Geometrical Operations

A set of geometrical operations applicable to rational Bezier curves and surfaces has been implemented. The ``split'' and ``extract'' operations are directly related to the fixation concept on the level of tensor-product polynomial entities. The ``expand'' operation serves during the surface forming if any of the bounding curves is of a lower order than the surface itself in the appropriate parametric direction. The individual operations are schematically sketched in Figures 2.3 - 2.8.

Expansion of a rational Bezier curve (Fig. 2.3) of degree given by the set of control points , , ..., with corresponding weights , , ..., to a curve of degree given by control points , , ..., , and weights , , ..., , is governed by the relations

(2.7) |

(2.8) |

where

(2.9) |

Splitting of a rational Bezier curve at point (Fig. 2.4) given by parameter yields two new curves of the same order as the original curve. Assuming that the original curve of degree is described by control points and weights ( ) the new curves, described by control points and weights (curve ) and control points and weights (curve ), are given by

The relations between the parameter on the original curve, , and parameters on the new curves, and , have the form

(2.12) |

Similarly, splitting of a rational Bezier surface at point (Fig. 2.5) given by the curvilinear coordinates and results in the creation of four new surfaces. Again, assuming that the original surface of degrees and in each parametric direction is given by the control points and weights the control points of the new surfaces ( for surface , for surface , for surface , for surface ) can be obtained as

(2.14) |

(2.15) |

where the corresponding weights are given by

(2.18) |

(2.19) |

The relations between the curvilinear coordinates on the original and new surfaces may be written as

(2.21) |

(2.22) |

(2.23) |

(2.24) |

Extraction of a curve from a curve (Fig. 2.6) is accomplished by two successive splitting operations. The original curve is firstly split at point to get the by-product curve (Eq. (2.10)) which is then split at point (Eq. (2.11)) yielding the curve . The extraction of a surface from a surface (Fig. 2.7) is performed in a similar way. In the first phase, the auxiliary surface is extracted from the original surface (Eqs (2.13) and (2.17)) which is then used to get the final surface (Eqs (2.16) and (2.20)). The extraction of a curve from a surface (Fig. 2.8) is realized in a different way which combines both extraction techniques. Firstly, a surface is extracted in such a way that the curve to be pull out coincides with its boundary. This boundary curve () is then used to extract the desired curve . Note that the entity to be extracted from a surface must be aligned with the parametric coordinate system of that surface.

*Daniel Rypl
2005-12-07*