uniform, non-rational B-Splines
B-Splines are piecewise, polynomial interpolation curves. A "single curve" is constructed by polynomial interpolation over a series of points, in a sliding window of a fixed number of points.
A “cubic” B-spline, which is defined by seven points, is a curve formed by a polynomial interpolation with four points. The curve can be regarded as a lot of different sections, each controlled by four points at a time. The full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, {3,4,5,6}, {4,5,6,7} four sections curves b1234, b2345, b3456, b4567.
It is important to note that B-splines are not the same as poly-Béziers or Catmull-Rom curves:
defining new sections based on new points, so that, for example, a cubic poly-Bézier curve with 12 points is not possible. Starting with a four-point curve and add three more points for each spline section, so that we can only have poly-Béziers with 4, 7, 10, 13, 16, etc. points.
Cubic B-splines, on the other hand, are smooth interpolations of a curve that involve four consecutive points, so that the segments along the curve, except for the start and end points, have the coordinates of the curve defined by four control points.
Note the difference:
It is important to note that B-splines are not the same as poly-Béziers or Catmull-Rom curves:
defining new sections based on new points, so that, for example, a cubic poly-Bézier curve with 12 points is not possible. Starting with a four-point curve and add three more points for each spline section, so that we can only have poly-Béziers with 4, 7, 10, 13, 16, etc. points.
Cubic B-splines, on the other hand, are smooth interpolations of a curve that involve four consecutive points, so that the segments along the curve, except for the start and end points, have the coordinates of the curve defined by four control points.
Note the difference:
- Bézier curves are defined as an interpolation of points, but:
- B-splines are defined as an interpolation of curves.