{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T08:24:43Z","timestamp":1663835083222},"reference-count":17,"publisher":"Association for Computing Machinery (ACM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[1974,4]]},"abstract":"\n The\n m<\/jats:italic>\n th degree Bernstein polynomial approximation to a function \u0192 defined over [0, 1] is \u03a3\n \n m<\/jats:italic>\n <\/jats:sup>\n \n \u03bc<\/jats:italic>\n <\/jats:sub>\n =0 \u0192(\n \u03bc<\/jats:italic>\n \/\n m<\/jats:italic>\n )\n \n \u03c6\n \u03bc<\/jats:sub>\n <\/jats:italic>\n (\n s<\/jats:italic>\n ), where the weights\n \n \u03c6\n \u03bc<\/jats:sub>\n <\/jats:italic>\n (\n s<\/jats:italic>\n ) are binomial density functions. The Bernstein approximations inherit many of the\n global<\/jats:italic>\n characteristics of \u0192, like monotonicity and convexity, and they always are at least as\n \u201csmooth\u201d<\/jats:italic>\n as \u0192, where \u201csmooth\u201d refers to the number of undulations, the total variation, and the differentiability class of \u0192.\n \n Historically, their relatively slow convergence in the L\n \u221e<\/jats:sub>\n -norm has tended to discourage their use in practical applications.\n <\/jats:italic>\n However, in a large class of problems the smoothness of an approximating function is of greater importance than closeness of fit. This is especially true in connection with problems of computer-aided\n geometric design<\/jats:italic>\n of curves and surfaces where aesthetic criteria and the\n intrinsic properties of shape<\/jats:italic>\n are major considerations. For this latter class of problems, P. B\u00e9zier of Renault has successfully exploited the properties of parametric Bernstein polynomials. The purpose of this paper is to analyze the B\u00e9zier techniques and to explore various extensions and generalizations. In a sequel, the authors consider the extension of the results contained herein to free-form curve and surface design using polynomial\n splines<\/jats:italic>\n . These B-spline methods have several advantages over the techniques described in the present paper.\n <\/jats:p>","DOI":"10.1145\/321812.321824","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:26:10Z","timestamp":1027769170000},"page":"293-310","source":"Crossref","is-referenced-by-count":91,"title":["Bernstein-B\u00e9zier Methods for the Computer-Aided Design of Free-Form Curves and Surfaces"],"prefix":"10.1145","volume":"21","author":[{"given":"William J.","family":"Gordon","sequence":"first","affiliation":[{"name":"Mathematics Department, Research Laboratories, General Motors Corporation, Warren, MI and Syracuse University, Syracuse, New York"}]},{"given":"Richard F.","family":"Riesenfeld","sequence":"additional","affiliation":[{"name":"Computer Science Department, University of Utah, Salt Lake City, UT and Syracuse University, Syracuse, New York"}]}],"member":"320","reference":[{"key":"e_1_2_1_1_2","volume-title":"transl, by A. R. Forrest. 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London A8~1","author":"FORREST A.R.","year":"1971"},{"key":"e_1_2_1_8_2","volume-title":"Coons' surfaces and multivariab)e functional interpolation","author":"FORREST A.R.","year":"1970"},{"key":"e_1_2_1_9_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/0708019","article-title":"Blending-function\" methods of bivariate and multivariate interpolation and ap'proximation","volume":"8","author":"GORDON W.","year":"1971","journal-title":"SIAM J. Numer. Anal."},{"key":"e_1_2_1_10_2","first-page":"223","volume-title":"Proceedings of the Symposium on Approximation with Special Emphasis on Splines, I. J. Schoenberg, Ed., U. of Wisconsin","author":"GORDON W. J.","year":"1969"},{"key":"e_1_2_1_11_2","volume-title":"General Motors Corp.","author":"GOI~DON W.J.","year":"1969"},{"key":"e_1_2_1_12_2","first-page":"3","article-title":"KELISKY, R. P., AND R~VL{N, T.J. Iterates of Bernstein polynomials","volume":"21","year":"1967","journal-title":"Pacific J. Math."},{"key":"e_1_2_1_13_2","volume-title":"Finite Markov Chains","author":"K~ MENY","year":"1960"},{"key":"e_1_2_1_14_2","volume-title":"of Toronto Press","author":"LORENTZ G.G.","year":"1953"},{"key":"e_1_2_1_15_2","first-page":"49","article-title":"des fonctions convex d'ordre sup~rieur","volume":"10","year":"1935","journal-title":"Mathematica"},{"key":"e_1_2_1_16_2","first-page":"249","volume-title":"On Numerical Approximation","author":"SC B~Q","year":"1959"},{"key":"e_1_2_1_17_2","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1090\/S0002-9904-1953-09695-1","article-title":"Smoothing operators and their generating functions","volume":"59","author":"SCHO~NB~ R~, I","year":"1953","journal-title":"Bull. Amer. Math. 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