The problem of reconstructing a multidimensional signal from its projection is uniquely multidimensional, having no 1-D counterpart. It has applications that range from computer-aided tomography to geophysical signal processing. It is a problem which can be explored from several points of view—as a deconvolution problem, a modeling problem, an estimation problem, or an interpolation problem. == Motivation and applications == Many fields in science and engineering use reconstruction from projections, especially in imaging. It is widely applied geophysical tomography, medical imaging and industrial radiography. For example, in a CT scanner, the 3D structure of the patient’s body being scanned is measured with beams going through the tissue and hitting a detector, giving a flat projection of the body from that angle. Multiple projections are put together to get an image of the position and shape of structures inside in 3D. == Problem statement and basics == A projection is a linear mapping of an M {\displaystyle M} dimensional signal into an N {\displaystyle N} dimensional one, where N ≤ M {\displaystyle N\leq M} . And the objective of reconstruction is to restore the M {\displaystyle M} dimensional signal based on the N {\displaystyle N} dimensional signal. The following case is a 2-D signal projected into 1D signal. The signal in the original coordinate is denoted as d ( u , v ) {\displaystyle d(u,v)} . Now consider a collimated beam of radiation coming from the opposite orientation of v ^ {\displaystyle {\hat {v}}} , producing a projection along u ^ {\displaystyle {\hat {u}}} . v ^ {\displaystyle {\hat {v}}} and u ^ {\displaystyle {\hat {u}}} are normal to each other, and the angle between u {\displaystyle u} and u ^ {\displaystyle {\hat {u}}} is theta. The signal obtained along u ^ {\displaystyle {\hat {u}}} axis is defined to be p θ ( u ^ ) {\displaystyle p_{\theta }({\hat {u}})} . The relationship between the original coordinate and the rotated coordinate is given by [ u ^ v ^ ] = [ cos θ sin θ − sin θ cos θ ] [ u v ] {\displaystyle {\begin{bmatrix}{\hat {u}}\\{\hat {v}}\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}u\\v\end{bmatrix}}} or inversely, [ u v ] = [ cos θ − sin θ sin θ cos θ ] [ u ^ v ^ ] {\displaystyle {\begin{bmatrix}u\\v\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}{\hat {u}}\\{\hat {v}}\end{bmatrix}}} Then we have p θ ( u ^ ) = ∫ − ∞ ∞ d ( u , v ) d v ^ = ∫ − ∞ ∞ d ( u ^ cos ( θ ) − v ^ sin ( θ ) , u ^ sin ( θ ) + v ^ cos ( θ ) ) d v ^ {\displaystyle p_{\theta }({\hat {u}})=\int _{-\infty }^{\infty }d(u,v)\,\mathrm {d} {\hat {v}}=\int _{-\infty }^{\infty }d({\hat {u}}\cos(\theta )-{\hat {v}}\sin(\theta ),{\hat {u}}\sin(\theta )+{\hat {v}}\cos(\theta ))\,\mathrm {d} {\hat {v}}} By varying theta, a large number of projections can be obtained. Given the projection-slice theorem, D ( Ω , θ ) {\displaystyle D(\Omega ,\theta )} ,the slice of the Fourier transform of d ( u , v ) {\displaystyle d(u,v)} at angle theta, is equivalent to P θ ( Ω ) {\displaystyle P_{\theta }(\Omega )} , the Fourier Transform of the projection p θ ( u ^ ) {\displaystyle p_{\theta }({\hat {u}})} . Therefore, the unknown d ( u , v ) {\displaystyle d(u,v)} can be obtained from its Fourier transform by means of the Fourier transform inversion integral d ( u , v ) = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ D ( Ω 1 , Ω 2 ) e j Ω 1 u e j Ω 2 v d Ω 1 , Ω 2 {\displaystyle \mathrm {d} (u,v)={\frac {1}{4\pi ^{2}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }D(\Omega _{1},\Omega _{2})e^{j\Omega _{1}u}e^{j\Omega _{2}v}\,\mathrm {d} \Omega _{1},\Omega _{2}} = 1 4 π 2 ∫ 0 ∞ ∫ − π π D ( Ω , θ ) e j Ω u cos ( θ ) e j Ω v s i n θ | Ω | d Ω d θ {\displaystyle ={\frac {1}{4\pi ^{2}}}\int _{0}^{\infty }\int _{-\pi }^{\pi }D(\Omega ,\theta )e^{j\Omega u\cos(\theta )}e^{j\Omega vsin\theta }{\begin{vmatrix}\Omega \end{vmatrix}}\,\mathrm {d} \Omega \mathrm {d} \theta } = 1 4 π 2 ∫ − π π ∫ 0 ∞ P θ ( Ω ) e j Ω ( u cos θ + v sin θ ) | Ω | d Ω d θ {\displaystyle ={\frac {1}{4\pi ^{2}}}\int _{-\pi }^{\pi }\int _{0}^{\infty }P_{\theta }(\Omega )e^{j}\Omega (u\cos \theta +v\sin \theta ){\begin{vmatrix}\Omega \end{vmatrix}}\,\mathrm {d} \Omega \mathrm {d} \theta } = 1 4 π 2 ∫ 0 π ( ∫ − ∞ ∞ P θ ( Ω ) | Ω | {\displaystyle ={\frac {1}{4\pi ^{2}}}\int _{0}^{\pi }(\int _{-\infty }^{\infty }P_{\theta }(\Omega ){\begin{vmatrix}\Omega \end{vmatrix}}} e j Ω u ^ d Ω ) d θ {\displaystyle e^{j\Omega {\hat {u}}}\mathrm {d} \Omega )\mathrm {d} \theta } By taking the inverse Fourier Transform and assuming g ( u ^ ) = F − 1 ( | Ω | 2 ) {\displaystyle g({\hat {u}})={\mathcal {F}}^{-1}({{\begin{vmatrix}\Omega \end{vmatrix}}^{2}})} , we get d ( u , v ) = ∑ i △ θ i [ p θ ( u ^ ) ∗ g θ i ( u ^ ) ] {\displaystyle d(u,v)=\sum _{i}\vartriangle \theta _{i}[p_{\theta }({\hat {u}})g_{\theta i}({\hat {u}})]} == Approaches == In practice, there are a wide variety of methods that are utilized, most of which are reconstruct 3-D information (volume) from 2-D signals (image). Typically used methods are CT, MRI, PET and SPECT. And the filtered back projection based on the principles introduced above are commonly applied. === Computed Tomography (CT) === In CT, a volume is formed by stacking the axial slices. The software cuts the volume in a different plane (usually orthogonal). Commonly, slice data is generated using an X-ray source that rotates around the object. X-ray sensors are positioned on the opposite side of the circle from the X-ray source. === Magnetic resonance imaging (MRI) === In MRI, energy from an oscillating magnetic field is temporarily applied to the patient at the appropriate resonance frequency. The protons (hydrogen atoms) emit a radio frequency signal which is measured by a receiving coil. The radio signal can be made to encode position information by varying the main magnetic field using gradient coils. === Positron emission tomography (PET) === The system detects pairs of gamma rays emitted indirectly by a positron-emitting radionuclide (tracer), which is introduced into the body on a biologically active molecule. Three-dimensional images of tracer concentration within the body are then constructed by computer analysis. In modern PET-CT scanners, three dimensional imaging is often accomplished with the aid of a CT X-ray scan performed on the patient during the same session, in the same machine. === Single-photon emission computed tomography (SPECT) === SPECT imaging is performed by using a gamma camera to acquire multiple 2-D images (projections) from multiple angles. Multiple projections are used to yield a 3-D data set. This data set may then be manipulated to show thin slices along any chosen axis of the body. SPECT is similar to PET in its use of radioactive tracer material and detection of gamma rays, while the tracers used in SPECT emit gamma radiation that is measured more directly.
