The Constraints: 0 = b1(0) 0 = b1'(0) b1(1) = b0(0) bob1'(1) = b0'(0) b0(1) = b-1(0) bob0'(1) = b-1'(0) b-1(1) = 0 bob-1'(1) = 0 r = 1 E B[i+r](ui) = b1(0) + b0(0) + b-1(0) r = -1 The Equations: b1 = c0,1 + c1,1ou + c2,1ousq b0 = c0,0 + c1,0ou + c2,0ousq b-1 = c0,-1 + c1,-1ou + c2,-1ousq b1' = c1,1 + 2c2,1ou b0' = c1,0 + 2c2,0ou b-1' = c1,-1 + 2c2,-1ou b1(0) = c0,1 b1(1) = c0,1 + c1,1 + c2,1 b0(0) = c0,0 b0(1) = c0,0 + c1,0 + c2,0 b-1(0) = c0,-1 b-1(1) = c0,-1 + c1,-1 + c2,-1 b1'(0) = c1,1 b1'(1) = c1,1 + 2c2,1 b0'(0) = c1,0 b0'(1) = c1,0 + 2c2,0 b-1'(0) = c1,-1 b-1'(1) = c1,-1 + 2c2,-1 b1(0) + b0(0) + b-1(0) = 1 The Derivation: (1) c0,1 = 0 c0,0 = c1,1 + c2,1 c0,-1 = c0,0 + c1,0 + c2,0 c1,1 = 0 c1,0 = 2bc2,1 9 equations in 9 unknowns c1,-1 = bc1,0 + 2bc2,0 0 = c0,-1 + c1,-1 + c2,-1 0 = c1,-1 + 2c2,-1 1 = c0,1 + c0,0 + c0,-1 (2) c0,0 = c2,1 c0,-1 = c0,0 + c1,0 + c2,0 c1,0 = 2bc2,1 7 equations in 7 unknowns, remove c0,1 and c1,1 c1,-1 = bc1,0 + 2bc2,0 0 = c0,-1 + c1,-1 + c2,-1 0 = c1,-1 + 2c2,-1 1 = c0,0 + c0,-1 (3) c0,-1 = c0,0 + c1,0 + c2,0 c1,0 = 2bc0,0 6 equations in 6 unknowns, remove c2,1 c1,-1 = bc1,0 + 2bc2,0 0 = c0,-1 + c1,-1 + c2,-1 c1,-1 = -2c2,-1 1 = c0,0 + c0,-1 (4) c0,-1 = c0,0 + c1,0 + c2,0 c1,0 = 2bc0,0 5 equations in 5 unknowns, remove c2,-1 c1,-1 = bc1,0 + 2bc2,0 0 = c0,-1 + (1/2)c1,-1 1 = c0,0 + c0,-1 (5) c0,-1 = c0,0 + c1,0 + c2,0 c1,0 = 2bc0,0 4 equations in 4 unknowns, remove c1,-1 -2c0,-1 = bc1,0 + 2bc2,0 1 = c0,0 + c0,-1 (6) 1 = 2c0,0 + c1,0 + c2,0 c1,0 = 2bc0,0 3 equations in 3 unknowns, remove c0,-1 -2 = -2c0,0 + bc1,0 + 2bc2,0 (7) 1 = (2 + 2b)c0,0 + c2,0 2 equations in 2 unknowns, remove c1,0 -2 = (-2 + 2bsq)c0,0 + 2bc2,0 (8) -2b = (-4b - 4bsq)c0,0 - 2bc2,0 -2 = (-2 + 2bsq)c0,0 + 2bc2,0 ----------------------------- -2(b + 1) = -2(1 + 2b + bsq)c0,0 (9) c0,0 = (b + 1)/(1 + 2b + bsq) = 1/(b + 1), let d = 1 + b (10) c2,0 = -1 c1,0 = 2b/d c0,-1 = b/d c1,-1 = -2b/d c2,-1 = b/d c2,1 = 1/d c0,1 = 0 c1,1 = 0 (11) . b1 = 1/dousq b0 = 1/do[1 + 2bu - usq] b-1 = b/do[1 - 2u + usq] ŠquadraticBetaSplines Copyright c 1984 by Xerox Corporation. All rights reserved. This file contains the derivation of the quadratic Beta-spline. Last Edited by: Tso, July 18, 1984 10:37:34 am PDT ŹŻ˜Jšœ™Jšœ Ļmœ1™