Abstract
A transition from the fixed basis in Bezier’s method to some class of base functions is proposed. A parameter vector of a basis function is introduced as additional information. This achieves a more universal form of presentation and analytical description of geometric objects as compared to the non-uniform rational B-splines (NURBS). This enables control of basis function parameters including control points, their weights and node vectors. This approach can be useful at the final stage of constructing and especially local modification of compound curves and surfaces with required differential and shape properties; it also simplifies solution of geometric problems. In particular, a simple elimination of discontinuities along local spline curves due to automatic tuning of basis functions is demonstrated.
Highlights
Let us have segment C of some curve
We can extend the class of transformations [3, 6] including the birational transformations, a special case of which is the inversion that has a property to transform a circle into a straight and vice versa
We demonstrate the possibility of a simpler and more efficient solution of the same problem by means of an automatic tuning of basis functions for each segment of a local spline
Summary
Let us have segment C of some curve. We represent this segment in Bezier form [1,2,3,4,5,6] by radiusvectors ri and weights wi (wi > 0) of the four vertices of characteristic polyline L. Conservation of the length of tangent vectors T1 and T2 in transformation of a curve segment is important in solving such problems as local modification of a compound surface when its smoothness should remain [6] and in construction of the line of intersection of two surfaces with a posteriori estimated accuracy, when isoparametric [4, 14] or birational [7] correspondence between points on the intersection line is represented in a 3-dimensional Euclidean space and in the parameters space of such surfaces. We demonstrate the possibility of a simpler and more efficient solution of the same problem by means of an automatic tuning of basis functions for each segment of a local spline
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