Head First 2D Geometry

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Self-assembly of organic molecules at solid-liquid interfaces is a route for developing novel functional materials on surfaces and modeling assembly phenomena in 3D. 5-Alkoxylated isophthalic acids (ISA) are known to self-assemble into two-dimensional (2D) lamellae at the interface between a surface of Au(111) or HOPG (highly oriented pyrolytic graphite) and a solvent. Presently, the self-assembly of 4,6-dialkoxylated isophthalic acid derivatives with variable alkyl chain length is investigated at Au(111)-water, Au(111)-tetradecane and HOPG-tetradecane interfaces with a particular focus on the first one. The main aspect of this study is to evaluate the role of the molecular geometry and different interactions in the 2D assembly of amphiphilic molecules. In contrast to 5-alkoxylated ISA, 4,6-dialkoxylated ISA derivatives self-assemble preferentially into arrays of cyclic pentameric/hexameric structures, which appear as 2D analogues of the inverted hexagonal phase of lipids. As a general trend, the derivatives bearing shorter alkyl chains show a higher level of ordering at Au(111)-liquid interfaces. In particular, at the Au(111)-water interface, the 4,6-diheptyloxy ISA derivative forms exclusively pentamers, which are arranged in a quasi-hexagonal lattice. Moreover, the cyclic pentameric features are not empty but host a single isophthalic acid residue which is found to be dynamic. Finally, the packing of the diheptyloxy derivative shows a distinct potential dependence: while at more negative potentials the pentameric arrangement is converted into lamellae, at more positive potentials a loosely packed zig-zag pattern is formed. The present results show that at different solid-liquid interfaces 4,6-dialkoxylated ISA derivatives tend to form cyclic structures that are 2D analogues of an inverted hexagonal phase, akin to lipids having two hydrophobic alkyl chains and a small polar head group. Moreover, the substrate potential at the Au(111)-water interface can tune the 2D molecular arrangement.

Methods for classifying and measuring orientations of objects, as nonlimiting examples, implants utilizing two-dimensional radiographs. One such method determines a three-dimensional orientation of an object based on its area projected onto a two-dimensional image and known or measured geometry. Another such method provides an automated solution to computationally determine the orientation and characterizing features of an implant based on two-dimensional radiographs. Orientations and characteristics of one or more objects in the vicinity of an object of interest may also be determined.

1. A method of determining a three-dimensional orientation of a first object that is partially obscured by a second object, the first and second objects having known or measured geometries and shapes, the method comprising:

2. The method of claim 1, further comprising determining other characteristics of the object, wherein the other characteristics of the object comprise one or more of make, model, and material of the object based on the area thereof projected onto the two-dimensional radiographic image and the known or measured geometry and radiographic opacity thereof.

10. The method of claim 8, further comprising determining other characteristics of the first object based on its area projected onto the two-dimensional radiographic image and the known or measured geometry thereof.

11. The method of claim 10, wherein the other characteristics of the first object comprise one or more of make, model, and material of the first object based on the area thereof projected onto the two-dimensional radiographic image and the known or measured geometry and radiographic opacity thereof.

12. The method of claim 10, further comprising determining the orientation and characteristics of the second object and determining therefrom a relative orientation of the first object to the second object.

I just followed the directions in the \"incomplete\" which said to add in explanations of the formulae ... Please feel free to edit to take out redundancy. However I did add in the following explanations:- the fact that the formula in the third figure is actually the same as the cross-section represented by the ellipse, which is why you may not get the joke after reading the first picture;- the use of 'd', 'r' and 'h' in the third figure, which adds to the confusion as they imply \"diameter\", \"radius\" and \"height\"- the fact that the area calculations must take into account the overlapping shapes (there were previously references to \"semi-ellipses\" which are extrapolations, not what's drawn there)Haven't yet done the last figure- pretty sure 'b' 'd' and 'h' are for 'breadth', 'depth' and 'height' and while 'height' is also used for 2D rectangles, 'breadth' less so in maths textbooks (usually 'width')- whoever pointed out that there is a theta as well, pretty sure it's only there because it's necessary for the area calculation, as 'depth' only really applies as labelled to rectangular prisms - if the base were not rectangular, 'd' would not be equal to the 'depth'Will try to come back later and shorten... 162.158.166.40 18:56, 1 September 2021 (UTC)

The first step of creating most 3D content is to build the physical representation of the object. The 3D object as a whole is referred as a model. The physical representation of the item is referred as the geometry, the skin, or the mesh. The act of building geometry is referred to as modeling, creating, sculpting, or authoring a 3D model.

