End Conditions#
Most spline types that are defined by a sequence of control points to be interpolated need some additional information to be able to draw their segments at the beginning and at the end. For example, cubic Catmull–Rom splines need four consecutive control points to define the segment between the middle two. For the very first and last segment, the fourth control point is missing. Another example are natural splines, which would require to solve an underdetermined system of equations when only the control points are given.
There are many ways to provide this missing information, here we will mention only a few of them.
- clamped
This means providing a fixed tangent (i.e. first derivative) at the beginning and end of a cubic spline. For higher degree splines, additional derivatives have to be specified.
- natural
For a cubic spline, this means setting the second derivative at the beginning and end of the spline to zero and calculating the first derivative from that constraint, see Natural End Conditions.
- closed
This problem can also be solved by simply not having a begin and an end. When reaching the last control point, the spline can just continue at the first control point. For non-uniform splines an additional parameter interval has to be specified for the segment that’s inserted between the end and the beginning.
For most splines in the splines module,
clamped, natural and closed end conditions are available
via the endconditions argument.
Except for closed,
the end conditions can differ between the beginning and end of the spline.
Additional information is available for end conditions of natural splines and monotone end conditions.