Curves and Surfaces Construction Based on New Basis with Exponential Functions

By incorporating two exponential functions into the cubic Bernstein basis functions, a new class of λμ-Bernstein basis functions is constructed. Based on these λμ-Bernstein basis functions, a kind of λμ-Bézier-like curve with two shape parameters, which include the cubic Bernstein-Bézier curve, is proposed. The C1 and C2 continuous conditions for joining two λμ-Bézier-like curves are given. By using tensor product method, a class of rectangular Bézier-like patches with four shape parameters is shown. The G1 and G2 continuous conditions for joining two rectangular Bézier-like patches are derived. By incorporating three exponential functions into the cubic Bernstein basis functions over triangular domain, a new class of λμη-Bernstein basis functions over triangular domain is also constructed. Based on the λμη-Bernstein basis functions, a kind of triangular λμη-Bézier-like patch with three shape parameters, which include the triangular Bernstein-Bézier cubic patch, is presented. The conditions for G1 continuous smooth joining two triangular λμη-Bézier-like patches are discussed. The shape parameters serve as tension parameters and have a predictable adjusting role on the curves and patches.

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