tsDeBoorNet Struct Reference

#include <tinyspline.h>

Public Attributes
struct tsDeBoorNetImpl *	pImpl

Detailed Description

Represents the output of De Boor's algorithm. De Boor's algorithm is used to evaluate a spline at a certain knot by iteratively computing a net of intermediate points until the resulting point is available:

https://en.wikipedia.org/wiki/De_Boor%27s_algorithm
https://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/de-Boor.html

All points of a net are stored in points (ts_deboornet_points). The resulting point is the last point in points and, for the sake of convenience, can be accessed with ts_deboornet_result.

Two dimensional points are stored as follows:

[x_0, y_0, x_1, y_1, ..., x_n-1, y_n-1]

Tree dimensional points are stored as follows:

[x_0, y_0, z_0, x_1, y_1, z_1, ..., x_n-1, y_n-1, z_n-1]

... and so on. The output also supports homogeneous coordinates (cf. tsBSpline).

There is a special case in which the evaluation of a knot u returns two results instead of one. It occurs when the multiplicity of u (s(u)) is equals to the order of the evaluated spline, indicating that the spline is discontinuous at u. This is common practice for B-Splines (and NURBS) consisting of connected Bezier curves where the endpoint of curve c_i is equal to the start point of curve c_i+1. Yet, the end point of c_i and the start point of c_i+1 may still be completely different, yielding to visible gaps (if distance of the points is large enough). In such case (s(u) == order), ts_deboornet_points stores only the two resulting points (there is no net to calculate) and ts_deboornet_result points to the first point in ts_deboornet_points. Since having gaps in splines is unusual, both points in ts_deboornet_points are generally equal, making it easy to handle this special case by simply calling ts_deboornet_result. However, one can access both points if necessary:

ts_deboornet_result(...)[0] ...       // Access the first component of
                                      // the first result.

ts_deboornet_result(...)[dim(spline)] // Access the first component of
                                      // the second result.

As if this wasn't complicated enough, there is an exception for this special case, yielding to exactly one result (just like the regular case) even if s(u) == order. It occurs when u is the lower or upper bound of the domain of the evaluated spline. For instance, if b is a spline with domain [0, 1] and b is evaluated at u = 0 or u = 1, then ts_deboornet_result is always a single point regardless of the multiplicity of u.

In summary, there are three different types of evaluation:

1. The regular case, in which all points of the net are returned.
2. A special case, in which two results are returned (required for splines with gaps).
3. The exception of 2., in which exactly one result is returned (even if s(u) == order).

All in all this looks quite complex (and actually it is), but for most applications you do not have to deal with this. Just use ts_deboornet_result to access the outcome of De Boor's algorithm.

Member Data Documentation

pImpl

struct tsDeBoorNetImpl* tsDeBoorNet::pImpl

The actual implementation.

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The documentation for this struct was generated from the following file:

• /tinyspline/src/tinyspline.h