¹Institute of Information Science & Technology, Hunan Agricultural University, Changsha, P. R. China
²Science College, Hunan Agricultural University, Changsha, P. R. China
³Institute of Computer Science ,Changsha University, Changsha, P. R. China

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Abstract—Based on a geometrical method, the internal relationships between locally parameterized curves and the local parameterized surfaces are analyzed. A necessary and sufficient condition is derived for the local convexity of parameterized surfaces and functional surfaces. A criterion for local convexity (concavity) of parameterized surfaces is found, also, the criterion condition of binary function convex surfaces is obtained. Finally, the relationships between a globally parameterized curves surfaces is discussed, a necessary condition is presented for the global convexity of parameterized surfaces , and it is proved that locally convex parameterized surfaces are also globally convex.

Index Terms—local convexity, global convexity, gauss curvature, the second fundamental form

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Cite: Kui Fang, Lu-Ming Shen, Xiang-Yang Xu, and Jing Song, "Convexity Conditions for Parameterized Surfaces," Journal of Software vol. 7, no. 6, pp. 1219-1226, 2012.