A simple method for drawing a rational curve as two Bézier segments

ABSTRACT

In this paper we give a simple method for drawing a closed rational curve specified in terms of control points as two Bézier segments. The main result is the following: For every affine frame (r,s) (where r<s), for every rational curve F(t) specified over [r,s] by some control polygon (&bgr;0, …, &bgr;m) (where the &bgr;zero are weighted control points or control vectors), the control points (&thgr;0,… ,&thgr;m (w.r.t.[r,s]) of the rational curve G(t) = F4t are given by qi=-1 ⁱbi, where 4:RP¹→RP¹ is the projectivity mapping [r,s] onto RP1−]r,s]. Thus, in order to draw the entire trace of the curve F over -∞,+∞ , we simply draw the curve segments F[(r,s]) and G([r,s]). The correctness of the method is established using a simple geometric argument about ways of partitioning the real projective line into two disjoint segments. Other known methods for drawing rational curves can be justified using similar geometric arguments.

REFERENCES

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.3 Picture/Image Generation
          Subjects: Line and curve generation

General Terms:
Design

Keywords:
Bézier curves, control points, de Casteljau algorithm, rational curves, subdivision, weights