B Spline functions

 

Bi(u) = (u^3)/6Bi-1(u) = (-3*(u^3) + 3*(u^2) + 3*u +1)/6
Bi-2(u) = (3*(u^3) - 6*(u^2) +4)/6
Bi-3(u) = (1 - u)^3/6
 
Bi(u) = (u^3)/6Bi-1(u) = (-3*((u-1)^3) + 3*((u-1)^2) + 3*(u-1) +1)/6
Bi-2(u) = (3*((u-2)^3) - 6*((u-2)^2) +4)/6
Bi-3(u) = (1 - (u-3))^3/6
 
Bi(u) = (u^3)/6
Bi-1(u) = (-3*((u-1)^3) + 3*((u-1)^2) + 3*(u-1) +1)/6
Bi-2(u) = (3*((u-2)^3) - 6*((u-2)^2) +4)/6
Bi-3(u) = (1 - (u-3))^3/6
       
B(u) = Bi(u) when 0 <= u < 1,
       Bi-1(u) when 1 <= u < 2,
       Bi-2(u) when 2 <= u < 3,
       Bi-3(u) when 3 <= u < 4
 
Bi(u) = (u^3)/6
Bi1(u) = (-3*((u-1)^3) + 3*((u-1)^2) + 3*(u-1) +1)/6
Bi2(u) = (3*((u-2)^3) - 6*((u-2)^2) +4)/6
Bi3(u) = (1 - (u-3))^3/6


B(u) = Bi(u) when 0 <= u < 1,
       Bi1(u) when 1 <= u < 2,
       Bi2(u) when 2 <= u < 3,
       Bi3(u) when 3 <= u < 4

B1(u) = B(u-1)
B2(u) = B(u-2)
B3(u) = B(u-3)
B4(u) = B(u-4)

 

To calculate the curve at any parameter t we place a gaussian curve over the parameter space. This curve is actually an approximation of a gaussian; it does not extend to infinity at each end, just to +/- 2 by using the following equations:

The Approximate Gaussian Curve


This curve peaks at a value of 2/3, and at +/- 1 its value is 1/6. When this curve is placed over the array of control points, it gives the weighting of each point. As the curve is drawn, each point will in turn become the heaviest weighted, therefore we gain more local control.