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Look up planar in Wiktionary, the free dictionary. Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Planar (computer...

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...

Planar transformers are high frequency transformers used in isolated switchmode power supplies operating at high frequency. As opposed to conventional...

Planarity is a 2005 puzzle computer game by John Tantalo, based on a concept by Mary Radcliffe at Western Michigan University. The name comes from the...

The Zeiss Planar is a photographic lens designed by Paul Rudolph at Carl Zeiss in 1896. Rudolph's original was a six-element symmetrical double Gauss lens...

Planar Systems, Inc. is an American digital display manufacturing corporation with a facility in Hillsboro, Oregon. Founded in 1983 as a spin-off from...

(9×12 cm fmt) Planar lenses (5 elements in 4 groups) Planar 80mm ƒ/2.8 (6×7 cm fmt) Planar 100mm ƒ/2.8 (6.5×9 cm fmt) Planar 135mm ƒ/3.5 Planar 135mm ƒ/3...

The planar process is a manufacturing process used in the semiconductor industry to build individual components of a transistor, and in turn, connect those...

In chemistry, trigonal planar is a molecular geometry model with one atom at the center and three atoms at the corners of an equilateral triangle, called...

The planar reentry equations are the equations of motion governing the unpowered reentry of a spacecraft, based on the assumptions of planar motion and...

In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can...

the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence...

In chemistry, the square planar molecular geometry describes the stereochemistry (spatial arrangement of atoms) that is adopted by certain chemical compounds...

In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has...

resulted in a planar chiral DMAP, which can be used as an enantioselective catalyst. Fu, Gregory C. (2004-08-01). "Asymmetric Catalysis with "Planar-Chiral"...

In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who...

The planar Hall sensor is a type of magnetic sensor based on the planar Hall effect of ferromagnetic materials. It measures the change in anisotropic...

Planar chirality, also known as 2D chirality, is the special case of chirality for two dimensions. Most fundamentally, planar chirality is a mathematical...

The plantar reflex is a superficial reflex elicited when the sole of the foot is stimulated with a blunt instrument. The reflex can take one of two forms...

A coral reef is an underwater ecosystem characterized by reef-building corals. Reefs are formed of colonies of coral polyps held together by calcium carbonate...

alternating planar algebras first appeared in the work of Hernando Burgos-Soto on the Jones polynomial of alternating tangles. Alternating planar algebras...

In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing...

The Rega Planar 3, together with its successors, the P3 and RP3, is a well-known budget audiophile turntable by British hi-fi manufacturer, Rega Research...

Planar transmission lines are transmission lines with conductors, or in some cases dielectric (insulating) strips, that are flat, ribbon-shaped lines....

In graph drawing, a clustered planar graph is a graph together with a hierarchical clustering on its vertices, such that the graph can be drawn together...

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate...

In computer graphics, planar is the method of arranging pixel data into several bitplanes of RAM. Each bit in a bitplane is related to one pixel on the...

In graph drawing, an upward planar drawing of a directed acyclic graph is an embedding of the graph into the Euclidean plane, in which the edges are represented...

A Sawyer motor or planar motor (also called area drive) is a multi-coordinate drive that can perform several independent movements in one plane. Goods...

three-dimensional space groups. A proof that there are only 17 distinct groups of such planar symmetries was first carried out by Evgraf Fedorov in 1891 and then derived...

planar straight-line graph. Any planar graph can be represented as a planar straight-line graph by Fáry's theorem. The planar straight-line graph is the special...

Planar projections are the subset of 3D graphical projections constructed by linearly mapping points in three-dimensional space to points on a two-dimensional...

In mathematics, a planar lamina (or plane lamina) is a plane figure representing a thin, usually uniform, flat layer of a solid. It serves also as an idealized...

forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain...

trigonometry, graph theory, and graphing are performed in a two-dimensional or planar space. In mathematics, a Euclidean plane is a Euclidean space of dimension...

In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar...

with DNA, the higher the salt concentration needed to elute that protein. Planar chromatography is a separation technique in which the stationary phase is...

higher-end models in the PS/2 line, such as the Model 80, used the term planar instead. Apple commonly uses logic board in its technical documentation...

Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and...

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into...

the molecule contains 2 lone pairs (6 − 4 = 2), resulting in a square planar molecular geometry. The overall geometry is further refined by distinguishing...

1993 200 Bulk – Planar – – – CS-24 500 1993 200 Bulk – Planar – – – 5L 500 1993 200 Bulk – Planar – – – 5S 500 1994 200 Bulk – Planar – – – SiGe 5HP 500...

{\displaystyle T\colon R^{3}\to R\,} may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used...

Planar cell polarity (PCP) is a type of cell polarity in which the cells are oriented in a coordinated way across the plane of an epithelial tissue. The...

name Worlds of Magic: Planar Conquest; which has been sometimes described as an improved sequel. In 2016 Worlds of Magic: Planar Conquest was also released...

In graph theory, a planar cover of a finite graph G is a finite covering graph of G that is itself a planar graph. Every graph that can be embedded into...

0 (planar graph) contribution survives. Planar Feynman diagrams are graphs in which no propagator cross over another one, in contrast to non-planar Feynman...

{\displaystyle N-1} is a prime number. The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries)...

panoramic image. The forward projection transforms spherical coordinates into planar coordinates. The reverse projection transforms from the plane back onto...

mathematics In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K5, the complete...

velocity center, instantaneous center, or pole of planar displacement) of a body undergoing planar movement is a point that has zero velocity at a particular...

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving...

