Earlier we built C continuous
Cardinal spline. Now we will make a piecewise
^{1}C continuous uniform B-spline of two cubic Bezier
segments ^{2} and V(t) with the control points
W(t)( and V_{0} , V_{1} , V_{2} ,
V_{3})( (see Fig.1).
Recall that for cubic Bezier spline
W_{0} , W_{1} ,
W_{2} , W_{3})
P '(0) = 3(P_{1} - P_{0}) ,
P '(1) = 3(P_{3} - P_{2}) ,
P "(0) = 6(P_{0} - 2P_{1}
+ P_{2}) ,
P "(1) = 6(P_{1} - 2P_{2}
+ P_{3}) .Note also, that for any two points and a
the point b(2
is "mirror a - b) =
a + (a - b) with respect to b".
a |

Continuity of the first derivative

i.e.

At last continuity of the second derivative

i.e.

We see, that only one control point

From Fig.2 it follows, that one can express all control points of Bezier
segments in terms of the *deBoor points P_{0,1,...,n}*
(the corners) , e.g.

Then Bezier segments points can be computed. But one can evaluate points of

where

Bezier and deBoor control points of quadratic B-spline are shown in Fig.3.

Note that I've found the Cox - beBoor formula in the Net and didn't check it (see Spline formulae standartization). I used basis function calculation instead.

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