# Triangulation

In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to it from known points.

Specifically in surveying, triangulation per se involves only angle measurements, rather than measuring distances to the point directly as in trilateration; the use of both angles and distance measurements is referred to as triangulateration.

## Applications

Optical 3D measuring systems use this principle as well in order to determine the spatial dimensions and the geometry of an item. Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object's surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3D coordinate, is calculated from the triangular relations.

## History

Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons.

The use of triangles to estimate distances dates to antiquity. In the 6th century BC, about 250 years prior to the establishment of the Ptolemaic dynasty, the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids of ancient Egypt. He measured the length of the pyramids' shadows and that of his own at the same moment, and compared the ratios to his height (intercept theorem).[1] Thales also estimated the distances to ships at sea as seen from a clifftop by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff.[2] Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances;[3] while Liu Hui (c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.[4][5]

## References

1. ^ Diogenes Laërtius, "Life of Thales", The Lives and Opinions of Eminent Philosophers, retrieved 2008-02-22 I, 27
2. ^ Proclus, In Euclidem
3. ^ Joseph Needham (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books Ltd. pp. 539–540
4. ^
5. ^ Kurt Vogel (1983; 1997), A Surveying Problem Travels from China to Paris, in Yvonne Dold-Samplonius (ed.), From China to Paris, Proceedings of a conference held July, 1997, Mathematisches Forschungsinstitut, Oberwolfach, Germany. ISBN 3-515-08223-9.
Anelloviridae

The Anelloviridae are a recently discovered family of viruses . They are classified as a vertebrate viruses and have a non-enveloped capsid, which is round with isometric, icosahedral symmetry and has a triangulation number of 3.

The type species is torque teno virus (genus Alphatorquevirus).Etymology-"Anello" is Italian for ring, which refers to the circular genome of anelloviridae.

Capsid

A capsid is the protein shell of a virus. It consists of several oligomeric structural subunits made of protein called protomers. The observable 3-dimensional morphological subunits, which may or may not correspond to individual proteins, are called capsomeres. The capsid encloses the genetic material of the virus.

Capsids are broadly classified according to their structure. The majority of viruses have capsids with either helical or icosahedral structure. Some viruses, such as bacteriophages, have developed more complicated structures due to constraints of elasticity and electrostatics. The icosahedral shape, which has 20 equilateral triangular faces, approximates a sphere, while the helical shape resembles the shape of a spring, taking the space of a cylinder but not being a cylinder itself. The capsid faces may consist of one or more proteins. For example, the foot-and-mouth disease virus capsid has faces consisting of three proteins named VP1–3.Some viruses are enveloped, meaning that the capsid is coated with a lipid membrane known as the viral envelope. The envelope is acquired by the capsid from an intracellular membrane in the virus' host; examples include the inner nuclear membrane, the golgi membrane, and the cell's outer membrane.Once the virus has infected a cell and begins replicating itself, new capsid subunits are synthesized using the protein biosynthesis mechanism of the cell. In some viruses, including those with helical capsids and especially those with RNA genomes, the capsid proteins co-assemble with their genomes. In other viruses, especially more complex viruses with double-stranded DNA genomes, the capsid proteins assemble into empty precursor procapsids that includes a specialized portal structure at one vertex. Through this portal, viral DNA is translocated into the capsid.Structural analyses of major capsid protein (MCP) architectures have been used to categorise viruses into lineages. For example, the bacteriophage PRD1, the algal virus Paramecium bursaria Chlorella virus (PBCV-1), mimivirus and the mammalian adenovirus have been placed in the same lineage, whereas tailed, double-stranded DNA bacteriophages (Caudovirales) and herpesvirus belong to a second lineage.

Causal dynamical triangulation

Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent.

This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.

The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called Quantum Einstein Gravity, and with other recent theoretical work. A brief article appeared in the February 2007 issue of Scientific American, which gives an overview of the theory, explained why some physicists are excited about it, and put it in historical perspective. The same publication gives CDT, and its primary authors, a feature article in its July 2008 issue.