In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces...

book IX, proposition 20. Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes...

with open boundary conditions, the planar code. As of 2025, Google Quantum AI has implemented a distance-7 planar code on their newest generation of superconducting...

graph drawing methods from planar graphs to graphs that are not planar, by embedding the non-planar graphs within a larger planar graph. Planarization may...

tweeters. Some loudspeaker designers use a planar magnetic tweeter, sometimes called a quasi-ribbon. Planar magnetic tweeters are generally less expensive...

color theorem was found only in 1976. First of all, one associates a simple planar graph G {\displaystyle G} to the given map, namely one puts a vertex in...

the graph is planar), χ (chromatic number) and χ' (chromatic index) are also sortable, allowing searching for a parameter. In the planarity column, T indicates...

coloring of the planar graph of adjacencies between regions. In graph-theoretic terms, the theorem states that for a loopless planar graph G {\displaystyle...

the special cases, planar or spherical simple open chain, M = ∑ i = 1 j   f i , {\displaystyle M=\sum _{i=1}^{j}\ f_{i},} planar or spherical simple...

called a planar mechanism. The kinematic analysis of planar mechanisms uses the subset of Special Euclidean group SE, consisting of planar rotations...

Krystek, Michael P. (2022), "On Angular Measures in Axiomatic Euclidean Planar Geometry", Measurement Science Review, 22 (4): 152–159, doi:10.2478/msr-2022-0019...

The planar quaternions make up a four-dimensional algebra over the real numbers. Their primary application is in representing rigid body motions in 2D...

In geology, a fault is a planar fracture or discontinuity in a volume of rock across which there has been significant displacement as a result of rock-mass...

of the first kind. A planar source such as this radiates sound uniformly for wavelengths longer than the dimensions of the planar source, and as frequency...

Additionally, Abrash enabled the VGA's planar memory mode (also called "unchained mode"). Even though planar memory mode is a documented part of the...

Symmetry (from Ancient Greek συμμετρία (summetría) 'agreement in dimensions, due proportion, arrangement') in everyday life refers to a sense of harmonious...

The Carl Zeiss Planar 50mm f/0.7 is one of the largest relative aperture (fastest) lenses in the history of photography. The lens was developed for the...

The first planar monolithic integrated circuit (IC) chip was demonstrated in 1960. The idea of integrating electronic circuits into a single device was...

The number density (symbol: n or ρN) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules,...

forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither...

{\displaystyle \mathbb {R} ^{3}} . A plane segment or planar region (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment...

In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open...

1950s as a bent-wire antenna. However, its most widespread use is as a planar inverted-F antenna (PIFA) in mobile wireless devices for its space saving...

planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph: every finite connected simple planar graph...

In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a book, a collection of half-planes all having the...

Planar Crown is a series of Blind mate electrical RF connectors manufactured by API Weinchel. It is fitted for example to the RF input connector of the...

the University of Amsterdam, who in 1970 used them to characterize rigid planar structures. However, this characterization, the Geiringer–Laman theorem...

the whole graph. Theorem—A 4-connected planar triangulation has a Hamiltonian cycle. Theorem—A 4-connected planar graph has a Hamiltonian cycle. An algebraic...

description of clitoral anatomy. MRI measurements, which provide a live and multi-planar method of examination, now complement the FFWHC's, as well as O'Connell's...

Cleavage, in structural geology and petrology, describes a type of planar rock feature that develops as a result of deformation and metamorphism. The degree...

Planar T* f1,4 50 mm Contax 139 Quartz with Distagon T* f4,0 18 mm Contax 159MM body Contax 167MT with Distagon T* f2,8 28 mm Contax S2 with Planar T*...

2009.12.071) F. Geiss (1987): Fundamentals of thin layer chromatography planar chromatography, Heidelberg, Hüthig, ISBN 3-7785-0854-7 Justus G. Kirchner...

types of defects are often characterized: point defects, line defects, planar defects, bulk defects. Topological homotopy establishes a mathematical method...

Planar laser-induced fluorescence (PLIF) is an optical diagnostic technique widely used for flow visualization and quantitative measurements. PLIF has...

assembly is called a planar four-bar linkage. Spherical and spatial four-bar linkages also exist and are used in practice. Planar four-bar linkages are...

The Schleicher ASK 18 is a single-seat sailplane that was built by the German manufacturer Alexander Schleicher GmbH & Co. It was designed to be a sturdy...

quadtrees with random insertion have been studied under the name weighted planar stochastic lattices. Point quadtrees are constructed as follows. Given the...

secrete oxidizable neurotransmitters by "loading" them with 5-HT or dopamine. Planar patch clamp is a novel method developed for high throughput electrophysiology...

mathematics, the left-right planarity test or de Fraysseix–Rosenstiehl planarity criterion is a characterization of planar graphs based on the properties...

fraction of an analyte in the mobile phase of a chromatographic system. In planar chromatography in particular, the retardation factor RF is defined as the...

carbon-based supercapacitors. Advanced hybrid designs include all-solid-state planar micro-supercapacitors based on 2D vanadium nitride nanosheets and cobalt...

In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices"...

isomers also occur in inorganic coordination compounds, such as square planar MX 2 Y 2 {\displaystyle {\ce {MX2Y2}}} complexes and octahedral MX 4 Y 2...

expert-level block consisting of the expansion sets Time Spiral (October 6, 2006), Planar Chaos (February 2, 2007), and Future Sight (May 4, 2007). It is set on the...

and dominating set problems for planar graphs are NP-complete, but can be solved in subexponential time using the planar separator theorem. "Each instance...

path of two adjacent edges. In Vizing's planar graph conjecture, Vizing (1965) states that all simple, planar graphs with maximum degree six or seven...

