B-spline curves

The disadvantage of Bezier's curves is that with increasing number of angles of the operative polygon in general the grade of the curve increases, too. It can be eliminated by segmentation of the curve, but it is time-demanding.These disadvantages are neutralized by B-spline curves.

B-spline curve of the grade k is defined by operative polygon V0,... ., Vn and by values of parameters t0 =< ... =< tn (n =< m). B-spline curve of the grade k is defined : ,
where Ni,k(t) are base functions defined recurrently :

               1 for t ∈ <ti,ti+1>
 1, Ni,k(t) = <
               0 else
              t - ti               ti+k - t
 2, Ni,k(t) = ---------- Ni,k-1(t) + --------- Ni+1,k-1(t), for k>1
              ti+k-1 - ti            ti+k - ti+1

For construction of B-spline curve there is modified Casteljau's algorithm, known as de Boor's algorithm.

Coons`s cubic

A special form of B-spline curve is Coons cubic. It is defined by its 4 operative dots V0 V1, V2 and V3 and by relation

       1     | -1  3 -3  1 | | V0 |
B(t)= --- T. |  3 -6  3  0 | | V1 |.
       6     | -3  0  3  0 | | V2 |
             |  1  4  1  0 | | V3 |

$$$APPLET

Coons cubic


This cubic does not cross outer dots of its operative polygon. The curve starts and ends in dots

        V0 + 4V1 +V2
B(0) = -----------------
              6
        V1 + 4V2 +V3
B(1) = -----------------
              6
Uniform non-rational cubic B-spline is termed also Coons cubic B-tangle. We obtain it by joining the following Coons kubics with particular operative angles:
Bi    : Vi-3,Vi-2,Vi-1 a Vi
Bi+1 : Vi-2,Vi-1,Vi aVi+1,
Bi+2 : Vi-1,Vi,Vi+1 a Vi+2,
where Coons cubic B-spline is given with n>=4 points and consist from n-3 segment's.

$$$APPLET

Coons cubic B-spline with 8 points and 5 segment's

$$$APPLET

Closed Coons cubic B-spline with 8 points and 5 segments, where first three and last three points are identical

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