Continuous curve with a kink in Fig.1 is called C 0 continuous.
A curve is C k continuous if all k derivatives of the
curve are continuous.
Interpolating piecewise Cardinal spline is composed of cubic Bezier
splines joined with C1 continuity (see Fig.2). The i-th
Bezier segment goes through two neighbouring points Pi ,
Pi+1 . Derivatives at a point Pi
are parallel to the Pi-1Pi+1 line, i.e.
P 'i = (Pi+1 -
Derivatives at the terminal points P0 ,
Pn go to the neighbouring nodes.
Remember, that a cubic Bezier spline is determined by 4 vectors (e.g.
four control points or two endpoints and two derivatives at the points).
For the i -th segment we have
B0 = Pi ,
B3 = Pi+1 .
As for Bezier spline
B'(0) = 3(B1 -
B'(1) = 3(B3 - B2) ,
therefore all control points B0,1,2,3 are
obtained easy for every segment.
Control polygon for the left Bezier segment is shown in Fig.2. Derivatives
at P1 are parallel to the P0P2
line. Derivative at the endpoint P0 goes to
One can take different αi
values for different points Pi and even different
α+,- for the "left" and "right"
segments at a point too. For α = 2 we get
the Catmul - Rom spline.
You can compare Cardinal (the left picture) and Bezier (the right one)
Cardinal spline applet Drag the mouse to move the nearest control point (a small blue
Drag up-down the small red square to
change α from 0 to 4
(you can see α value in the Status bar).
Click mouse + "Shift"("Ctrl") to add (remove) a point.
Unfortunately the Cardinal curve cannot be subdivided in two sub-curves
which coincide with the initial one. As since Fig.3 shows, that in common case
the line P0P3 is not parallel to the
P02P12 one and
the derivative at P(1/2) is not (P3 -