Projective transformations of the parameter of a Bernstein-Bézier curve
ACM Transactions on Graphics (TOG)
Curves and surfaces for computer aided geometric design: a practical guide
Curves and surfaces for computer aided geometric design: a practical guide
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Rasterization of nonparametric curves
ACM Transactions on Graphics (TOG)
Numerically stable implicitization of cubic curves
ACM Transactions on Graphics (TOG)
Oriented projective geometry
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
NURB curves and surfaces: from projective geometry to practical use
NURB curves and surfaces: from projective geometry to practical use
Curves and surfaces in geometric modeling: theory and algorithms
Curves and surfaces in geometric modeling: theory and algorithms
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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), 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 segmentsF[(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.