After a model is created, it is classified into one of two categories as either a static or skeletal mesh. The classification depends on how you want to use your model. A static mesh is just a geometry, while a skeletal mesh has a movement system or a 'rig' built to go with it. For example, a human character can have a representation of a skeleton (head bones, arm bones, leg bones etc.).

For example: RAS means that the first dimension (X) points towards the righthand side of the head, the second dimension (Y) points towards the Anterioraspect of the head, and the third dimension (Z) points towards the top of thehead.The directions are considered to be from the subject's perspective.For example, in the RAS coordinate system, a point to the subject's leftwill have a negative x value.

The \"space\" and all coordinates expressed in this space are by design atransformation of the real world geometry, and nearly always different from theindividual subject space that it stems from.An example is theTalairach-Tournoux space, which is constructed by piecewise linear scaling of anindividual's brain to that of the Talairach-Tournoux 1988 atlas. In theTalairach-Tournoux space, the origin of the coordinate system is at the AC andunits are expressed in mm.

The coordinate systems below all relate to neuroscience and therefore to thehead or brain coordinates.$Please be aware that all data acquisition starts with\"device coordinates\" (scanner), which does not have to be identical to theinitial \"file format coordinates\" (DICOM), which are again different from the\"head\" coordinates (for example, NIFTI).Not only do device coordinate vary betweenhardware manufacturers, but also the head coordinates differ, mostly due todifferent conventions used in specific software packages developed by different(commercial or academic) groups.

Generally, across the MEG, EEG, and iEEG modalities, the first two pieces ofinformation for a coordinate system (origin and orientation) are specified inCoordinateSystem.The third piece of information for a coordinate system (units) are specified inCoordinateUnits.Here, can be one of of the following,depending on the data that is supposed to be documented:

The transformation of the real world geometry to an artificial frame ofreference is described in CoordinateSystem.Unless otherwise specified below, the origin is at the AC and the orientation ofthe axes is RAS.Unless specified explicitly in the sidecar file in theCoordinateUnits field, the units are assumed to be mm.

When considering how wavy layers interact with one another in a 3D textile composite, and how such layers might alter overall laminate mechanical properties, one can expect difficulties in studying the dynamic behavior of such textile composites. Waved layers will create interlaminar stresses and strains, which may vary the geometry of the unit cell of the laminate, increasing the damping and causing variations in the dynamic response. To obtain significant data, instead of fabricating and analyzing a costly series of samples, finite element models can be used to predict the behavior under different scenarios. The required number of degrees of freedom for models to accurately capture the possible geometry variations, fiber distortion and contact among adjacent fiber tows is unknown, and it will vary with assumptions for the tow geometry. As such, the analytical approach is also expensive in terms of required modeling time. Simpler analytical models will enable less expensive studies on dynamic behavior. In order to determine how detailed a model should be, experimental data and analytical data are compared.

The three IM7/PR520 panels, identified as panel02, panel03 and panel04, had identical as-woven geometry but, due to different compaction and cure pressures, exhibit different fiber volumes and different thickness (Table 1).

Because of the known idealized as-woven geometry of the samples, it has been possible to focus the attention on understanding if the increased fiber (tow) distortion at higher cure pressure affects the dynamic response of the panels.

One scan of the entire panel was then made for each of the resonant frequencies noted. From this scan a collection of the first six vibration modes of the panel were identified (Figure 5). The above procedure was conducted for all three composite panels.

If neither of these two approaches showed a good relation to the experimental laminate, then a more complex model would need to be created, focusing on the reproduction of the contact regions between fiber tows and the related deformations in tow geometry. However, if there are not significant differences between the two models, we can assume that in first approximation the deformation effects, due to different fiber distortions, can be neglected for dynamic response. 59ce067264