Class Notes _ B-spline curves and Nubrs

Why Study B-spline

Bezier curve/surface does not support local modification and editing ;
It is very difficult to meet the geometric continuous conditions when the Bezier curve/surface is spliced. B-Spline history

In 1946, Schoenberg proposed a spline-based approach to approximate the curve;

The motive of B-spline stems from the Runge-kutta phenomenon in interpolation: high-order polynomial can easily produce unstable upper and lower jitter.

Why not use piecewise low-order polynomial to replace high-order polynomial with successive connections ? This is the idea of splines (piecewise low-order polynomial) .

In 1972, based on the work of Schoenberg, Gordon and Riesenfeld presented a B-spline and a series of corresponding geometric algorithms. How to solve B-spline

The interpolation of spline functions can be solved by solving a three diagonal equation.
For a given interval division, the interpolation of splines can be computed similarly.
All spline functions on a given interval form a linear space. The basic function of this linear space is called the B-spline basis function. B-spline basis function

P (t) =∑i=04pini,4 p (t) = \sum_{i = 0}^{4}p_{i}n_{i,4}
1.B Spline basis function ni,k (t) n_{i,k} (t) of the non-0 interval is what. (Ti,ti+k t_{i},t_{i+k})
-a small portion of 0, mostly 0, guarantees the locality of the B-spline.

2. How many nodes are required altogether.
P0 p_{0} corresponds to n0,4 n_{0,4}, n0,4 n_{0,4} corresponding to a non-0 interval (t0,t4 t_{0},t_{4})
P1 P_{1} corresponds to n1,4 n_{1,4}, n1,4 n_{1,4} corresponding to a non-0 interval (t1,t5 t_{1},t_{5})
P2 p_{2} corresponds to n2,4 n_{2,4}, n2,4 n_{2,4} corresponding to a non-0 interval (