Journal ArticleDOI
Tight linear envelopes for splines
David Lutterkort,Jörg Peters +1 more
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TLDR
A sharp bound on the distance between a spline and its B-spline control polygon is derived and the bound yields a piecewise linear envelope enclosingspline and polygon that can be easily and efficiently implemented.Abstract:
A sharp bound on the distance between a spline and its B-spline control polygon is derived. The bound yields a piecewise linear envelope enclosing spline and polygon. This envelope is particularly simple for uniform splines and splines in Bernstein-Bezier form and shrinks by a factor of 4 for each uniform subdivision step. The envelope can be easily and efficiently implemented due to its explicit and constructive nature.read more
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On-Line Path Generation for Unmanned Aerial Vehicles Using B-Spline Path Templates
Dongwon Jung,Panagiotis Tsiotras +1 more
TL;DR: The proposed algorithm provides a complete solution to the obstacle-free path-generation problem for an unmanned aerial vehicle in a computationally efficient manner, which is suitable for real-time implementation.
Dissertation
Hierarchical Path Planning and Control of a Small Fixed-wing UAV: Theory and Experimental Validation
Journal ArticleDOI
SLEVEs for planar spline curves
Jörg Peters,Xiaobin Wu +1 more
TL;DR: General criteria, that certify correctness of a global, polygonal enclosure built from a sequence of individual bounding boxes by extending and intersecting their edges, are developed that prove correctness of the sleve construction.
Journal ArticleDOI
Optimized refinable enclosures of multivariate polynomial pieces
David Lutterkort,Jörg Peters +1 more
TL;DR: An enclosure is a two-sided approximation of a uni- or multivariate function b@?B by a pair of typically simpler functions b^+,b^-@?H B such that b^-=.
Journal ArticleDOI
Bi-3 C2 polar subdivision
Ashish Myles,Jörg Peters +1 more
TL;DR: In this paper, the authors present a simple-to-implement C2 subdivision algorithm for generating surfaces of good shape and piecewise degree bi-3 in the polar setting.
References
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Journal ArticleDOI
Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon
TL;DR: This paper provides a straightforward proof of quadratic convergence of the sequence of control polygons to the Bezier segment under subdivision or degree-fold degree-raising, and establishes the explicit convergence constants, and allows analyzing the optimal choice of the subdivision parameter for adaptive refinement of Quadratic and cubic segments.
B(asic)-Spline Basics.
TL;DR: This report contains the lecture notes for the first of four lectures which comprise the course The extension of B-spline curve algorithms to surfaces given at SIGRAPH'86, an elaboration and extension of the MRC report 2896 by de Boor and Hollig, in which the basic B- Spline theory is developed from the reccurence relation rather than the original definition in terms of divided differences of the truncated power.
Journal ArticleDOI
Rates of convergence of control polygons
Elaine Cohen,Larry L. Schumaker +1 more
TL;DR: This paper presents a simple general method for treating such convergence questions which actually provides precise rates of convergence and illustrates the method by applying it to B-spline curves which are refined by increasing the degrees and/or refining the knot sequences.
Journal ArticleDOI
Subdivision algorithms converge quadratically
TL;DR: In this paper, the exact convergence rate of subdivision algorithms for various settings of polynomial and spline curve or surface representations is investigated. But the convergence rate is not known.