Bspline values at end points

Question

Using for example Wolfram alpha definition of Bspline basis functions, the basis functions of order p=3 (cubic) and knotvector = [0,0,0,0, 0.2, 0.5, 0.5, 0.5 , 0.8, 1,1,1,1] looks like this:

As you can see, the last basis function approaches one as x approaches 1, but when x equal 1, the last basis function becomes 0.

Why is this. I have copied the definition exactly, so it should work in the end points also, right?

Furthermore, if i insert one more knotpoint, 0.5, to get a discontinous basefunction in the middle, I get the same problem (knotvector is now [0,0,0,0, 0.2, 0.5, 0.5, 0.5, 0.5, 0.8, 1,1,1,1]):

Here is my code if anyone is intrested (p=order, i=basefunction (starts from 1)):

function _bspline_basis_value_debug(p::Int, knot::Vector{Float64}, i::Int, xi)

if p>0
    _N1_1 = (xi - knot[i])*_bspline_basis_value_debug(p-1, knot, i, xi)
    _N1_2 = (knot[i+p]-knot[i])

    if _N1_2 == 0.0 && _N1_1 == 0.0
        _N1 = 0.0
    else
        _N1 = (_N1_1/_N1_2)
    end

    _N2_1 = (knot[i+p+1] - xi)*_bspline_basis_value_debug(p-1, knot, i+1, xi)
    _N2_2 = (knot[i+p+1] - knot[i+1])

    if _N2_2 == 0.0 && _N2_1 == 0.0
        _N2 = 0.0
    else
        _N2 = (_N2_1/_N2_2)
    end
    return _N1+_N2
else

    if knot[i] <= xi < knot[i+1]
        return 1.0
    else
        return 0.0
    end
end


end

Answer 1

The problem with the right most basis function going to zero is answered here Behaviour of a clamped Bspline curve at t=1.0

The second problem happens becuase one is not supposed to have p+1 multiple knots in the knotvector...