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### Minimum-Length Polygons in Simple Cube-Curves [chapter]

Reinhard Klette, Thomas Bülow
2000 Lecture Notes in Computer Science
The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve fully contained and complete in the tube of this digital curve.  ...  A critical edge is a grid edge contained in three consecutive cubes of a simple cube-curve.  ...  A non-flat simple cube-curve in R 3 specifies exactly one minimum-length polygonal simple curve (MLP, minimum-length polygon) which is contained and complete in its tube [11] .  ...

### Minimum-Length Polygons of First-Class Simple Cube-Curves [chapter]

Fajie Li, Reinhard Klette
2005 Lecture Notes in Computer Science
The curve's length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve.  ...  We consider simple cube-curves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set.  ...  A minimum-length polygon (MLP) of a simple cube-curve g is a shortest simple curve P which is contained and complete in tube g.  ...

### Minimum-Length Polygon of a Simple Cube-Curve in 3D Space [chapter]

Fajie Li, Reinhard Klette
2004 Lecture Notes in Computer Science
The curve's length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve.  ...  We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set.  ...  A minimum-length polygon (MLP) of a simple cube-curve g is a shortest simple curve P which is contained and complete in tube g.  ...

### Approximation of 3D Shortest Polygons in Simple Cube Curves [chapter]

Thomas Bülow, Reinhard Klette
2001 Lecture Notes in Computer Science
One possible definition of the length of a digitized curve in 3D is the length of the shortest polygonal curve lying entirely in a cube curve.  ...  One possible de nition of the length of a digitized curve i n 3D is the length of the shortest polygonal curve lying entirely in a cube curve.  ...  A non-at simple cube-curve in R 3 speci es exactly one minimum-length polygonal simple curve (MLP) which i s c o n tained and complete in its tube 10].  ...

### Shortest Paths in a Cuboidal World [chapter]

Fajie Li, Reinhard Klette
2006 Lecture Notes in Computer Science
The shortest path through a simple cube-curve in the orthogonal 3D grid is a minimum-length polygonal curve (MLP for short).  ...  to move in form a simple cube-curve.  ...  A minimum-length curve of a simple cube-curve g is a shortest simple curve P which is contained and complete in tube g. The length L(g) of g is defined to be the length L(P ).  ...

### Analysis of the rubberband algorithm

Fajie Li, Reinhard Klette
2007 Image and Vision Computing
The curve's length is defined to be that of the minimum-length polygonal curve (MLP) contained and complete in the tube of the curve.  ...  We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set.  ...  The length of a simple cube-curve S in 3D Euclidean space can be defined by the (Euclidean) length of the minimum-length polygonal curve (MLP for short) contained and complete in the polyhedrally bounded  ...

### Euclidean Shortest Paths in Simple Cube Curves at a Glance [article]

Fajie Li, Reinhard Klette
2007 arXiv   pre-print
>0, and in time complexity κ(ϵ) · O(n), where κ(ϵ) is the length difference between the path used for initialization and the minimum-length path, divided by ϵ.  ...  This paper reports about the development of two provably correct approximate algorithms which calculate the Euclidean shortest path (ESP) within a given cube-curve with arbitrary accuracy, defined by ϵ  ...  it is of minimum Euclidean length.  ...

### Page 6382 of Mathematical Reviews Vol. , Issue 98J [page]

1998 Mathematical Reviews
Let y be a simple closed curve of length < 2z on the unit sphere S?.  ...  98}:52011 that a regular simplex of edge-length / in E” has minimum width 1\/2/(n +1) if n is odd, and /\/2(n + 1)/n(n + 2) if n is even. G. D.  ...

### Polyhedra generation from lattice points [chapter]

Yukiko Kenmochi, Atsushi Imiya, Norberto F. Ezquerra
1996 Lecture Notes in Computer Science
If a digital polygon has n holes, the inside of the digital polygon is encircled by n + 1 simple closed curves because each hole is encircled by a simple closed curve.  ...  to the inside of the simple closed curve.  ...

### The Class of Simple Cube-Curves Whose MLPs Cannot Have Vertices at Grid Points [chapter]

Fajie Li, Reinhard Klette
2005 Lecture Notes in Computer Science
The curve's length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve.  ...  We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set.  ...  A minimum-length polygon (MLP) of a simple cube-curve g is a shortest simple curve P which is contained and complete in tube g.  ...

### Polygonization of implicit surfaces

Jules Bloomenthal
1988 Computer Aided Geometric Design
With a polygonal representation of the surface available, certain computational economies result. In particular, the roots to the function need not be solved each time the surface is rendered.  ...  This paper discusses a numerical technique that approximates an implicit surface with a polygonal representation.  ...  Given the polygon vertices, P i , their unit length normals, N i , and the unit length normal N at the polygon center, the planarity of the polygon can be estimated by: 1, nPoints] and V i the unit length  ...

### Finding the Shortest Path Between Two Points in a Simple Polygon by Applying a Rubberband Algorithm [chapter]

Fajie Li, Reinhard Klette
2006 Lecture Notes in Computer Science
Let p and q be two points in a simple polygon Π.  ...  An open problem in computational geometry asks to devise a simple linear-time algorithm for computing a shortest path between p and q, which is contained in Π, such that the algorithm does not depend on  ...  The original rubberband algorithm was published in [1] and [6] , aiming at an (approximative) calculation of a minimum-length polygonal path (MLP) contained and complete in a simple cube-curve (subsequent  ...

### Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

Anne Driemel, Sariel Har-Peled, Carola Wenk
2012 Discrete & Computational Geometry
We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in R d .  ...  We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.  ...  Acknowledgements The authors thank Mark de Berg and Marc van Kreveld for insightful discussions on the problems studied in this paper and related problems.  ...

### Approximating the Fréchet distance for realistic curves in near linear time

Anne Driemel, Sariel Har-Peled, Carola Wenk
2010 Proceedings of the 2010 annual symposium on Computational geometry - SoCG '10
We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in R d .  ...  We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.  ...  Acknowledgements The authors thank Mark de Berg and Marc van Kreveld for insightful discussions on the problems studied in this paper and related problems.  ...

### MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1995 Lecture Notes Series on Computing
We especially focus on optimal triangulations of geometric domains in two-and three-dimensions.  ...  We brie y survey the heuristic algorithms used in some practical mesh generators. remove the rst edge e from the queue. If Q e is not reversed, we simply continue to the next edge.  ...  Conclusions We have described work in computational geometry motivated by nite element mesh generation.  ...
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