### 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* ∈ <*t*_{i},t_{i+1}>
1, *N*_{i,k}(t) = <
0 else
* t - ti* * t*_{i+k} - t
2, *N*_{i,k}(t) = ---------- *N*_{i,k-1}(*t*) + --------- *N*_{i+1,k-1}(*t*), for *k*>1
* t*_{i+k-1 }- t_{i} * t*_{i+k} - t_{i+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 *V*_{0} V_{1},
V_{2} and V_{3} and by relation

1 | -1 3 -3 1 | | V_{0} |
B(t)= --- T. | 3 -6 3 0 | | V_{1} |.
6 | -3 0 3 0 | | V_{2} |
| 1 4 1 0 | | V_{3} |

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

V_{0} + 4V_{1} +V_{2}
B(0) = -----------------
6
V_{1} + 4V_{2} +V_{3}
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 : V*_{i-3},V_{i-2},V_{i-1}
a V_{i}
*B*_{i+1} : V_{i-2},V_{i-1},V_{i}
aV_{i+1},
*B*_{i+2} : V_{i-1},V_{i},V_{i+1
}a
V_{i+2,}
where

**Coons cubic B-spline** is given with

*n*>=4 points
and consist from

*n*-3 segment's.

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

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