Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
Cylindrical algebraic decomposition II: an adjacency algorithm for the plane
SIAM Journal on Computing
Introduction to mathematical morphology
Computer Vision, Graphics, and Image Processing
Offsetting operations in solid modelling
Computer Aided Geometric Design
The approximation of non-degenerate offset surfaces
Computer Aided Geometric Design
Application of vector sum operator
Computer-Aided Design
Algebraic decomposition of regular curves
Journal of Symbolic Computation
A polynomial-time algorithm for the topological type of a real algebraic curve
Journal of Symbolic Computation
A mathematical model for shape description using Minkowski operators
Computer Vision, Graphics, and Image Processing
Analytic properties of plane offset curves
Computer Aided Geometric Design
Sweeping of three-dimensional objects
Computer-Aided Design
Geometric representation of swept volumes with application to polyhedral objects
International Journal of Robotics Research
Curves and surfaces in geometrical optics
Mathematical methods in computer aided geometric design II
Approximate general sweep boundary of a 2D curved object
CVGIP: Graphical Models and Image Processing
An algebraic algorithm to compute the exact general sweep boundary of a 2D curved object
Information Processing Letters
Computational methods for evaluating swept object boundaries
The Visual Computer: International Journal of Computer Graphics
Computing Minkowski sums
An efficient method for analyzing the topology of plane real algebraic curves
Selected papers presented at the international IMACS symposium on Symbolic computation, new trends and developments
Swept volume: a retrospective and prospective view
Neural, Parallel & Scientific Computations - computer aided geometric design
A new approach to the surface intersection problem
Computer Aided Geometric Design
Polynomial/rational approximation of Minkowski sum boundary curves
Graphical Models and Image Processing
Journal of Computational and Applied Mathematics - Special issue on computational methods in computer graphics
The sweep-envelope differrential equation algorithm for general deformed swept volumes
Computer Aided Geometric Design
An algorithm for planning collision-free paths among polyhedral obstacles
Communications of the ACM
Modern Control Systems
Polygon decomposition for efficient construction of Minkowski sums
Computational Geometry: Theory and Applications - Special issue on: Sixteenth European Workshop on Computational Geometry (EUROCG-2000)
Comparing Offset Curve Approximation Methods
IEEE Computer Graphics and Applications
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Algorithms for the Shape of Semialgebraic Sets. A New Approach
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Minkowski Roots of Complex Sets
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
Topologically reliable display of algebraic curves
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
Combinatorics, Probability and Computing
Protein similarity from knot theory and geometric convolution
RECOMB '04 Proceedings of the eighth annual international conference on Resaerch in computational molecular biology
Computer Aided Geometric Design
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Root neighborhoods, generalized lemniscates, and robust stability of dynamic systems
Applicable Algebra in Engineering, Communication and Computing
A Hierarchy of Geometric Forms
IEEE Computer Graphics and Applications
Geometric Modeling for Swept Volume of Moving Solids
IEEE Computer Graphics and Applications
Approximate swept volumes of NURBS surfaces or solids
Computer Aided Geometric Design
Computing the Minkowski Value of the Exponential Function over a Complex Disk
Computer Mathematics
Reparameterization of curves and surfaces with respect to their convolution
MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces
Approximate convolution with pairs of cubic Bézier LN curves
Computer Aided Geometric Design
Hi-index | 7.29 |
Algorithms are developed, based on topological principles, to evaluate the boundary and ''internal structure'' of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the ''internal'' envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R^3, and other forms of geometrical convolution, are briefly discussed.