4-Colored Triangulation of 3-Maps

2017 ◽  
Vol 27 (04) ◽  
pp. 297-325
Author(s):  
Lucas Moutinho Bueno ◽  
Jorge Stolfi
Keyword(s):  
Data Structure ◽  
Barycentric Subdivision ◽  
3 Dimensional ◽  
Arbitrary Space ◽  
Standard Solution ◽  
Gem Data

We describe an algorithm to triangulate a general 3-dimensional-map on an arbitrary space in such way that the resulting 3-dimensional triangulation is vertex-colorable with four colors. (Four-colorable triangulations can be efficiently represented and manipulated by the GEM data structure of Montagner and Stolfi.) The standard solution to this problem is the barycentric subdivision (BCS) of the map. Our algorithm yields a 4-colored triangulation that is provably smaller than the BCS, and in practice is often a small fraction of its size. When the input map is a shellable triangulation of a 3-ball, in particular, we can prove that the output size is less than [Formula: see text] times the size of the BCS.

3-Colored Triangulation of 2D Maps

2016 ◽  
Vol 26 (02) ◽  
pp. 111-133 ◽  
Author(s):  
Lucas Moutinho Bueno ◽  
Jorge Stolfi
Keyword(s):  
Data Structure ◽  
Euler Characteristic ◽  
Upper Bounds ◽  
Experimental Results ◽  
Barycentric Subdivision ◽  
Standard Solution ◽  
Gem Data

We describe an algorithm to triangulate a general map on an arbitrary surface in such way that the resulting triangulation is vertex-colorable with three colors. (Three-colorable triangulations can be efficiently represented and manipulated by the GEM data structure of Montagner and Stolfi.) The standard solution to this problem is the barycentric subdivision, which produces [Formula: see text] triangles when applied to a map with [Formula: see text] edges, such that [Formula: see text] of them are border edges (adjacent to only one face). Our algorithm yields a subdivision with at most [Formula: see text] triangles, where [Formula: see text] is the Euler Characteristic of the surface; or at most [Formula: see text] triangles if all [Formula: see text] faces of the map have the same degree [Formula: see text]. Experimental results show that the resulting triangulations have, on the average, significantly fewer triangles than these upper bounds.

scholarly journals THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLS

2012 ◽  
Vol 22 (04) ◽  
pp. 341-364 ◽  
Author(s):  
TIMOTHY M. CHAN
Keyword(s):  
Data Structure ◽  
Convex Hull ◽  
Query Time ◽  
Convex Hulls ◽  
Lower Envelope ◽  
Dynamic Data ◽  
Line Segments ◽  
3 Dimensional ◽  
Small Constant ◽  
Dynamic Data Structure

We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O( log 1+εn) for an arbitrarily small constant ε > 0. This improves the previous bound of O( log 3/2n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O( log n+k) time with O( polylog n) expected amortized update time. A similar result holds for 3-dimensional orthogonal range reporting. For 3-dimensional halfspace range reporting, the query time increases to O( log 2n/ log log n + k). • A semi-online dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O( log n) and amortized update time O(nε). As a corollary, we can solve the following problem in O(n1+ε) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.

scholarly journals Persistent triangulations

2001 ◽  
Vol 11 (5) ◽  
pp. 441-466 ◽  
Author(s):  
GUY BLELLOCH ◽  
HAL BURCH ◽  
KARL CRARY ◽  
ROBERT HARPER ◽  
GARY MILLER ◽  
...  
Keyword(s):  
Data Structure ◽  
Convex Hull ◽  
Randomized Algorithm ◽  
Computation Time ◽  
Simplicial Complexes ◽  
Constant Factor ◽  
Running Time ◽  
3 Dimensional ◽  
Small Constant ◽  
Triangulated Surfaces

Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These representations of triangulations make it difficult to support persistence, including ‘multiple futures’, the ability to use a data structure in several unrelated ways in a given computation; ‘time travel’, the ability to move freely among versions of a data structure; or parallel computation, the ability to operate concurrently on a data structure without interference. We present a purely functional interface and representation of triangulated surfaces, and more generally of simplicial complexes in higher dimensions. In addition to being persistent in the strongest sense, the interface more closely matches the mathematical definition of triangulations (simplicial complexes) than do interfaces based on mutable representations. The representation, however, comes at the cost of requiring O(lg n) time for traversing or adding triangles (simplices), where n is the number of triangles in the surface. We show both analytically and experimentally that for certain important cases, this extra cost does not seriously affect end-to-end running time. Analytically, we present a new randomized algorithm for 3-dimensional Convex Hull based on our representations for which the running time matches the Ω(n lg n) lower-bound for the problem. This is achieved by using only O(n) traversals of the surface. Experimentally, we present results for both an implementation of the 3-dimensional Convex Hull and for a terrain modeling algorithm, which demonstrate that, although there is some cost to persistence, it seems to be a small constant factor.

