'" , DN+I ; Ul , U2 , " ' " , UN) N~-I f = ~ ui ui--1 ( g ( x ) -- h ( x , a j , b j ) ) 2 dx, j=l ui_l (1} p r o v i d e d t h a t we know how to m i n i m i z e the f u n c t i o n b f uN ( g ( z ) -- h(X, a N , b ~ ) ) 2 dx (2} over aN a n d b ~ , or the m i n i m u m of (1) hr+l max (g(x) - - a i - - bi x ) 2 dx, j~l Ui_l~X~U j l g(x) - h(x, aj,bj)l . (3} http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications of the ACM Association for Computing Machinery

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