Maureen C. Stone
Abstract
In this paper, we analyze planar parametric cubic curves to determine conditions for loops, cusps, or inflection points. By expressing the curve to be analyzed as a linear combination of control points, it can be transformed such that three of the control points are mapped to specific locations on the plane. We call this image curve the canonical curve. Affine maps do not affect inflection points, cusps, or loops, so the analysis can be applied to the canonical curve instead of the original one. Since the first three points are fixed, the canonical curve is completely characterized by the position of its fourth point. The analysis therefore reduces to observing which region of the canonical plane the fourth point occupies. We demonstrate that for all parametric cubes expressed in this form, the boundaries of these regions are tonics and straight lines. Special cases include Bézier curves, B-splines, and Beta-splines. Such a characterization forms the basis for an easy and efficient solution to this problem.
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Index Terms
- A geometric characterization of parametric cubic curves
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Reviews
Patrick Gilles Maillot
The authors present an analysis of planar parametric cubic curves to determine conditions for loops, cusps, or inflection points. In an introduction, the authors present the main justification and application for their research: automobile body design and aerodynamics. Another domain of interest for such a characterization, not mentioned in this paper, is the rendering of “fat curves” or curves with an arbitrary width, where the characterization of the curve is used to determine how the curve has to be decomposed into pieces that can be rendered separately. The authors also present some previous related work and specify that their results can be applied to non-rational curves. They do not give a reason for this limitation, and I regret that they did not give a characterization of the rational curves as well, or at least a study of the complexity involved. Stone and DeRose present their study in three steps. The first step is a review of previous work that gives an easily understandable base for nonspecialists. The limitation of their study to non-rational curves also helps. The second step proposes a characterization of Bezier curves in detail; it is followed by step three, a study of the region decomposition of the curve. A separate paragraph treats the degenerate forms of Bezier curves in a very descriptive way. The use of figures is helpful in visualizing the effect of the displacement of one control point of the curve. The authors generalize their study to characterize other curve types, including B-splines, Catmull-Rom splines, and Beta-splines. They show that all variations of a fourth point, the three other remaining fixed, can be obtained by a projective transformation of a figure studied in the case of Bezier curves. The mathematical study leads to a two-dimensional projective transformation, which is more intuitive for graphics users than the previous nonlinear characterization schemes. This paper presents a very intuitive and interesting approach to the characterization of curve complexity. The mathematical notation used in this work is consistent. The use of figures, as well as the organization of the different sections of the paper, make it easy to understand. It represents a good contribution to the graphics community, and provides a clear vision of non-rational curve complexity characterization.
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