Three-Dimensional Reconstruction Using Stereo Pairs from Serial Thick Sections

1977 ◽  
Vol 35 ◽  
pp. 562-563
Author(s):  
Robert Glaeser ◽  
Thomas Bauer ◽  
David Grano
Keyword(s):  
Three Dimensional ◽  
Dimensional Structure ◽  
Serial Sections ◽  
Thin Sections ◽  
3 Dimensional ◽  
Transmission Electron ◽  
Freeze Dried ◽  
Dimensional Reconstruction ◽  
Stereo Imagery ◽  
Whole Mounts

In transmission electron microscopy, the 3-dimensional structure of an object is usually obtained in one of two ways. For objects which can be included in one specimen, as for example with elements included in freeze- dried whole mounts and examined with a high voltage microscope, stereo pairs can be obtained which exhibit the 3-D structure of the element. For objects which can not be included in one specimen, the 3-D shape is obtained by reconstruction from serial sections. However, without stereo imagery, only detail which remains constant within the thickness of the section can be used in the reconstruction; consequently, the choice is between a low resolution reconstruction using a few thick sections and a better resolution reconstruction using many thin sections, generally a tedious chore. This paper describes an approach to 3-D reconstruction which uses stereo images of serial thick sections to reconstruct an object including detail which changes within the depth of an individual thick section.

Structural preservation and absolute contrast of catalase crystal sections prepared with tannic acid

1983 ◽  
Vol 41 ◽  
pp. 448-449
Author(s):  
C.W. Akey ◽  
M. Szalay ◽  
S.J. Edelstein
Keyword(s):  
Tannic Acid ◽  
Beef Heart ◽  
X Rays ◽  
Screw Axis ◽  
General Applicability ◽  
Thin Sections ◽  
Beef Liver ◽  
3 Dimensional ◽  
Dimensional Reconstruction ◽  
Structural Preservation

Three methods of obtaining 20 Å resolution in sectioned protein crystals have recently been described. They include tannic acid fixation, low temperature embedding and grid sectioning. To be useful for 3-dimensional reconstruction thin sections must possess suitable resolution, structural fidelity and a known contrast. Tannic acid fixation appears to satisfy the above criteria based on studies of crystals of Pseudomonas cytochrome oxidase, orthorhombic beef liver catalase and beef heart F1-ATPase. In order to develop methods with general applicability, we have concentrated our efforts on a trigonal modification of catalase which routinely demonstrated a resolution of 40 Å. The catalase system is particularly useful since a comparison with the structure recently solved with x-rays will permit evaluation of the accuracy of 3-D reconstructions of sectioned crystals.Initially, we re-evaluated the packing of trigonal catalase crystals studied by Longley. Images of the (001) plane are of particular interest since they give a projection down the 31-screw axis in space group P3121. Images obtained by the method of Longley or by tannic acid fixation are negatively contrasted since control experiments with orthorhombic catalase plates yield negatively stained specimens with conditions used for the larger trigonal crystals.

Deviations from Ideal Decagonal Symmetry in Electron Diffraction Patterns of the “Decagonal” Phase in the Al-Co-Cu Alloy System

1991 ◽  
Vol 49 ◽  
pp. 914-915
Author(s):  
Atul S. Ramani ◽  
Earle R. Ryba ◽  
Paul R. Howell
Keyword(s):  
Electron Microscope ◽  
Alloy System ◽  
Nominal Composition ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Decagonal Phase ◽  
Cu Alloy ◽  
Transmission Electron ◽  
Lattice Periodicity ◽  
Diffraction Patterns

The “decagonal” phase in the Al-Co-Cu system of nominal composition Al65CO15Cu20 first discovered by He et al. is especially suitable as a topic of investigation since it has been claimed that it is thermodynamically stable and is reported to be periodic in the dimension perpendicular to the plane of quasiperiodic 10-fold symmetry. It can thus be expected that it is an important link between fully periodic and fully quasiperiodic phases. In the present paper, we report important findings of our transmission electron microscope (TEM) study that concern deviations from ideal decagonal symmetry of selected area diffraction patterns (SADPs) obtained from several “decagonal” phase crystals and also observation of a lattice of main reflections on the 10-fold and 2-fold SADPs that implies complete 3-dimensional lattice periodicity and the fundamentally incommensurate nature of the “decagonal” phase. We also present diffraction evidence for a new transition phase that can be classified as being one-dimensionally quasiperiodic if the lattice of main reflections is ignored.