This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International...

reality. Optical waveguides can be classified according to their geometry (planar, strip, or fiber waveguides), mode structure (single-mode, multi-mode),...

reaching sufficient pressures during such an event. Planar fractures are parallel sets of multiple planar cracks or cleavages in quartz grains; they develop...

In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

characteristics over time. The planar transistor was developed by Dr. Jean Hoerni at Fairchild Semiconductor in 1959. The planar process used to make these...

slightly more stable than those of copper(III). For instance, the square planar periodate [Ag(IO5OH)2]5− and tellurate [Ag{TeO4(OH)2}2]5− complexes may...

monolithic IC was fabricated using the planar process, developed by his colleague Jean Hoerni in early 1959. In turn, the planar process was based on Carl Frosch...

cosmology from that of its contemporary AD&D game, which is a more open planar system that is less regulated than that of its counterpart. The planes were...

architecture (planar vs. non-planar design) and the number of channels/gates (2, 3, or 4). A planar double-gate MOSFET (DGMOS) employs conventional planar (layer-by-layer)...

theory, a planar straight-line graph (PSLG), also called a straight-line plane graph or plane straight-line graph, is an embedding of a planar graph in...

four graphs is planar; however, the Erdős–Gyárfás conjecture is now known to be true for the special case of 3-connected cubic planar graphs (Heckman...

The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without...

contents) Ahmed, Sherif Sayed (2014). Electronic Microwave Imaging with Planar Multistatic Arrays. Logos Verlag Berlin. p. 1. ISBN 978-3-8325-3621-3. Extract...

In chemistry, the pentagonal planar molecular geometry describes the shape of compounds where five atoms, groups of atoms, or ligands are arranged around...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

of the world permits planar, convex, and/or aspheric mirrors on either side of the vehicle. American research suggests non-planar driver-side mirrors may...

Planar hexacoordinate carbon in chemistry describes a molecular geometry featuring a planar arrangement of carbon with six surrounding atoms. No actual...

these imposing structures have earned them the nickname "elephant cages". Planar array – a flat two-dimensional array of antennas. Since an array of omnidirectional...

multicoated) 503 with Zeiss Planar 2.8/80 (CF black barrel, T*) 2000FC/M with Zeiss Planar 2.8/80 (F black barrel, T*) 500C with Zeiss S-Planar 5.6/120 (C chrome...

Planar deformation features, or PDFs, are optically recognizable microscopic features in grains of silicate minerals (usually quartz or feldspar), consisting...

are not Cartesian because the measurements are angles and are not on a planar surface. A full GCS specification, such as those listed in the EPSG and...

as revolutes, as is necessary for the links to be two-force members. A planar truss is one where all members and nodes lie within a two-dimensional plane...

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight...

projection surfaces are cylindrical (e.g., Mercator), conic (e.g., Albers), and planar (e.g., stereographic). Many mathematical projections, however, do not neatly...

The planar hydroid (Sertularella arbuscula) is a branching colonial hydroid in the family Sertulariidae. Planar hydroids are bright yellow and branched...

This is a list of tessellations. Tessellation Uniform tiling Convex uniform honeycombs List of k-uniform tilings List of Euclidean uniform tilings Uniform...

made of sheet metal and wire that Jane Fluegel terms a "three-dimensional planar counterpart of Cubist painting" that marks a "revolutionary departure from...

not a planar graph. Multiple proofs of this impossibility are known, and form part of the proof of Kuratowski's theorem characterizing planar graphs...

contains the query point (e.g. the Voronoi diagram). In the planar case, we are given a planar subdivision S, formed by multiple polygons called faces, and...

degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case...

formed from a set of letters (or words) in such a way that it has a non-planar spelling net. Gary S. Bloom, Allan Gewirtz, John W. Kennedy, and Peter J...

a linear molecular shape. [citation needed] Trigonal planar: Molecules with the trigonal planar shape are somewhat triangular and in one plane (flat)...

may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked...

Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that...

special cases become planar or spherical simple open chain, M = ∑ i = 1 j   f i , {\displaystyle M=\sum _{i=1}^{j}\ f_{i},} planar or spherical simple...

centers in Au(III) complexes, like other d8 compounds, are typically square planar, with chemical bonds that have both covalent and ionic character. Gold(I...

graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors. According to the four-color...

The Sony Zeiss Planar T* FE 50mm F1.4 ZA is a large-aperture, standard full-frame prime lens for the Sony E-mount. It was released by Sony in July 2016...

thin layer of platinum and comes in both heated and unheated forms. The planar-style sensor entered the market in 1990 and significantly reduced the mass...

A plane joint (arthrodial joint, gliding joint, plane articulation) is a synovial joint which, under physiological conditions, allows only gliding movement...

embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality...

quadratic forms are admitted, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

every planar graph have an integral Fáry embedding? More unsolved problems in mathematics In mathematics, Harborth's conjecture states that every planar graph...

graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by Saunders Mac Lane (1937);...

