Convexity Conditions for Parameterized Surfaces

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.