Tiki Wiki CMS Groupware
Tiki Wiki CMS Groupware or simply Tiki, originally known as TikiWiki, is a free and open source Wiki-based content management system and online office suite written primarily in PHP and distributed under the GNU Lesser General Public License (LGPL-2.1-only) license. In addition to enabling websites and portals on the internet and on intranets and extranets, Tiki contains a number of collaboration features allowing it to operate as a Geospatial Content Management System (GeoCMS) and Groupware web application. Tiki includes all the basic features common to most CMSs such as the ability to register and maintain individual user accounts within a flexible and rich permission / privilege system, create and manage menus, RSS-feeds, customize page layout, perform logging, and administer the system. All administration tasks are accomplished through a browser-based user interface. Tiki features an all-in-one design, as opposed to a core+extensions model followed by other CMSs. This allows for future-proof upgrades (since all features are released together), but has the drawback of an extremely large codebase (more than 1,000,000 lines). Tiki can run on any computing platform that supports both a web server capable of running PHP 5 (including Apache HTTP Server, IIS, Lighttpd, Hiawatha, Cherokee, and nginx) and a MySQL/MariaDB database to store content and settings. == Major components == Tiki has four major categories of components: content creation and management tools, content organization tools and navigation aids, communication tools, and configuration and administration tools. These components enable administrators and users to create and manage content, as well as letting them communicate to others and configure sites. In addition, Tiki allows each user to choose from various visual themes. These themes are implemented using CSS and the open source Smarty template engine. Additional themes can be created by a Tiki administrator for branding or customization as well. == Internationalization == Tiki is an international project, supporting many languages. The default interface language in Tiki is English, but any language that can be encoded and displayed using the UTF-8 encoding can be supported. Translated strings can be included via an external language file, or by translating interface strings directly, through the database. As of 29 September 2005, Tiki had been fully translated into eight languages and reportedly 90% or more translated into another five languages, as well as partial translations for nine additional languages. Tiki also supports interactive translation of actual wiki pages and was the initial wiki engine used in the Cross Lingual Wiki Engine Project. This allows Tiki-based web sites to have translated content — not just the user interface. == Implementation == Tiki is developed primarily in PHP with some JavaScript code. It uses MySQL/MariaDB as a database. It will run on any server that provides PHP 5, including Apache and Microsoft's IIS. Tiki components make extensive use of other open source projects, including Zend Framework, Smarty, jQuery, HTML Purifier, FCKeditor, Raphaël, phpCAS, and Morcego. When used with Mapserver Tiki can become a Geospatial Content Management System. == Project team == Tiki is under active development by a large international community of over 300 developers and translators, and is one of the largest open-source teams in the world. Project members have donated the resources and bandwidth required to host the tiki.org website and various subdomains. The project members refer to this dependence on their own product as "eating their own dogfood", which they have been doing since the early days of the project. Tiki community members also participate in various related events such as WikiSym and the Libre Software Meeting. == History == Tiki has been hosted on SourceForge.net since its initial release (Release 0.9, named Spica) in October 2002. It was primarily the development of Luis Argerich (Buenos Aires, Argentina), Eduardo Polidor (São Paulo, Brazil), and Garland Foster (Green Bay, WI, United States). In July 2003, Tiki was named the SourceForge.net July 2003 Project of the Month. In late 2003, a fork of Tiki was used to create Bitweaver. In 2006, Tiki was named to CMS Report's Top 30 Web Applications. In 2008, Tiki was named to EContent magazine's Top 100 In 2009, Tiki adopted a six-month release cycle and announced the selection of a Long Term Support (LTS) version and the Tiki Software Community Association was formed as the legal steward for Tiki. The Tiki Software Association is a not-for-profit entity established in Canada. Previously, the entire project was run entirely by volunteers. In 2010, Tiki received Best of Open Source Software Applications Award (BOSSIE) from InfoWorld, in the Applications category. In 2011, Tiki was named to CMS Report's Top 30 Web Applications. In 2012, Tiki was named "Best Web Tool" by WebHostingSearch.com, and "People's Choice: Best Free CMS" by CMS Critic. In 2016, Tiki was named as one of the "10 Best Open Source Collaboration Software Tools" by Small Business Computing. == Name == The name TikiWiki is written in CamelCase, a common Wiki syntax indicating a hyperlink within the Wiki. It is most likely a compound word combining two Polynesian terms, Tiki and Wiki, to create a self-rhyming name similar to wikiwiki, a common variant of wiki. A backronym has also been formed for Tiki: Tightly Integrated Knowledge Infrastructure. == Release Information and History == In general, the Tiki Software Community Association releases a new major version of Tiki Wiki every 8 months where prior, non-LTS, major versions are supported until the first minor version release of the next major version (i.e., 16.0 ⇒ 17.1). Starting with version 12.x, Tiki Wiki LTS is supported for 5 years where it enters a security/maintenance release cycle upon the release of the next LTS version. Tiki Wiki's release history is outlined below.
