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]



����Ð��quadraticBetaSplines
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
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