Rolleiflex SLX system: Lens hood Rollei-HFT Planar 80 mm f/2.8 lens Film holder Camera body (with back opened) Battery Folding waist-level finder Fast...

conjecture: every planar graph can be drawn with integer edge lengths Negami's conjecture on projective-plane embeddings of graphs with planar covers The strong...

is the Hughes AN/AWG-9 X band radar; the antenna is a 36 in (91 cm)-wide planar array, and has integrated Identification friend or foe antennas. The AWG-9...

camera than their imitators.[citation needed] The highly regarded Zeiss Planar f2.8 and Schneider Xenotar lenses, both 80mm focal length and fast in comparison...

lightning channel exceed 30,000 K, with sufficient pressure to produce planar deformation features in SiO2, a kind of polymorphism. This is also known...

bombs, sometimes in place of more conventional control surfaces, such as planar fins. They were developed in the 1950s by a team led by Sergey Belotserkovskiy [ru]...

other. Interferometers are typically used to accurately measure distances. Planar lightwave circuits are either optical integrated circuits (ICs) or optical...

In computational geometry, the gift wrapping algorithm is an algorithm for computing the convex hull of a given set of points. In the two-dimensional case...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

intersection lineations, the trace of a planar feature on another planar surface). The inclination of a planar structure in geology is measured by strike...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

Planar Handbook is an optional supplemental source book for the 3.5 edition of the Dungeons & Dragons fantasy roleplaying game. It contains updates for...

In structural geology, a fold is a stack of originally planar surfaces, such as sedimentary strata, that are bent or curved ("folded") during permanent...

Digital planar holography (DPH) is a method for designing and fabricating miniature components for integrated optics. It was invented by Vladimir Yankov...

transistor pioneer, and a member of the "traitorous eight". He developed the planar process, an important technology for reliably fabricating and manufacturing...

aromatic, however, because various types of ring strain destabilize an all-planar geometry. Although not aromatic itself, [10]annulene can transition between...

Extra-planar nebulae...

follow along a planar surface. In these cases the rake can be used to describe the line's orientation in three dimensions relative to that planar surface. One...

oscillator Inductor Relay reed relay mercury relay Other devices Hall effect sensor SQUID Spin valve Magnetic tunnel junction (MTJ) Planar Hall sensor ggNMOS...

(or the mesh current method) is a circuit analysis method for planar circuits; planar circuits are circuits that can be drawn on a plane surface with...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

between the planar and staggered forms) may be attributed to electrostatic interactions as unfavorable O−O repulsion is maximized in the planar form. Oxalate...

brain scanning or where images are needed very rapidly, is called echo-planar imaging (EPI): In this case, each RF excitation is followed by a train of...

invented the planar integrated circuit. The industry preferred Fairchild's invention over Texas Instruments' because the transistors in planar ICs were interconnected...

decades. Non-planar wingtips provide the wake control benefit of winglets, with less parasitic drag penalty, if designed carefully. The non-planar wing tip...

algorithms are known that solve the problem for planar graphs in linear time. By the Euler characteristic of planar graphs, m ≤ 3n - 6 ∈ O(n), so this is in...

volume V of the body. The polar moment of inertia of a body undergoing planar movement can be computed relative to any reference point S, I S = ∫ V ρ...

In remote sensing, ground sample distance (GSD) in a digital photo of the ground from air or space is the distance between pixel centers measured on the...

comes from the Greek word for ‘near’ and so periplanar means “approximately planar”. In their article “Periplanar or Coplanar?” Kane and Hersh point out that...

the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least...

radar program. Accordingly, the N019 was originally intended to have a flat planar array antenna and full digital signal processing, for a detection and tracking...

planar magnetic loudspeakers and in 1987 introduced the world's first full-range push-pull (with electromagnets on both sides of the membrane) planar...

In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer...

IEMs exhibiting common weaknesses of planar magnetic drivers, such as their size and high price, innovations in planar magnetic drivers now allow for newer...

also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional...

generated from two to exactly one. Mac Lane's planarity criterion, named after Saunders Mac Lane, characterizes planar graphs in terms of their cycle spaces and...

In organic chemistry, Hückel's rule predicts that a planar ring molecule will have aromatic properties if it has (4n + 2)π-electrons, where n is a non-negative...

longitude; and x {\displaystyle x} and y {\displaystyle y} are the projected (planar) coordinates for a given ( φ , λ ) {\displaystyle (\varphi ,\lambda )} coordinate...

In mathematics, the sumset of two subsets A and B of an (additive) abelian group is formed by adding each element of A to each element of B: A + B = {...

planar graph has a Hamiltonian cycle. It strengthens an earlier theorem of Hassler Whitney according to which every 4-vertex-connected maximal planar...

exhibit significantly faster switching times and higher current density than planar CMOS (complementary metal–oxide–semiconductor) technology, resulting in...

played on planar graphs. The following proof is from theorem 3 of. Since planar GG is a special case of GG, and GG is in PSPACE, so planar GG is in PSPACE...

Planar Doppler Velocimetry (PDV), also referred to as Doppler Global Velocimetry (DGV), determines flow velocity across a plane by measuring the Doppler...

special cases become planar or spherical simple open chain, M = ∑ i = 1 j   f i , {\displaystyle M=\sum _{i=1}^{j}\ f_{i},} planar or spherical simple...

An ideal solid surface is flat, rigid, perfectly smooth, and chemically homogeneous, and has zero contact angle hysteresis. Zero hysteresis implies the...

the aim of providing a straightedge at a right angle or other particular planar angle to a baseline. The simplest form of set square is a triangular piece...

commonly use short pieces of transmission line, usually in the form of printed planar transmission lines, arranged in certain patterns to build circuits such...

behind V IV: below H, in front of V These labels are the same as used in 2D planar geometry (), as seen from infinitely far to the "left", taking H and V to...

C6H6. The benzene molecule is composed of six carbon atoms joined in a planar hexagonal ring with one hydrogen atom attached to each. As it contains only...

a hydrogen (chemical formula of (CH)n). Furthermore, while graphene is planar due to its double-bonded nature, graphane is rugged, with the hexagons adopting...