Delaunay triangulation

In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934.For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.

By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean distance. However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique.

Piecewise linear manifold

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.An isomorphism of PL manifolds is called a PL homeomorphism.

Planar graph

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection.

Plane graphs can be encoded by combinatorial maps.

The equivalence class of topologically equivalent drawings on the sphere is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map have a particular status.

Planar graphs generalize to graphs drawable on a surface of a given genus. In this terminology, planar graphs have graph genus 0, since the plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics.

Polygon triangulation

In computational geometry, polygon triangulation is the decomposition of a polygonal area (simple polygon) P into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is P.

Triangulations may be viewed as special cases of planar straight-line graphs. When there are no holes or added points, triangulations form maximal outerplanar graphs.

Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.

Stereotypes of Asians

Stereotypes of Asians may refer to:

Stereotypes of East Asians in the United States, ethnic stereotypes of Eastern and Southeast Asians found in American society as well as other similar societies.

Stereotypes of South Asians, oversimplified ethnic stereotypes of South Asian people in Western societies.

TWiT.tv

TWiT.tv, which is the operating trade name of TWiT LLC, is a podcast network (although TWiT uses the term "netcast") founded by technology broadcaster and author Leo Laporte and run by his wife and company CEO Lisa Laporte. The network began operation in April 2005 with the launch of This Week in Tech. Security Now was the second podcast on the network, debuting in August of that year. Currently, the network hosts 13 podcasts and live streaming shows from a network high of 27 in 2014. Podcasts including The Tech Guy, This Week in Tech, Security Now, FLOSS Weekly, MacBreak Weekly, and eight other podcasts covering various topics including technology companies, Ham Radio and current technology news.TWiT founder and owner Laporte, in an October 2009 speech, stated that it grossed revenues of \$1.5 million per year, while costs were around \$350,000. In November 2014, during an interview with American Public Media's Marketplace Leo Laporte stated that TWiT makes \$6 million in ad revenue a year from 5 million TWiT podcasts downloaded each month, mostly in the form of audio, and that 3,000 to 4,000 people watch its live-streamed shows. On March 18, 2015, prior to the filming of This Week in Google, Leo Laporte stated that TWiT expects to make \$7 million in revenue in fiscal year 2015, and made "almost" \$10 million in revenue in 2016.TWiT gets its name from its first and flagship podcast, This Week in Tech. The logo design originated from a traditional logic gate symbol of an "AND gate" turned on its side. Voiceovers are provided by Jim Cutler.

Tetraviridae

The Tetraviridae were a family of viruses so named because they have a T=4 symmetry (T is the triangulation number), are extremely host specific, and infect moths and butterflies of the following genera:

Genus Betatetravirus; type species: Nudaurelia capensis β virus (see Nudaurelia cytherea capensis)

Genus Omegatetravirus; type species: Nudaurelia capensis ω virusThis family was divided into three families by the ICTV in 2011: Alphatetraviridae, Carmotetraviridae and Permutotetraviridae.

Triangulation (chess)

Triangulation is a tactic used in chess to put one's opponent in zugzwang (a position when it is a disadvantage to move). Triangulation is also called losing a tempo or losing a move.

Triangulation occurs most commonly in endgames with only kings and pawns when one king can maneuver on three adjacent squares in the shape of a triangle and maintain the basic position while the opposing king only has two such squares. Thus, if one king triangulates by using three moves to return to the original square and the opposing king cannot do the same, he has lost a crucial tempo and reached the same position with the other player to move. Triangulation can occur in other endgames and even in some middlegames (Flear 2004:15).

Triangulation (geometry)

In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it into tetrahedra ("pyramids" of various shapes and sizes) packed together.

In most instances, the triangles of a triangulation are required to meet edge-to-edge and vertex-to-vertex.

