A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area

and Applied Analysis 3 where E = ⟨x u , x u ⟩ , F = ⟨x u , xV⟩ , G = ⟨xV, xV⟩ (3) are the first fundamental coefficients, and e = ⟨N, x uu ⟩ , f = ⟨N, x uV⟩ , g = ⟨N, xV⟩ (4) are the second fundamental coefficients with N (u, V) = xu × xV 󵄨󵄨󵄨󵄨xu × xV 󵄨󵄨󵄨󵄨 , (5) being the unit normal to the surface x(u, V). For a minimal surface [6, 33] the mean curvature (2) is identically zero. For minimization we use only the numerator part of mean curvature H given by (2), as done in [34] following [16] which explicitly mentions that “for a locally parameterized surface, the mean curvature vanishes when the numerator part of the mean curvature is equal to zero.” We call the numerator part of (2) asH 0 corresponding the initial surface x 0 (u, V) that is used in the ansatz equation (31) to get firstorder variationally improved surface x 1 (u, V) of lesser area; we chose our initial surface to be a Coons patch described later. This process could be continued as an iterative process until a minimal surface is achieved. But due to complexity of the calculations required for obtaining the second-order improvement x 2 (u, V), we have been able to calculate the firstorder surface x 1 (u, V) only.The numerator partH 0 is denoted by H 0 = e 0 G 0 − 2F 0 f 0 + g 0 E 0 , (6) where E 0 , F 0 , G 0 , e 0 , f 0 and g 0 denote the fundamental magnitudes, given by (3) and (4), for the initial surface x 0 (u, V), and N 0 (u, V) is the unit normal, given by (5), to this initial surface. We call the root mean square (rms) of this H 0 , for 0 ≤ u ≤ 1 and 0 ≤ V ≤ 1, as μ 0 . That is,