# Is a B-Spline always made up of Bezier curve segments?

According to what I have read, a B-Spline curve is made up of segments, with each segment controlled by 'k' control points (where k is the order of the curve). Also, a B-Spline curve can be formed by gluing Bezier curves together and constraining some of its control points.

More specifically, 'S' Bezier curves, each of order 'k', can be glued together to form a 'C^a' continuous B-Spline with order 'k' and kS-(a+1)(S-1) control points.

My question is:
Is the converse true? Is every segment of a B-Spline (defined by the parameter between two consecutive knot values) an individual Bezier curve, or can be represented as one?

Also, a B-Spline curve can be formed by gluing Bezier curves together and constraining some of its control points

You don't need no constraining. Given $$n$$ arbitrary Bezier curves there exist a (probably discontinuous) BSpline curve which coincide with the union of the Bezier curves in any given order.

Is the converse true? Is every segment of a B-Spline (defined by the parameter between two consecutive knot values) an individual Bezier curve, or can be represented as one

Yes, if the spline degree is $$d$$, you can perform knot insertion until all distinct knots has multiplicity $$d+1$$. The control points of the resulting BSpline curve coincides with those of the individual Bezier curves.

• Sorry, I'm new to this topic, but how do we prove that those individual curves are Bezier curves? Oct 17, 2018 at 14:02

A B-spline curve is basically a piecewise continuos polynomial curve. So, each segment of a B-spline curve is a polynomial curve of a certain degree. A Bezier curve is also a polynomial curve and any polynomail curve can be represented as a Bezier curve. Therefore, each segment of a B-spline curve can be represented as a Bezier curve.