Creepy treehouse
Creepy treehouse is a social media term, or internet slang, referring to websites or technologies that are used for educational purposes but regarded by students as an invasion of privacy. == History == The term was first described in 2008 by Utah Valley University instructional-design services director Jared Stein as "institutionally controlled technology/tool that emulates or mimics pre-existing [sic] technologies or tools that may already be in use by the learners, or by learners' peer groups." This was when social media such as Facebook was starting to become mainstream and professors would try and get students to interact with them on the site for educational purposes. Some professors would require their students to use Facebook or Twitter as part of class assignments. == Usage == The term was first described as "technological innovations by faculty members that make students’ skin crawl." The term also refers to online accounts and websites that users tend to avoid, especially young people who avoid visiting the pages of educators and other adults. Author Martin Weller defines creepy treehouse as a digital space where authority figures are viewed as invading younger people's privacy. One such example is a professor giving his students an option to use a popular video game to learn about history instead of writing an essay. Students in that class chose to write the essay instead as the method was previously unmentioned and it was not an unnatural method of interaction. Another example given was Blackboard Sync, a feature that was used to connect the school website Blackboard with students' Facebook accounts. == Solutions == University of Regina professor Alec Couros suggests that instead of "forcing" student participation with their own digital platforms, professors should use methods like online forums. Jason Jones of chronicle.com suggested letting students create social media groups for the class themselves and explaining why using technologies is required and important.
List of search appliance vendors
A search appliance is a type of computer which is attached to a corporate network for the purpose of indexing the content shared across that network in a way that is similar to a web search engine. It may be made accessible through a public web interface or restricted to users of that network. A search appliance is usually made up of: a gathering component, a standardizing component, a data storage area, a search component, a user interface component, and a management interface component. == Vendors of search appliances == Fabasoft Google InfoLibrarian Search Appliance™ Maxxcat Searchdaimon Thunderstone == Former/defunct vendors of search appliances == Black Tulip Systems Google Search Appliance Index Engines Munax Perfect Search Appliance
GPU switching
GPU switching is a mechanism used on computers with multiple graphic controllers. This mechanism allows the user to either maximize the graphic performance or prolong battery life by switching between the graphic cards. It is mostly used on gaming laptops which usually have an integrated graphic device and a discrete video card. == Basic components == Most computers using this feature contain integrated graphics processors and dedicated graphics cards that applies to the following categories. === Integrated graphics === Also known as: Integrated graphics, shared graphics solutions, integrated graphics processors (IGP) or unified memory architecture (UMA). This kind of graphics processors usually have much fewer processing units and share the same memory with the CPU. Sometimes the graphics processors are integrated onto a motherboard. It is commonly known as: on-board graphics. A motherboard with on-board graphics processors doesn't require a discrete graphics card or a CPU with graphics processors to operate. === Dedicated graphics cards === Also known as: discrete graphics cards. Unlike integrated graphics, dedicated graphics cards have much more processing units and have its own RAM with much higher memory bandwidth. In some cases, a dedicated graphics chip can be integrated onto the motherboards, B150-GP104 for example. Regardless of the fact that the graphics chip is integrated, it is still counted as a dedicated graphics cards system because the graphics chip is integrated with its own memory. == Theory == Most Personal Computers have a motherboard that uses a Southbridge and Northbridge structure. === Northbridge control === The Northbridge is one of the core logic chipset that handles communications between the CPU, GPU, RAM and the Southbridge. The discrete graphics card is usually installed onto the graphics card slot such as PCI-Express and the integrated graphics is integrated onto the CPU itself or occasionally onto the Northbridge. The Northbridge is the most responsible for switching between GPUs. The way how it works usually has the following process (refer to the Figure 1. on the right): The Northbridge receives input from Southbridge through the internal bus. The Northbridge signals to CPU through the Front-side bus. The CPU runs the task assignment application (usually the