quadraticBetaSplines

Copyright © 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

The Constraints:
  0 = b1(0)    0 = b1'(0)
  b1(1) = b0(0)   bb1'(1) = b0'(0)
  b0(1) = b-1(0)  bb0'(1) = b-1'(0)
  b-1(1) = 0   bb-1'(1) = 0
  
  r = 1
   E B[i+r](ui) = b1(0) + b0(0) + b-1(0)
  r = -1
  
  
The Equations:
  b1 = c0,1 + c1,1u + c2,1usq
  b0 = c0,0 + c1,0u + c2,0usq
  b-1 = c0,-1 + c1,-1u + c2,-1usq
  
  b1' = c1,1 + 2c2,1u
  b0' = c1,0 + 2c2,0u
  b-1' = c1,-1 + 2c2,-1u
  
  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/dusq
  b0 = 1/d[1 + 2bu - usq]
  b-1 = b/d[1 - 2u + usq]