Computation on Parametric Curves. Yan-Bin Jia. Department of Computer Science Iowa State University Ames, IA 50011-1040, USA [email protected] Dec 16, 2002. Why Curved Objects?. Frequent subjects of maneuver (pen, mouse, cup, etc.).
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Frequent subjects of maneuver (pen, mouse, cup, etc.)
Actions and mechanics are inherently continuous /
differential and subject to local geometry of bodies
Shape localization, recognition, reconstruction
Not well studied compared to polygonal and polyhedral
1. Constructing common tangents of two curves
2. Computing all pairs of antipodal points
Primitive used by other algorithms
Hong et al. 1990; Chen & Burdick 1992; Ponce et al. 1993;
Blake & Taylor 1993
Grasping & Fixturing
Salisbury & Roth 1983; Mishra et al. 1987; Nguyen 1988;
Markenscoff et al. 1992; Trinkle 1992;Brost & Goldberg 1994;
Bicchi 1995, Kumar & Bicchi 2000
Goodman 1991; Kriegman & Ponce 1991;
Manocha & Canny 1992; Sakai 1999; Jia 2001
Preparata & Hong 1977; Yao 1982; Chazelle et al. 1993;
Matousek & Schwarzkopf 1996; Ramos 1997; Bespamyatnikh 1998Previous Work
Two segments satisfying conditions (i)—(iv)
b) are always on the same side of every common
tangent or always on different sides.
(N, translation of tangents, etc)
How to distinguish them?
Van Dis & Jia 2002
Bisection over [s , s ].
bOpposite Angle Sign at Endpoints
antipodal angleincreases monotonicallyas s increases.
no antipodal points otherwise.
Both segments are concave.
s ,s , …
t , t , …
Marching II (convex-concave)
Same convergence result
Efficiency – O(n + m)two-level calls of numerical primitivesConclusion
Design of algorithms for curve computing
Dissecting the curve into monotone segments
(preprocessing of global geometry)
Interleaving marching with numerical bisection
(exploiting of local geometry)
Provable convergence rates
(depending on curvatures)
Completeness – up to numerical resolution
Optimization along Curves, Surfaces