Page 18: B-Splines

B-Splines (which is short for “Basis Splines”) are an important type of curve in computer graphics.

They are very general and mathematically elegant. However, this makes them hard to learn about. The Curves Notes provides a pretty extensive introduction. Here, I will try to convey some of the basic intuitions of a specific type of B-Spline: uniform cubic B-Splines.

This page seeks to give you some intuition about B-Splines. That’s about all we’ve had time for in previous years as well.

A Cubic B-Spline Segment

A cubic B-Spline segment is a cubic polynomial that is defined by 4 points. It is influenced by these points, but it does not interpolate them.

Try this…

Initially, the curve is a single segment; a B-Spline segment is controlled by four control points. Move the points around and notice how it is influenced by them; it does not interpolate them.

Shift-click to add another control point. With 5 points, there are two segments: the blue segment (its control points are marked blue), and the red segment (red control points). The segments share 3 control points (they are both red and blue). Notice how the segments stay attached with C(2) continuity as the points are moved.

Shift-click again to add yet another control point, and another segment. Each segment is shown in a different color, and the control points that influence it have a circle of that color. Each segment is controlled by 4 consecutive points - and it shares 3 points with its neighbors.

Add more points… notice how any point influences at most 4 segments. And each segment is influenced by 4 control points.

More Points

Here is the same demo, with some more points.

Move the points around to see:

1. The curve stays C(2)
2. Control is local (each point influences up to 4 segments)
3. The points influence the curve, but the curve doesn’t necessarily go through them

Try “doubling” the points (putting two consecutive points in the same place). Notice how this puts a “kink” in the curve (the 2nd derivative vanishes, so it acts like a C(2) discontinuity; it is like a C(1) curve there).

If you triple the points, the curve will interpolate this spot (with 3 points on it).

This is particularly useful to make the curve interpolate its endpoints.

On the Next: Page  19 - Arc-Length Parameterizations, we will consider a general problem with parametric curves.