Orthogonal Basis In Cubic Spline Space

In spline space, the B-spline curve is a kind of quite popular spline curve, and it isone of main methods of constructing free form curve/surface at present, as well, it is avery important technique and significant tool in CAGD system.However, the B-spline basis is not orthogonal, this thesis discusses the orthogonalbasis in 3-degree spline space.Usually, a basis in a space, not orthogonal, will be transmitted to an orthogonal basisby Gram-Schmidt orthogonal method. But Gram-Schmidt method needs integralcomputation many times, and the result is quite troublesome. In this thesis, We proposea strategy to compute an orthogonal basis in possession of elegant representation in3-degree spline space, and analyses the case of multi-knots.This thesis focuses on the problems as follows:Firstly, the orthogonal basis is constructed step by step, not wholly obtained atone time. Inserting only one knot, one corresponding basis could be structured. In thisway, the whole basis over spline space will be obtained in the end.Secondly, a group of assistant functions, which are linear combination of6-degree B-spline basis, are defined previously in order to construct the orthogonalbasis. The thesis analyses the properties of the assistant functions and presents3-degree orthogonal basis in the differential form.Next, the transformation matrices between B-spline bases and orthogonal basesare derived from the properties of the two bases, escaping the difficulties in theprocess of general solution of the inverse matrix.Finally, applying the orthogonal basis to degree-reduction approximation of B-splinecurves, adopting least square method with respect to L₂ norms, the thesis propose thealgorithm of the optimal multi-degree reduction of the B-spline curves. The algebraicapproximation algorithms not only can obtain the control points of the approximationcurves directly, but also simple and straightforward.