graphics card driver) to determine which GPU core to use. The CPU passes down the command to the Northbridge. The Northbridge passes down the command to the according GPU core. The GPU core processes the command and returns the rendered data back to the Northbridge. The Northbridge sends the rendered data back to Southbridge. === Southbridge control === The Southbridge is a set of integrated circuits such Intel's I/O Controller Hub (ICH). It handles all of a computer's I/O functions, such as receiving the keyboard input and outputting the data onto the screen. The way how it usually works usually has two steps: Take in the user input and pass it down to the Northbridge. (Optional) Receive the rendered data from the Northbridge and output it. The reason why the second step can be optional is that sometimes the rendered the data is outputted directly from the discrete graphics card which is located on the graphics card slot so there is no need to output the data through the Southbridge. == Main purpose == GPU switching is mostly used for saving energy by switching between graphic cards. The dedicated graphics cards consume much more power than integrated graphics but also provides higher 3D performances, which is needed for a better gaming and CAD experience. Following is a list of the TDPs of the most popular CPU with integrated graphics and dedicated graphics cards. The dedicated graphics cards exhibit much higher power consumption than the integrated graphics on both platforms. Disabling them when no heavy graphics processing is needed can significantly lower the power consumption. == Technologies == === Nvidia Optimus === Nvidia Optimus™ is a computer GPU switching technology created by Nvidia that can dynamically and seamlessly switch between two graphic cards based on running programs. === AMD Enduro === AMD Enduro™ is a collective brand developed by AMD that features many new technologies that can significantly save power. It was previously named as: PowerXpress and Dynamic Switchable Graphics (DSG). This technology implements a sophisticated system to predict the potential usage need for graphics cards and switch between graphics cards based on predicted need. This technology also introduces a new power control plan that allows the discrete graphics cards consume no energy when idling. == Manufacturers == === Integrated graphics === In personal computers, the IGP (integrated graphics processors) are mostly manufactured by Intel and AMD and are integrated onto their CPUs. They are commonly known as: Intel HD and Iris Graphics - also called HD series and Iris series AMD Accelerated Processing Unit (APU) - also formerly known as: fusion === Dedicated graphics cards === The most popular dedicated graphics cards are manufactured by AMD and Nvidia. They are commonly known as: AMD Radeon Nvidia GeForce == Drivers and OS support == Most common operating systems have built-in support for this feature. However, the users may download the updated drivers from Nvidia or AMD for better experience. === Windows support === Windows 7 has built-in support for this feature. The system automatically switches between GPUs depending on the program that's running. However, the user may switch the GPUs manually through device manager or power manager. === Linux === Modern Linux systems handle hybrid graphics in two parts: power/control for the inactive GPU, and optional render offloading for individual applications. vga_switcheroo (in the kernel since 2.6.34) coordinates power and mux control on systems with multiple GPUs. It was designed primarily for muxed designs (hardware display switch), and on muxless laptops it is typically used only for power control. A display server restart is no longer required for offloading on muxless systems. DRI PRIME (Mesa) enables per-process render offload on muxless systems: an app renders on the discrete GPU and the integrated GPU presents the result. Users can opt in via the DRI_PRIME environment variable (e.g., DRI_PRIME=1) or desktop integration. On GNOME, the switcheroo-control service exposes the discrete GPU to the shell, adding a “Launch using Discrete Graphics Card” entry to app menus on supported systems (Wayland or Xorg), which invokes render offload under the hood. With the proprietary Nvidia driver, render offload is provided as PRIME Render Offload (supported since driver 435.xx). Distributions commonly ship a helper like prime-run or desktop menu entries that set the required environment for offloading. ==== Notes and limitations (Linux) ==== On muxless systems the internal display is hard-wired to the integrated GPU; the discrete GPU cannot directly drive that panel and instead renders offscreen for composition by the iGPU. External displays connected to the dGPU may allow direct output depending on the laptop’s wiring. Power-saving behavior varies by driver and distro defaults. Some setups need explicit configuration to power down the inactive GPU when idle. Desktop integrations (e.g., GNOME's menu item) simply opt an app into offload; they do not "auto-switch" the whole session. Users can still launch apps on either GPU as needed.
