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Natural The Second
Spline derivative of the
spline is zero at the
endpoints.
Parabolic The spline reduces
Runout to a parabolic curve
Spline on the first and last
intervals.
Cubic The spline is a
Runout single cubic curve
Spline on the first two and
last two intervals.
The Natural Spline
The two simplest mathematical conditions we can impose are
These conditions together with 15 result in an linear system for , , …, , which can be written in matrix form as
For numerical calculations it is more convenient to eliminate and from this system and write
(16)
together with
(17)
(18)
Thus, the linear system can be solved for the coefficients , , …, , and and are
determined by 17 and 18.
Physically, the natural spline results when the ends of a drafting spline extend freely beyond the interpolating points without
constraint. The end portions of the spline outside the interpolating points will fall on straight line paths, causing to vanish at
