B-spline parametric curve demonstration

A B-spline is defined by an ordered set of control points or control polygon which determines what path the curve will follow and consequently how the curve will look. A longer B-spline is actually made up of several curve segments, each one of which is defined by some number of control points in its viscinity. For a cubic B-spline, this number is four. A point on a particular curve segment is calculated by simply adding up the coordinate values of its defining control points after they have been multiplied by a weighting factor. The weighting factor is calculated using a set of parametric basis or blending functions. Each control point is weighted by the value of just one blending function for a specific curve segment. For each curve segment, the value of the parameter varies from zero to one in the blending functions. The value of the blending functions across the range of the parameter multiplied by the control point's coordinates define a number of intermediate points, which form a curve when connected.

The applet below will let you play with some splines yourself to see:

• how the control points determine the path of the spline and
• how the weight (i.e. influence on the shape of the spline) of a particular control point changes over the length of the spline.

• Sun's  Hot Java browser.
• Netscape Navigator 4.06+.
• Microsoft's  Internet Explorer 4.
• Sun's Applet viewer that comes with the JDK 1.1 .  (This is pretty clunky).

How to use the applet:

• The buttons on the left select the mode.
• First add some points in add mode (note that no curve will be drawn until at least four points have been added). Adjacent curve segments are drawn in different colors.
• Move the control points around using move mode to see how the curve changes shape.
• Move the scroll bar to change the trace point position. The trace point is shown as a small asterisk on the curve. As the trace point moves along the curve, the four control points currently contributing to its position will be highlighted in red. Their relative weights are given next to the point number as a percentage.
• You can delete points one at a time with the delete function, or clear all the control points at once with the clear function.

Things to notice:

B-splines have a couple of features that make them popular curve representations.

• Locality.  This means that changing the position of one control point only changes the sections of the curve near it.  The farther away a curve segment is from the control point being moved, the smaller the change.  Try moving one of the fat red control points in the applet.  The position of  the trace point changes a lot because this point is weighted heavily for the trace position (hence the fat dot).  Move one of the smaller red control points and the position of the trace point changes less.  If you move one of the black control points, the trace point doesn't move at all.  Those control points aren't contributing anything to the trace point's position.
• Continuity.  Because adjacent curve segments share control points by definition, B-splines have built in 2nd order continuity.  That means that every point has a first and a second derivative.  More intuitively it means that there are no sharp spikes or inflection points within a curve segment.  You can simulate a discontinuity by stacking several control points on top of one another.  Three control points at the same location suffice to create a sharp spike that interpolates (passes through) that location.  Try it in the applet!
• B-splines are easily extended to three dimensional curves and surfaces.