Rationalization of a class of space B-spline curves based on piecewise projective transform

Abstract

This paper presents a rationalization of a class of space B-spline curves whose base functions can be arbitrary (algebraic polynomial or non-algebraic polynomial). Based on the theory of the three-dimensional projective transform and the construction of an appropriate sequence of projective transforms, we perform a piecewise rationalization of this class of space B-spline curves such that the resulting generalized rational B-spline curves preserve piecewise geometric invariance in the sense of projective transform. By introducing a geometric point (called the weight centre) composed of weight coefficients and using it as the basic point and four control vertexes as coordinate points, a projective coordinate system is constructed piecewise. Thus a non-homogeneous coordinate (i.e. rational coordinate) representation of the curve in the projective coordinate system is also obtained, where the weight coefficient groups are the gravity centre coordinates of the weight centre about the coordinate quadri-point figure and the base function groups are homogeneous projective coordinates of the curve. Furthermore, we give the curve’s projective coordinate sequence corresponding to a new weight coefficient sequence when the weight coefficients are changed, show that the new rational curve preserves piecewise projective invariance and finally conclude that space cubic non-uniform rational B-spline (NURBS) curves are special case of the above conclusions.

Keywords:

• Projective transform
• NURBS
• Rationalization
• Homogeneous coordinate
• Geometric invariance
• Generalized B-spline curve

Acknowledgements

The research of the first author was supported by the National Science Foundation of China (grant no. 10071060). The second author received support from the National Defense Pre-research Projects (grant no. 431318.1.1.7 and no. 41801070704).