Why B-Spline are defined only where basis function sum to 1?

Question

I'm trying to understand b-spline. It's not clear why << The curve is defined only where order basis functions overlap>>, where order is degree+1 (for a cubic the order is 4). I found also that where a number equal to the order of basis functions overlap the sum is 1, and maybe is linked to the fact that curve starts here.

The first phrase comes from: http://www-evasion.imag.fr/~Francois.Faure/doc/inventorMentor/sgi_html/ch08.html in "Knot Sequence" section. I pasted it for you:

Answer 1

The definition of the curve states, that the NURBS factors of the basis function sum up to 1. Outside the interval, their sum is lower than 1.

E.g. take two points p1 and p2 (and give them some coordinates as you like). The combination q = 0.5*p1 + 0.5 * p2 gives us the point q in the middle of p1 and p2 (as 0.5 + 0.5 = 1). But where does the point q' = 0.2 * p1 + 0.2 * p2 sit? Try it out...