A Novel Reconstitution Method for Inducing the Formation of Regular 2-Dimensional Arrays of Membrane Proteins and Lipids

1988 ◽  
Vol 46 ◽  
pp. 152-153 ◽  
Author(s):  
A. Engel ◽  
A. Holzenburg ◽  
K. Stauffer ◽  
J. Rosenbusch ◽  
U. Aebi
Keyword(s):  
Membrane Proteins ◽  
Flow Rate ◽  
Lipid Membranes ◽  
Functional Characterization ◽  
Dimensional Structure ◽  
Electron Crystallography ◽  
Peltier Element ◽  
Protein Ratio ◽  
Precise Control ◽  
3 Dimensional

Reconstitution of solubilized and purified membrane proteins in the presence of phospholipids into vesicles allows their functions to be studied by simple bulk measurements (e.g. diffusion of differently sized solutes) or by conductance measurements after transformation into planar membranes. On the other hand, reconstitution into regular protein-lipid arrays, usually forming at a specific lipid-to-protein ratio, provides the basis for determining the 3-dimensional structure of membrane proteins employing the tools of electron crystallography.To refine reconstitution conditions for reproducibly inducing formation of large and highly ordered protein-lipid membranes that are suitable for both electron crystallography and patch clamping experiments aimed at their functional characterization, we built a flow-dialysis device that allows precise control of temperature and flow-rate (Fig. 1). The flow rate is generated by a peristaltic pump and can be adjusted from 1 to 500 ml/h. The dialysis buffer is brought to a preselected temperature during its travel through a meandering path before it enters the dialysis reservoir. A Z-80 based computer controls a Peltier element allowing the temperature profile to be programmed as function of time.

Structure and Interaction of Protamine by Dark Field Electron Microscopy

1976 ◽  
Vol 34 ◽  
pp. 160-161
Author(s):  
D.P. Bazett-Jones ◽  
F.P. Ottensmeyer
Keyword(s):  
Electron Microscopy ◽  
Dark Field ◽  
Amino Acid Sequences ◽  
Dimensional Structure ◽  
3 Dimensional ◽  
Field Electron ◽  
Proposed Model ◽  
Field Electron Microscopy ◽  
Dark Field Electron ◽  
Positively Charged

It has been shown for some time that it is possible to obtain images of small unstained proteins, with a resolution of approximately 5Å using dark field electron microscopy (1,2). Applying this technique, we have observed a uniformity in size and shape of the 2-dimensional images of pure specimens of fish protamines (salmon, herring (clupeine, Y-l) and rainbow trout (Salmo irideus)). On the basis of these images, a model for the 3-dimensional structure of the fish protamines has been proposed (2).The known amino acid sequences of fish protamines show stretches of positively charged arginines, separated by regions of neutral amino acids (3). The proposed model for protamine structure (2) consists of an irregular, right-handed helix with the segments of adjacent arginines forming the loops of the coil.

Design and calibration of an IVEM for general 3-dimensional imaging of non-diffracting materials

1992 ◽  
Vol 50 (2) ◽  
pp. 1040-1041
Author(s):  
Neil Rowlands ◽  
Jeff Price ◽  
Michael Kersker ◽  
Seichi Suzuki ◽  
Steve Young ◽  
...  
Keyword(s):  
3D Imaging ◽  
Tomographic Reconstruction ◽  
Three Dimensional ◽  
3 Dimensional ◽  
3D Microstructure ◽  
Image Translation ◽  
Digital Correlation ◽  
On Line ◽  
Mechanical Stage ◽  
Projection Angle

Three-dimensional (3D) microstructure visualization on the electron microscope requires that the sample be tilted to different positions to collect a series of projections. This tilting should be performed rapidly for on-line stereo viewing and precisely for off-line tomographic reconstruction. Usually a projection series is collected using mechanical stage tilt alone. The stereo pairs must be viewed off-line and the 60 to 120 tomographic projections must be aligned with fiduciary markers or digital correlation methods. The delay in viewing stereo pairs and the alignment problems in tomographic reconstruction could be eliminated or improved by tilting the beam if such tilt could be accomplished without image translation.A microscope capable of beam tilt with simultaneous image shift to eliminate tilt-induced translation has been investigated for 3D imaging of thick (1 μm) biologic specimens. By tilting the beam above and through the specimen and bringing it back below the specimen, a brightfield image with a projection angle corresponding to the beam tilt angle can be recorded (Fig. 1a).

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