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### Polygonal Boundary Evaluation of Minkowski Sums and Swept Volumes

Marcel Campen, Leif Kobbelt
2010 Computer graphics forum (Print)
It is exact for Minkowski sums and approximates swept volumes polygonally.  ...  By using plane-based geometry in conjunction with hierarchical arrangement computations we avoid the necessity of arbitrary precision arithmetics and extensive special case handling.  ...  This, of course, limits the accuracy of the result. Swept Volumes The exact boundary of a swept volume consists of patches of ruled surfaces and developable surfaces in general.  ...

### Computing Minkowski Sum of Periodic Surface Models

Yan Wang
2009 Computer-Aided Design and Applications
This implicit surface model helps create super-porous nano structures parametrically and support crystal packing. In this paper, we study construction methods of Minkowski sums for periodic surfaces.  ...  A numerical approximation approach based on the Chebyshev polynomials is developed and can be applied in the formulations of surface normal direction matching and volume translations.  ...  The problem of computing the Minkowski sum boundary       can be transformed to the problem of computing the convolution between  and  because of             .  ...

### Polynomial/Rational Approximation of Minkowski Sum Boundary Curves

In-Kwon Lee, Myung-Soo Kim, Gershon Elber
1998 Graphical Models and Image Processing
The Minkowski sum of two planar objects is closely related to the convolution curve of the two object boundary curves. That is, the convolution curve is a superset of the Minkowski sum boundary.  ...  The Minkowski sum can be used in various important geometric computations, especially for collision detection among planar curved objects.  ...  The most important steps in the Minkowski sum computation include a compatible subdivision of input curves/surfaces, the generation of convolution curves/surfaces, and the elimination of redundant parts  ...

### Minkowski sum computation for planar freeform geometric models using \$\$G^1\$\$ G 1 -biarc approximation and interior disk culling

Sangjun Han, Seung-Hyun Yoon, Myung-Soo Kim, Gershon Elber
2019 The Visual Computer
We present an efficient algorithm for computing the Minkowski sum of two planar geometric models bounded by B-spline curves.  ...  We demonstrate a real-time performance and the stability of circle-based Minkowski sum computation using a large set of test data.  ...  For some special types of input, the Minkowski sum boundary can be computed exactly. Mühlthaler and Pottmann  represented the Minkowski sum boundary of ruled surfaces as rational surfaces.  ...

### Minkowski sum boundary surfaces of 3D-objects

Martin Peternell, Tibor Steiner
2007 Graphical Models
Given two objects A and B with piecewise smooth boundary we discuss the computation of the boundary Γ of the Minkowski sum A + B.  ...  Additionally we point to some relations between Minkowski sums and kinematics, and compute local quadratic approximations of the envelope.  ...  This work has been carried out partially within the Kplus competence center Advanced Computer Vision and funded partially by the Austrian Science Fund FWF within the research network S92.  ...

### Visualization of Minkowski Operations by Computer Graphics Techniques [chapter]

J. B. T. M. Roerdink, G. S. M. Blaauwgeers
1994 Computational Imaging and Vision
We consider the problem of visualizing 3D objects de ned as a Minkowski addition or subtraction of elementary objects.  ...  Applications of the method are found in solid modelling and shape description.  ...  So the former alternative is the most workable. Thus, the purpose is to derive a list of replacement rules for every possible Minkowski sum of two Minkowski primitives.  ...

### A GPU-based voxelization approach to 3D Minkowski sum computation

Wei Li, Sara McMains
2010 Proceedings of the 14th ACM Symposium on Solid and Physical Modeling - SPM '10
We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum.  ...  The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms for computing Minkowski sums of objects with curved surfaces  ...  Instead of computing the exact union of pairwise Minkowski sums, they compute a signed distance field and extract its zero iso-surface.  ...

### Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries [article]

Sipu Ruan, Gregory S. Chirikjian
2020 arXiv   pre-print
The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each body surface.  ...  This paper proposes a closed-form parametric formula of the Minkowski sum boundary for broad classes of convex bodies in d-dimensional Euclidean space.  ...  When the authors were at Johns Hopkins University, they were supported by U.S National Science Foundation grant IIS-1619050. The ideas expressed in this article are solely those of the authors.  ...

### Voxelized Minkowski sum computation on the GPU with robust culling

Wei Li, Sara McMains
2011 Computer-Aided Design
The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.  ...  We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum, analyzing and adaptively bounding the rounding errors of the culling algorithm to solve the floating  ...  The authors are supported in part by UC Discovery and the National Science Foundation under CAREER Grant No. 0547675 and Grant No. 0621198.  ...

### Problem Reduction to Parameter Space [chapter]

Myung-Soo Kim, Gershon Elber
2000 The Mathematics of Surfaces IX
An important advantage of our approach is that equations represented in the parameter space have degrees signi cantly lower than those of geometric solution spaces in the original work space.  ...  This paper presents a problem reduction scheme that converts geometric constraints in work space to a system of equations in parameter space.  ...  Acknowledgements The authors thank Hee-Seok Heo for collaboration on the work for intersecting ruled surfaces and ringed surfaces.  ...

### Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

Rida T. Farouki, Joel Hass
2007 Journal of Computational and Applied Mathematics
Extensions of the algorithm to the computation of Minkowski sums in R 3 , and other forms of geometrical convolution, are briefly discussed.  ...  Algorithms are developed, based on topological principles, to evaluate the boundary and "internal structure" of the Minkowski sum of two planar curves.  ...  Acknowledgements This work was supported in part by the National Science Foundation under grants CCR-0202179 and DMS-0138411.  ...

### Page 2744 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews
Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention.  ...  The Minkowski sums and products offer basic ‘shape operators’ for applications such as computer-aided design and mathematical morphology, and may also prove useful in other contexts where complex variables  ...

### Rational surfaces with linear normals and their convolutions with rational surfaces

Maria Lucia Sampoli, Martin Peternell, Bert Jüttler
2006 Computer Aided Geometric Design
We discuss the geometric properties of these surfaces. In particular, using the dual representation it is shown that the convolution with general rational surfaces yields again rational surfaces.  ...  Similar results hold in the case of curves.  ...  Costantini for supporting this visit within the frame of the FIRB project, contract n. RBAU0128CL.  ...

### Simple offset algorithm for generating workpiece solid model for milling simulation

Masatomo INUI, Nobuyuki UMEZU, Masashi TSUKAHARA
2017 Journal of Advanced Mechanical Design, Systems, and Manufacturing
Simplification level of the offset shape can be controlled by changing the resolution of the grids for sampling the surface points.  ...  In this method, surface points of the offset shape are sampled using 3 mutually perpendicular grids in the model space. The final offset shape is obtained by properly connecting the sampled points.  ...  Their method first generates possible surface elements of the Minkowski sum shape of two objects. Voxels corresponding to the Minkowski sum shape are then selected according to the surface elements.  ...

### Minkowski sums of point sets defined by inequalities

A Pasko, O Okunev, V Savchenko
2003 Computers and Mathematics with Applications
The existing approaches support Minkowski sums for the boundary, set-theoretic, and ray representations of solids.  ...  In this paper, we consider the Minkowski sum operation in the context of geometric modeling using real functions.  ...  The Minkowski sum is applied to the corresponding elements of the skeletons of the operands, resulting in an intermediate skeleton that generates a new implicit surface.  ...
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