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Polygonal Boundary Evaluation of Minkowski Sums and Swept Volumes

2010
*
Computer graphics forum (Print)
*

It is exact for

doi:10.1111/j.1467-8659.2010.01770.x
fatcat:hsc57ijjbbdkzlbr4siwe52h44
*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

2009
*
Computer-Aided Design and Applications
*

This implicit

doi:10.3722/cadaps.2009.825-837
fatcat:zpybwee7bvc5vbluaqzfk5z6hm
*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

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

2019
*
The Visual Computer
*

We present an efficient algorithm for

doi:10.1007/s00371-019-01687-6
fatcat:vakynai4o5h7zbipcparlsjwv4
*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 [34] represented*the**Minkowski**sum*boundary*of**ruled**surfaces*as rational*surfaces*. ...##
###
Minkowski sum boundary surfaces of 3D-objects

2007
*
Graphical Models
*

Given two objects A and B with piecewise smooth boundary we discuss

doi:10.1016/j.gmod.2007.01.001
fatcat:vo4bmx2pevdypfdrnz6kazqx6u
*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]

1994
*
Computational Imaging and Vision
*

We consider

doi:10.1007/978-94-011-1040-2_37
dblp:conf/ismm/RoerdinkB94
fatcat:vzah2ss35rcdzoluldsabnpylm
*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

2010
*
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling - SPM '10
*

We first cull out

doi:10.1145/1839778.1839783
dblp:conf/sma/LiM10
fatcat:a673qifuxvf4fmrk6f5sfnf7ty
*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]

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

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]

2000
*
The Mathematics of Surfaces IX
*

An important advantage

doi:10.1007/978-1-4471-0495-7_6
fatcat:zxtyze6qgbgfhl4qr75ez4ixsu
*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

2007
*
Journal of Computational and Applied Mathematics
*

Extensions

doi:10.1016/j.cam.2006.11.006
fatcat:xqlpmdnsgzaxpn5qodho3qg33q
*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

2006
*
Computer Aided Geometric Design
*

We discuss

doi:10.1016/j.cagd.2005.07.001
fatcat:gv3iqxmvcbgejibq3tfxq76roi
*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

2017
*
Journal of Advanced Mechanical Design, Systems, and Manufacturing
*

Simplification level

doi:10.1299/jamdsm.2017jamdsm0042
fatcat:2eggaovokngvjpk7fbsr24bsce
*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

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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