Geometric primitive
In vector computer graphics, CAD systems, and geographic information systems, a geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segments, which were all that early vector graphics systems had. In constructive solid geometry, primitives are simple geometric shapes such as a cube, cylinder, sphere, cone, pyramid, torus. Modern 2D computer graphics systems may operate with primitives which are curves (segments of straight lines, circles and more complicated curves), as well as shapes (boxes, arbitrary polygons, circles). A common set of two-dimensional primitives includes lines, points, and polygons, although some people prefer to consider triangles primitives, because every polygon can be constructed from triangles (polygon triangulation). All other graphic elements are built up from these primitives. In three dimensions, triangles or polygons positioned in three-dimensional space can be used as primitives to model more complex 3D forms. In some cases, curves (such as Bézier curves, circles, etc.) may be considered primitives; in other cases, curves are complex forms created from many straight, primitive shapes. == Common primitives == The set of geometric primitives is based on the dimension of the region being represented: Point (0-dimensional), a single location with no height, width, or depth. Line or curve (1-dimensional), having length but no width, although a linear feature may curve through a higher-dimensional space. Planar surface or curved surface (2-dimensional), having length and width. Volumetric region or solid (3-dimensional), having length, width, and depth. In GIS, the terrain surface is often spoken of colloquially as "2 1/2 dimensional," because only the upper surface needs to be represented. Thus, elevation can be conceptualized as a scalar field property or function of two-dimensional space, affording it a number of data modeling efficiencies over true 3-dimensional objects. A shape of any of these dimensions greater than zero consists of an infinite number of distinct points. Because digital systems are finite, only a sample set of the points in a shape can be stored. Thus, vector data structures typically represent geometric primitives using a strategic sample, organized in structures that facilitate the software interpolating the remainder of the shape at the time of analysis or display, using the algorithms of Computational geometry. A Point is a single coordinate in a Cartesian coordinate system. Some data models allow for Multipoint features consisting of several disconnected points. A Polygonal chain or Polyline is an ordered list of points (termed vertices in this context). The software is expected to interpolate the intervening shape of the line between adjacent points in the list as a parametric curve, most commonly a straight line, but other types of curves are frequently available, including circular arcs, cubic splines, and Bézier curves. Some of these curves require additional points to be defined that are not on the line itself, but are used for parametric control. A Polygon is a polyline that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior. Some data models allow for a single feature to consist of multiple polylines, which could collectively connect to form a single closed boundary, could represent a set of disjoint regions (e.g., the state of Hawaii), or could represent a region with holes (e.g., a lake with an island). A Parametric shape is a standardized two-dimensional or three-dimensional shape defined by a minimal set of parameters, such as an ellipse defined by two points at its foci, or three points at its center, vertex, and co-vertex. A Polyhedron or Polygon mesh is a set of polygon faces in three-dimensional space that are connected at their edges to completely enclose a volumetric region. In some applications, closure may not be required or may be implied, such as modeling terrain. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior. A triangle mesh is a subtype of polyhedron in which all faces must be triangles, the only polygon that will always be planar, including the Triangulated irregular network (TIN) commonly used in GIS. A parametric mesh represents a three-dimensional surface by a connected set of parametric functions, similar to a spline or Bézier curve in two dimensions. The most common structure is the Non-uniform rational B-spline (NURBS), supported by most CAD and animation software. == Application in GIS == A wide variety of vector data structures and formats have been developed during the history of Geographic information systems, but they share a fundamental basis of storing a core set of geometric primitives to represent the location and extent of geographic phenomena. Locations of points are almost always measured within a standard Earth-based coordinate system, whether the spherical Geographic coordinate system (latitude/longitude), or a planar coordinate system, such as the Universal Transverse Mercator. They also share the need to store a set of attributes of each geographic feature alongside its shape; traditionally, this has been accomplished using the data models, data formats, and even software of relational databases. Early vector formats, such as POLYVRT, the ARC/INFO Coverage, and the Esri shapefile support a basic set of geometric primitives: points, polylines, and polygons, only in two dimensional space and the latter two with only straight line interpolation. TIN data structures for representing terrain surfaces as triangle meshes were also added. Since the mid 1990s, new formats have been developed that extend the range of available primitives, generally standardized by the Open Geospatial Consortium's Simple Features specification. Common geometric primitive extensions include: three-dimensional coordinates for points, lines, and polygons; a fourth "dimension" to represent a measured attribute or time; curved segments in lines and polygons; text annotation as a form of geometry; and polygon meshes for three-dimensional objects. Frequently, a representation of the shape of a real-world phenomenon may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This dimensional generalization correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood, but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines). == In 3D modelling == In CAD software or 3D modelling, the interface may present the user with the ability to create primitives which may be further modified by edits. For example, in the practice of box modelling the user will start with a cuboid, then use extrusion and other operations to create the model. In this use the primitive is just a convenient starting point, rather than the fundamental unit of modelling. A 3D package may also include a list of extended primitives which are more complex shapes that come with the package. For example, a teapot is listed as a primitive in 3D Studio Max. == In graphics hardware == Various graphics accelerators exist with hardware acceleration for rendering specific primitives such as lines or triangles, frequently with texture mapping and shaders. Modern 3D accelerators typically accept sequences of triangles as triangle strips.
