# Uniform non-rational cubic B-spline

## Uniform non-rational cubic B-spline

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 |

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
**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.

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