دانلود رایگان مقاله تشخیص تقارن منحنی فضای گویا از انحنا و پیچ خوردگی آنها

1. Introduction

The problem of detecting the symmetries of curves and surfaces has attracted the attention of many researchers throughout the years, because of the interest from fields like Pattern Recognition (Boutin, 2000; Calabi et al., 1998; Huang and Cohen, 1996; Lebmeir and Richter-Gebert, 2008; Lebmeir, 2009; Suk and Flusser, 1993, 2005; Tarel and Cooper, 2000; Taubin and Cooper, 1992; Weiss, 1993), Computer Graphics (Berner et al., 2008; Bokeloh et al., 2009; Lipman et al., 2010; Martinet et al., 2006; Mitra et al., 2006; Podolak et al., 2006; Schnabel et al., 2008; Simari et al., 2006), and Computer Vision (Alt et al., 1988; Brass and Knauer, 2004; Jiang et al., 1996; Li et al., 2008, 2010; Loy and Eklundh, 2006; Tate and Jared, 2003; Sun and Sherrah, 1997). The introduction in Alcázar et al. (2014b) contains an extensive account of the variety of approaches used in the above references. A common characteristic in most of these papers is that the methods focus on computing approximate symmetries more than exact symmetries, which is perfectly reasonable in many applications, where curves and surfaces often serve as merely approximate representations of a more complex shape. Some exceptions appear here: If the object to be considered is discrete (e.g. a polyhedron), or is described by a discrete object, like for instance a control polygon or a control polyhedron, then the symmetries can be determined exactly (Alt et al., 1988; Brass and Knauer, 2004; Jiang et al., 1996; Li et al., 2008). Examples of the second class are Bézier curves and tensor product surfaces. Furthermore, in these cases the symmetries of the curve or surface follow from those of the underlying discrete object. Another exception appears in Lebmeir and Richter-Gebert (2008), where the authors provide a deterministic method to detect rotation symmetry of an implicitly defined algebraic plane curve and to find the exact rotation angle and rotation center. The method uses a complex representation of the curve and is generalized in Lebmeir (2009) to detect mirror symmetry as well.

6. Conclusion

We have presented a new, deterministic, and efficient method for detecting whether a rational space curve is symmetric. The method combines ideas in Alcázar (2014), Alcázar et al. (2014b) with the use of the curvature and torsion as differential invariants of space curves. The complexity analysis and experiments show a good theoretical and practical performance, clearly beating the performance obtained in Alcázar et al. (2014b). The algorithm also improves in scope on the algorithm of Alcázar et al. (2014b), which can only be applied to find the symmetries for Pythagorean-hodograph curves and involutions of other curves. Finally Algorithm Symm± is simpler than the algorithm in Alcázar et al. (2014b), which imposes certain conditions on the parametrization that often lead to a reparametrized, non-sparse curve whose coefficients have a large bitsize. By contrast, the algorithm in this paper has fewer requirements and is efficient even with high degrees.