Triax is widely used for the connection of cameras in television studios. Planar format transmission lines are flat conductors manufactured by a number of...

theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex...

crystal. These planes, which show up as lines under a microscope, are called planar deformation features (PDFs), or shock lamellae. Shocked quartz was discovered...

mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line...

Helias configuration. It is mainly a toroid, consisting of 50 non-planar and 20 planar superconducting magnetic coils, 3.5 m high, which induce a magnetic...

simple solution processed layers. However, both techniques can result in planar thin film layers or for use in mesoscopic designs, such as coatings on a...

time. In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds...

systems can produce large induced-drag reductions relative to a conventional planar wing, there is no significant aerodynamic advantage that uniquely accrues...

a molecule adopting a non-planar geometry to avoid the destabilization that results from antiaromaticity. If it were planar, it would have a single eight-electron...

trigonal planar molecular geometry. The structure of the trimer of the metaborate ion ([B3O6]3−). This anion is a cyclic molecule and has a trigonal planar molecular...

moving parallel to the axis; a helix. The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane;...

composite number with two prime factors. 267 is the number of planar partitions of the number 12. Planar partitions are the number of ways in which the given number...

The Vortex of Madness and Other Planar Perils is an adventure module for the 2nd edition of the Advanced Dungeons & Dragons fantasy role-playing game....

Graphical projection Ingrid Carlbom; Joseph Paciorek; Dan Lim (December 1978). "Planar Geometric Projections and Viewing Transformations". ACM Computing Surveys...

that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs. Let G = (V,E) be a digraph and let M be an abelian group. A map...

undirected planar graphs of maximum degree three, directed planar graphs with indegree and outdegree at most two, bridgeless undirected planar 3-regular...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

counting version of Planar 3SAT. The hardness reduction from 3SAT to Planar 3SAT given by Lichtenstein is parsimonious. This implies that Planar #3SAT is #P-complete...

regular octahedron is a special case in which all three quadrilaterals are planar squares. Schönhardt polyhedron, a non-convex polyhedron that cannot be partitioned...

{\displaystyle O(n\log n)} time, for instance by using integer sorting algorithms, planar convex hulls can also be computed more quickly: the Graham scan algorithm...

_{n}={\frac {(2n-1)\pi }{l}}.} When h / l ≪ 1 {\displaystyle h/l\ll 1} , the planar Couette flow is recovered, as shown in the figure. Taylor–Couette flow is...

Retrieved April 22, 2010. Shor, Peter Williston (September 1985). Random Planar Matching and Bin Packing (Ph.D. thesis). MIT. OCLC 14107348. Shor, Peter...

ended in 1993. The exterior of the building consists of sharply angled planar surfaces. The architecture of the building has been likened to Deconstructivism...

In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational...

#P-complete as well. This is the counting version of Planar 3SAT. The hardness reduction from 3SAT to Planar 3SAT given by Lichtenstein has the additional property...

co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these...

geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by definition is the least upper...

monolithic integrated circuit chip was enabled by the inventions of the planar process by Jean Hoerni and of p–n junction isolation by Kurt Lehovec. Hoerni's...

also several new categories (although the gem dragons did not return): "planar dragons" which are defined as dragons that were warped by living on a plane...

Kasteleyn, and Neville Temperley, counts the number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs...

is a red liquid with the formula CSCl2. It is a molecule with trigonal planar geometry. There are two reactive C–Cl bonds that allow it to be used in...

the Su-27. In this original N011 form, it deployed a mechanically scanned planar array with 960 mm diameter, ±85 degrees scan sector. The peak output power...

a reference wave, holographic principles enabled the use of a compact, planar feeding structure and suppression of undesired lobes. This was extended...

the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists a collection of k planar graphs, all having the...

board to form the "windings" of a planar transformer, replacing the turns of wire used to make other types. Some planar transformers are commercially sold...

(also Benioff–Wadati zone or Benioff zone or Benioff seismic zone) is a planar zone of seismicity corresponding with the down-going slab in a subduction...

affected only by self-cancelling tangential forces. Gaudí evolved from planar to spatial geometry, and then to ruled geometry. These forms allowed the...

that supports the sculpture from below. A plinth is defined as a flat and planar support which separates the sculpture from the environment. A pedestal,...

Planar reflectarray fed by a horn antenna. Structure showing unit cells. This antenna has fixed cell delays, configured by different cutout sizes in each...

circuit boards and these filters have the advantage of largely being planar. Precision planar filters are manufactured using a thin-film process. Higher Q factors...

Outlands (a Planescape campaign setting guide), Morte's Planar Parade (a bestiary of planar creatures), and Turn of Fortune's Wheel (an adventure module)...

In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This...

not have a five-vertex complete graph as a minor are the 3-clique-sums of planar graphs with the eight-vertex Wagner graph; this structure theorem can be...

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...

Planar approximation of angular distance on sky...

study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry. An equilateral triangle is a triangle that has three...

The effect seen on two co-planar polygons...

"awesome powers" to "plenty of horrors to encounter in space". Morte's Planar Parade was the bestiary book contained within the Planescape: Adventures...

Brown University. He is known for his work on atomic gold pyramids and planar boron clusters. Wang obtained a B.S. degree in chemistry from Wuhan University...

graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs. The analogue concept for polytopes of general dimension are the...

the atoms within the pyrimidinedione ring system are sp2 hybridized and planar. The imidazole ring also has a resonance. Therefore, the fused 5,6 ring...

is equal to zero, which are the values 0 and −1. Cassini ovals appear as planar sections of tori, but only when the cutting plane is parallel to the axis...

