A B-spline is bounded by the polygon formed by the B-spline control
points;
??? The ease of specifying the range of a multi-valued curve;
??? The decoupling of x and y (and z in 3D) coordinates, with each having its
parametric representation;
??? Local controllability, which implies that local changes in shape are only
confined to the B-spline parameters local to that change.
4.2 B-Spline Curves
The B-splines are piecewise polynomial functions that provide local approximations
to contours/surfaces using a small number of parameters (control
points). A pth order B-spline is Cp??’1 continuous, i.e., is continuous and has
(p??’1) continuous derivatives. A pth order closed B-spline with n+1 parameters
P0, P1, . . . , Pn (control points) consists of n+1 connected curve segments
ri(t) = (xi(t), yi(t)), each of which is a linear combination of (p + 1) polynomials
of order p in the parameter t, where t is normalized between 0 and 1,
(0 ?· t ?· 1). The parameter t may be thought of as time and the curve may
be thought of as the trajectory of a particle moving in 2-D or 3-D space with
a speed of ||dri(t)/dt||.
The recursive equations of the B-Spline basis function are shown in equations
5 and 6, and Figure 3, where Pi are the control points, U is the knot
vector, and p is the B-Spline order.
R(t) =
n
Xi=0
Ni,p(t)Pi, 0 ?· t ?· 1, (5)
Ni,p(t) =
t ??’ Ui
Ui+p ??’ Ui
Ni,p??’1(t) +
Ui+p+1 ??’ t
Ui+p+1 ??’ Ui+1
Ni+1,p??’1(t), (6)
Ni,0(t) = ??1 Ui ?· t < Ui+1,
0 otherwise.
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