Triangulation (politics)

In politics, triangulation is the strategy in which a political candidate presents their ideology as being above or between the left and right sides (or "wings") of a traditional (e.g. American or British) democratic political spectrum. It involves adopting for oneself some of the ideas of one's political opponent. The logic behind it is that it both takes credit for the opponent's ideas, and insulates the triangulator from attacks on that particular issue.

Triangulation (social science)

In the social sciences, triangulation is often used to indicate that two (or more) methods are used in a study in order to check the results of one and the same subject and is a popular method of study in sociology. "The concept of triangulation is borrowed from navigational and land surveying techniques that determine a single point in space with the convergence of measurements taken from two other distinct points." The idea is that one can be more confident with a result if different methods lead to the same result.

Triangulation is a powerful technique that facilitates validation of data through cross verification from two or more sources. In particular, it refers to the application and combination of several research methods in the study of the same phenomenon.

It can be used in both quantitative (validation) and qualitative (inquiry) studies.

It is a method-appropriate strategy of founding the credibility of qualitative analyses.

It becomes an alternative to traditional criteria like reliability and validity.

It is the preferred line in the social sciences.By combining multiple observers, theories, methods, and empirical materials, researchers hope to overcome the weakness or intrinsic biases and the problems that come from single method, single-observer and single-theory studies.

Triangulation (topology)

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h : K → X.Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories.

Triangulation sensor

Optical Triangulation Sensors are commonly used to provide door mounted safety detection on swinging automatic doors.

In North America, these types of sensors are very common and manufactured in very high volumes. When automatic swinging doors open and close, it is important that they do not come into contact with pedestrians passing through the door.

Sensors used in the automatic door industry typically fall into four categories:

Microwave sensors used to detect motion of a person as they approach an automatic door,

Reflective optical sensors that are mounted on the door header and detect the presence of a person in the door path

Camera based sensors that are also mounted on the door header and similarly detect presence; and

Triangulation sensors which are mounted on the door and move with the door to provide safety.Reflective and camera technologies do not perform well on moving swing doors, because they typically depend upon detection of changes in the background. A sensor mounted on a swing door system is constantly in motion, so the background (i.e. floor beneath the door) changes continuously. A triangulation sensor overcomes this problem.

Triangulation station

A triangulation station, also known as a triangulation pillar, trigonometrical station, trigonometrical point, trig station, trig beacon, or trig point, and sometimes informally as a trig, is a fixed surveying station, used in geodetic surveying and other surveying projects in its vicinity. The nomenclature varies regionally: they are generally known as trigonometrical or triangulation stations in North America, trig points in the United Kingdom, trig pillars in Ireland, trig stations or points in Australia and New Zealand, and trig beacons in South Africa; triangulation pillar is the more formal term for the concrete columns found in the UK.

Wi-Fi positioning system

Wi-Fi positioning system (WPS) or WiPS/WFPS is a geolocation system that uses the characteristics of nearby Wi-Fi hotspots and other wireless access points to discover where a device is located. It is used where satellite navigation such as GPS is inadequate due to various causes including multipath and signal blockage indoors, or where acquiring a satellite fix would take too long. Such systems include indoor positioning systems. Wi-Fi positioning takes advantage of the rapid growth in the early 21st century of wireless access points in urban areas.

The most common and widespread localization technique used for positioning with wireless access points is based on measuring the intensity of the received signal (received signal strength indication or RSSI) and the method of "fingerprinting". Typical parameters useful to geolocate the wireless access point include its SSID and MAC address. The accuracy depends on the number of nearby access points whose positions have been entered into the database. The Wi-Fi hotspot database gets filled by correlating mobile device GPS location data with Wi-Fi hotspot MAC addresses. The possible signal fluctuations that may occur can increase errors and inaccuracies in the path of the user. To minimize fluctuations in the received signal, there are certain techniques that can be applied to filter the noise.

In the case of low precision, some techniques have been proposed to merge the Wi-Fi traces with other data sources such as geographical information and time constraints (i.e., time geography).

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.