A planar Fourier capture array (PFCA) is a tiny camera that requires no mirror, lens, focal length, or moving parts. It is composed of angle-sensitive...

In computational geometry, a CC system or counterclockwise system is a ternary relation pqr introduced by Donald Knuth to model the clockwise ordering...

In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles...

Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable. The Grötzsch graph is a member of...

Property of a planar simple closed curve...

NAND-like) 1T-1C capacitor arrays Density High (potential Tb/mm2) Low (Mb range, planar limits) Read Non-destructive Destructive (write-after-read) Endurance 109–1012...

Zeiss HFT Planar 7 6 0.45 m (1 ft 6 in) 47 mm (1.9 in) 62 mm (2.4 in) 230 g (8.1 oz) 49 mm Voigtländer Color-Ultron Identical to HFT Planar 50 f/1.8–16...

Rolleiflex 600x series Rolleiflex 6008 SRC 1000 professional, with Zeiss 4/120 Planar HFT lens Overview Type Medium format single-lens reflex camera Focusing...

Butterworth-Heinemann. p. 55. ISBN 978-0-08-052335-4. Hibbeler, R. C. (2007). "Planar kinematics of a rigid body: Instantaneous center of zero velocity". Engineering...

in its service life, with the most significant one being adopting a new planar array radar, Emerson AN/APQ-159 with a range of 20 nmi to replace the original...

has a weight of 250 kg and uses a 680 mm electronically scanned slotted planar array antenna which offers a detection range of 90 km against a target with...

version of the game took the form of a flat wooden board inscribed with a planar graph with the same combinatorial structure as the dodecahedron (a Schlegel...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

a regular projection of a knot, or knot diagram is thus a quadrivalent planar graph with over/under-decorated vertices. The local modifications of this...

Contarex Contarex Special with Planar 50 mm f/2; note bayonet lugs are on the body instead of the lens The Contarex with Planar 55 mm f/1.4 From left, Sonnar...

coordinate system – also called a projected coordinate reference system, planar coordinate system, or grid reference system – is a type of spatial reference...

conventional speakers in relying not on cones mounted in an enclosure, but on a planar driver system mounted in a panel. A Magneplanar's reproducing mechanism...

spherical aberration, which could lower peripheral contrast. Double Gauss/Planar tweaks formed the basis for most normal and near-normal prime lens designs...

(5-10 GPa); uneven darkening of olivine as seen under polarized light; planar and irregular fractures (breaks in other than a natural cleavage plane.)...

Wiktionary, the free dictionary. Fault commonly refers to: Fault (geology), planar rock fractures showing evidence of relative movement Fault (law), blameworthiness...

ligands. Other common coordination geometries are tetrahedral and square planar. Crystal field theory may be used to explain the relative stabilities of...

{i+j+k-1}{i+j+k-2}}=\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {i+j+t-1}{i+j-1}}.} The planar case (when t = 1) yields the binomial coefficients: B ( r , s , 1 ) = (...

butterfly graph (also called the bowtie graph and the hourglass graph) is a planar, undirected graph with 5 vertices and 6 edges. It can be constructed by...

Bangladesh University of Engineering and Technology. He is an author of the book Planar Graph Drawing. He is known for his contribution in graph drawing, graph...

dimensions also has a complete skeleton. K1 through K4 are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must...

them) either is a polygon or resembles a polygon, but is not necessarily planar, and the vertices of the two end polygons are joined by either a curve or...

processing Bare Ti ~50 - - None TiO2 TiO Ti4O7 TiO4 (Planar) ~33 2.2 700 Oxidation TiO2 TiO Ti4O7 (Planar) ~45 4 700 Oxidation TiO2 TiO Ti4O7 TiO4 (Hollow)...

symmetric and planar structure consists of four fused benzene rings. Triphenylene has delocalized 18-π-electron systems based on a planar structure, corresponding...

T H ′ {\displaystyle T_{G}=T_{H}\cdot T_{H'}} If G {\displaystyle G} is planar and G ∗ {\displaystyle G^{*}} denotes its dual graph then T G ( x , y )...

(hypothetical) crystals. A planar cover of a graph is a finite covering graph that is itself a planar graph. The property of having a planar cover may be characterized...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

between them. The design can be realised in planar format but it has a more natural implementation in coax – in planar, the two lines have to be kept apart so...

(diphos)Pd(η2-CSSe) (dpmb)Pd(η2-CSSe) P21/c a=9.585 b=15.02 c=21.50 Å β=103.1° planar Os(η2-CSSe)(CO)(CN-p-tol)(PPh3)2 Os(η2-CSeS)(CO)(CN-p-tol)(PPh3)2...

foliation of a rock is a planar feature that often contains a linear feature called lineation. Similarly, a fault plane is a planar feature that may contain...

one another entirely, in which case their paths will diverge along other planar conic sections. If one object is very much heavier than the other, it will...

two-dimensional shape of the largest area that can be maneuvered through an L-shaped planar region with legs of unit width. The area thus obtained is referred to as...

in biological systems. The simplest case of the instability arises at a planar interface within a porous medium or Hele-Shaw cell, and was treated by Saffman...

without changing its number of crossings. In particular, every planar graph has a planar arc diagram. However, this embedding may need to use more than...

compounds anthracene and phenanthrene. PAHs are uncharged, non-polar and planar. Many are colorless. Many of them are also found in fossil fuel deposits...

and modified for use in planar structures (such as MMICs and RFICs) at frequencies up to at least 100 GHz. These types of planar baluns work exciting a...

Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion into...

Barrier Reef Marine Park, and in the Swain Reefs (20–22 degrees south). Planar reefs are found in the northern and southern parts, near Cape York Peninsula...

or planar bedding consists of cross-bedded units that are extensive horizontally relative to the set thickness and that have essentially planar bounding...

in which no edges cross. Every matchstick graph is a planar graph, but some otherwise-planar unit distance graphs (such as the Moser spindle) have a...

systems built from an appropriate collection of UTM zones cover a region with planar maps with well-controlled, minimal distortion. For this reason, UTM coordinates...

geometry that most easily facilitates this is a planar surface. In the case of an infinitely extended planar surface, there are two dimensions in which the...

that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or region) so that no two faces that...

on the bearing. Some turntables, such as the newer versions of the Rega Planar series, use a fixed plinth with a low vibration motor and bearing attached...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

equals are subtracted from equals, then the remainders are equal" (among planar shapes, those of the same area were called "equal"). The propositions in...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

Wetting is the ability of a liquid to maintain contact with a solid surface by displacing another substance or material – either a gas, or other liquid...

study of planar graphs, blossom trees are trees with additional directed half edges. Each blossom tree is associated with an embedding of a planar graph...

dimers. At higher temperatures those increasingly dissociate into trigonal planar AlCl3 monomers similar to the structure of BCl3. Aluminium tribromide and...

sound: a construction of aural surfaces and regular surfaced planes, a planar conception, the product of a mechanistic discipline, with an emphasis upon...

Jansen's linkage is a planar leg mechanism designed by the kinetic sculptor Theo Jansen to generate a smooth walking motion. Jansen has used his mechanism...

{p^{2}}{16a^{3}}}\sin 2\theta } Similar to the conducting plane, the case of a planar interface between two different dielectric media can be considered. If a...

This allows the description of a rotation as the angular position of a planar reference frame M relative to a fixed F about this shared z-axis. Coordinates...

Hart's inversors are two planar mechanisms that provide a perfect straight line motion using only rotary joints. They were invented and published by Harry...

will be able to interact with the boundary. If the boundary is essentially planar, then this fraction will be given by N i n t e r a c t = 2 r N t o t a l...

edges, screws, vacancies, self inter-stitials, and more that are linear, planar, and three dimensional types of defects. New and advanced materials that...

The Campanella in Campanella 3, though the controls are different (with planar 2D motion to dodge obstacles as a common theme). Eggs are a recurring theme...

The Debye sheath (also electrostatic sheath) is a layer in a plasma which has a greater density of positive ions, and hence an overall excess positive...

smooth Riemann mapping theorem can be generalized to multiply connected planar regions with smooth boundary. The Beltrami coefficient in these cases is...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

remainder of the material was recovered as a mixture of monoacetate. The planar-chiral DMAP catalyst was also shown to be effective at kinetically resolving...

NP-complete, implying that no simple characterization is likely to exist. Every planar graph is a string graph: one may form a string graph representation of an...

probability theory and stochastic processes, including graphons, random planar maps, the Schramm–Loewner evolution, and their applications to quantum gravity...

mathematics, the Y-Δ transform plays an important role in theory of circular planar graphs. The Y-Δ transform is known by a variety of other names, mostly based...

(34 oz) Bay VI f/2.8–22 Planar HFT SLX, PQ, PQS 7 5 0.90 m (2 ft 11 in) 81.5×63 mm (3.2×2.5 in) 590 g (21 oz) Bay VI f/2.8–22 Planar HFT EL 7 5 0.90 m (2 ft...

efficiently on them. The planar graphs do not have bounded treewidth, because the n × n {\displaystyle n\times n} grid graph is a planar graph with treewidth...

Finite-difference time-domain electromagnetic solver Open-source Momentum MoM-based planar EM simulation in Keysight ADS Commercial Numerical Electromagnetics Code...

rhombohedral system. The carbonate ions (CO2− 3) are arranged in a triangular planar configuration, and the manganese ions (Mn2+) are surrounded by six oxygen...

with d8 metal ions, such as Ni(salen), typically have a low-spin square planar molecular geometry in the coordination sphere. Other metal–salen complexes...

from the Latin word folium, meaning "leaf", and refers to its sheet-like planar structure. It is caused by shearing forces (pressures pushing different...

three sides and one of the angles of a spherical triangle, analogous to the planar law of cosines for a triangle in the Euclidean plane. A spherical triangle...

August 2010. Retrieved 22 August 2010. Ingrid Carlbom, Joseph Paciorek, Planar Geometric Projections and Viewing Transformations, ACM Computing Surveys...

Unsolved problem in mathematics Do planar graphs have universal point sets of subquadratic size? More unsolved problems in mathematics In graph drawing...

antennas. Radomes can be constructed in several shapes – spherical, geodesic, planar, etc. – depending on the particular application, using various construction...

unstable, non-planar compound which is non-aromatic. The dianion, however, satisfies Hückel's rule, is thermally stable, and is planar. Benzocyclooctatetraene...

approach to generating 3-dimensional, optimized stellarator fields using only planar coils (Thea Energy) In December 2025, Thea Energy completed a milestone...

measurement convention used to describe the plane orientation or attitude of a planar geologic feature. A feature's strike is the azimuth of an imagined horizontal...

possible distance from the centroid of the I-beam's cross-section. The planar second moment of area provides insight into a beam's resistance to bending...

stereoelectronic effects. The spiro transition state is favored over the planar due to the non-bonding orbitals of the superior oxygen donating into the...

