From the seminal paper by Hughes I learnt that any B-Spline curve (or surface) can be described by a linear combination of B-Spline basis and control points, i.e.

C(x) = sum_{i=1}^n N_{i,p} B_i

where N_{i,p} denotes the i-th node of the B-Spline curve with order p, and B_i the coefficients, or the control points.

This works well if we have a pre-defined plane, and summing up the parts give the desired B-Spline curve/surface.

My question is: is there a way to express the B-Spline curve in the same manner where it is not necessarily a function (vertical line can have multiple intersections with the curve), for example how to write the B-Spline curve in the form above for the following curve?

{
    "discretization": 100,
    "degree": 2,
    "controlPoints": [
        -6.386, 4.902, 0, 1,
        -3.693, 0.932, 0, 1,
        -8.215, -4.66, 0, 1,
        -2.899, -5.626, 0, 1,
        0.814, -2.881, 0.359, 1,
        7.024, -5.433, 0.804, 1
    ],
    "knots": [
        0, 0, 0, 0.375, 0.5, 0.625, 1, 1, 1
    ]
}

I am sorry that I cannot post the image because of low reputation. You can see the curve via importing the above data into the nurbs calculator. This is basically a curve with both vertical and horizontal components, with zigzags on both of them (there exists vertical/horizontal lines intersect with the curve at two points, respectively)

Clearly for a curve that is not necessarily a function (such as the one above) it cannot be decomposed directly into the bases. I am wondering how, then, can it be expressed explicitly using the same formulation.

Both analytic and numerical expressions will be very useful. Thanks in advance!