General time- and transfer constant analysis
The general time- and transfer-constants (TTC) analysis is the generalized version of the Cochran-Grabel (CG) method, which itself is the generalized version of zero-value time-constants (ZVT), which in turn is the generalization of the open-circuit time constant method (OCT). While the other methods mentioned provide varying terms of only the denominator of an arbitrary transfer function, TTC can be used to determine every term both in the numerator and the denominator. Its denominator terms are the same as that of Cochran-Grabel method, when stated in terms of time constants (when expressed in Rosenstark notation). however, the numerator terms are determined using a combination of transfer constants and time constants, where the time constants are the same as those in CG method. Transfer constants are low-frequency ratios of the output variable to input variable under different open- and short-circuited active elements. In general, a transfer function (which can characterize gain, admittance, impedance, trans-impedance, etc., based on the choice of the input and output variables) can be written as: H ( s ) = a 0 + a 1 s + a 2 s 2 + … + a m s m 1 + b 1 s + b 2 s 2 + … + b n s n {\displaystyle H(s)={\frac {a_{0}+a_{1}s+a_{2}s^{2}+\ldots +a_{m}s^{m}}{1+b_{1}s+b_{2}s^{2}+\ldots +b_{n}s^{n}}}} == The denominator terms == The first denominator term b 1 {\textstyle b_{1}} can be expressed as the sum of zero value time constants (ZVTs): b 1 = ∑ i = 1 N τ i 0 {\displaystyle b_{1}=\sum _{i=1}^{N}\tau _{i}^{0}} where τ i 0 {\textstyle \tau _{i}^{0}} is the time constant associated with the reactive element i {\textstyle i} when all the other sources are zero-valued (hence the superscript '0'). Setting a capacitor value to zero corresponds to an open circuit, while a zero-valued inductor is a short circuit. So for calculation of the τ i 0 {\textstyle \tau _{i}^{0}} , all other capacitors are open-circuited and all other inductors are short-circuited. This is the essence of the ZVT method, which reduces to OCT when only capacitors are involved. All independent sources are also zero-valued during the time constant calculations (voltage sources short-circuited and current source open-circuited). In this case, if the element in question (element i {\textstyle i} ) is a capacitor, the time constant is given by τ i 0 = R i 0 C i {\displaystyle \tau _{i}^{0}=R_{i}^{0}C_{i}} and when element i {\textstyle i} is an inductor is it given by: τ i 0 = L i / R i 0 {\displaystyle \tau _{i}^{0}=L_{i}/R_{i}^{0}} . where in both cases, the resistance R i 0 {\textstyle R_{i}^{0}} , is the resistance seen by elements i {\textstyle i} (denoted by subscript), when all the other elements are zero-valued (denoted by the zero superscript). The second-order denominator term is equal to: b 2 = ∑ i = 1 N − 1 ∑ j = i + 1 N τ i 0 τ j i = ∑ i 1 ⩽ i ∑ j < j ⩽ N τ i 0 τ j i {\displaystyle b_{2}=\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}\tau _{i}^{0}\tau _{j}^{i}=\sum _{i}^{1\leqslant i}\sum _{j}^{