I am confused by the locality of B-spline. For example, if we remove the second point of a 4-point spline, then the new t1 would be the original t2 and the new t2 would be the original t3. Then, the value of B0,2 should be influenced even if we have not changed the position of first point. My primary understanding of locality is that change of position of a local point should not globally influence other splines not connected to the point. This might be incorrect. Could someone please give a more detailed explanation on the locality of B-spline?
@EdCat Two things:
Locality refers to how changing the values of a node will affect the interpolated function. Adding or removing points is another question, which we haven't really discussed in class. For instance, one important feature of many splines is that knots can be inserted without changing the interpolated function. This property is quite important in practice, especially for things like subdivision surfaces. On the flip side, removing a knot will definitely change the curve (in general), because you're losing information. In fact, it's not entirely clear how to do this without changing the whole curve (e.g., do you shift all knots to the right of the removed knot left by one index?)
Locality is also a property of piecewise splines. Any single piece will be "global" in the sense that the shape of that one piece will always depend on all its knots. The locality question is, when it comes to assembling these pieces, do you have to solve some kind of global equation to get the values? Or can you locally determine each polynomial?
Hopefully that helps---let me know if something still isn't clear.