professional markets; in 1961, the retail price (including the 50 mm f/2.0 Planar lens) was $499. The camera bodies are complex; for example, the first model...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in the form of a circular cone can be projected onto...

two papers a mile-stone in PLED development. Their device have similar planar structure, i.e. the active layer (perovskite) is sandwiched between two...

disilenes each silicon atom has planar coordination but the substituents are twisted so that the molecule as a whole is not planar. In diplumbenes the Pb=Pb...

profile, usually consisting of a printed circuit board. It consists of a planar rectangular or circular sheet or "patch" of metal, mounted over a larger...

axes, so the planar Chebyshev distance can be viewed as equivalent by rotation and scaling to (i.e. a linear transformation of) the planar Manhattan distance...

given a planar Boolean circuit (i.e. a Boolean circuit whose underlying graph is planar) containing only NAND gates with exactly two inputs. Planar Circuit...

a wire in the shape of a spiral or helix which could be a solenoid or a planar structure. One field where these are found is nuclear magnetic resonance...

diagram. Unsatisfiable core Satisfiability modulo theories Counting SAT Planar SAT Karloff–Zwick algorithm Circuit satisfiability The SAT problem for arbitrary...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

Isodynamics, a company from Ukraine, which has been at the forefront of planar magnetic development since the 1980s. Combining Meze's aesthetic and user-centered...

The chiral Potts model is a spin model on a planar lattice in statistical mechanics studied by Helen Au-Yang Perk and Jacques Perk, among others. It may...

In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k {\displaystyle k} concentric...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

achieve a goal, and is sometimes represented by a diagram. For spatial or planar topologic or topographic sets see map. Plans can be formal or informal:...

1955, the first planar silicon dioxide transistors by Frosch and Derick in 1957, the MOSFET demonstration by a Bell Labs team, the planar process by Jean...

deploying few-modes optical planar waveguides. Especially, if modes can be easily converted from conventional single-mode planar waveguides into few-mode...

the energy splitting will be lower than in the octahedral case. Square planar and other complex geometries can also be described by CFT. The size of the...

3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented...

are linkages that can have one or more actuators, and can perform simple planar or complex motion. Compared to a wheel, a leg mechanism is potentially better...

and the stationary part away from the user that contains a portion of the planar pantograph. The function of the bridge is to form a base for the vertical...

electrophilic reagents. As confirmed by X-ray crystallography, nitrobenzene is a planar molecule. Nitrobenzene is prepared by nitration of benzene with a mixture...

elements from a Planar-type double Gauss lens, resulting in the designs shared by the Schneider Kreuznach Xenotar (1952) and Zeiss Planar (1953) & Biometar...

Cambridge Research Centre in the 1950s. Stripline is the earliest form of planar transmission line. A stripline circuit uses a flat strip of metal which...

Stanford University Thesis Flatland Photonics: Circumventing Diffraction with Planar Plasmonic Architectures (2009, Awarded Francis and Milton Clauser Prize...

In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four...

In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into...

double bonds. This unsaturated arrangement means the bicyclic molecule is planar. Guanine, along with adenine and cytosine, is present in both DNA and RNA...

of a complete graph All Hamiltonian paths Non-planarity game All edges of a complete graph All non-planar sub-graphs Arithmetic progression game The numbers...

embedding. If the graph isn't planar, it raises an exception. Planar embeddings exactly preserve the topology of planar networks—useful for circuit schematics...

For 4-regular planar graphs, additional necessary conditions can be derived from Grinberg's theorem. An example of a 4-regular planar graph that does...

Traditional implementations assumed the buffer was being created on a planar cartesian coordinate space (i.e., created by a map projection) using Euclidean...

Kosmos-2542 ejected from Kosmos-2543 Kosmos-2558 14F150 2022 Aug 1 in orbit co-planar with USA-326 Kosmos-2562 14F164 or 14F172 2022 Oct 21 re-entered RPO with...

by the German chemist Arthur Rudolf Hantzsch (1857–1935). Imidazole is a planar 5-membered ring, that exists in two equivalent tautomeric forms because...

simplest anastigmatic lens designs. It was designed by H. Dennis Taylor. A Planar (1896) design is one with 4 groups of 6 elements and a flat field design...

example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be...

Projection of a tetrahedral molecule onto a planar surface....

The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the...

Momentum is 3-D planar EM simulation software for electronics and antenna analysis, a partial differential equation solver of Maxwell's equations based...

incidence poset of a planar map (planar graph with fixed plane embedding) is at most four. Felsner later proved in "The order dimension of planar maps revisited...

vertex or edge or the empty set and so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke requires...

A flow-based generative model is a generative model used in machine learning that explicitly models a probability distribution by leveraging normalizing...

Isochrons and a stable limit cycle of the planar system x ˙ = x − y − x ( x 2 + y 2 ) ; y ˙ = x + y − y ( x 2 + y 2 ) {\displaystyle {\dot {x}}=x-y-x(x^{2}+y^{2});{\dot...

however in this case the polyhedral symbol specifies that the complex is non-planar. There are two digits. The first digit is the priority number of the ligand...

angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the...

proteoglycan binding coreceptor activity involved in Wnt signaling pathway, planar cell polarity pathway Cellular component membrane plasma membrane integral...

tectonics, the movement of large regions of the Earth's crust in the manner of planar rigid bodies, are key to understanding the formation of the Himalayas. The...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

tangential to it at the equator. The cylinder is then unrolled to give the planar map.[page needed] The fraction ⁠R/a⁠ is called the representative fraction...

conventions such as tapering provide this information more effectively. Upward planar drawing uses the convention that every edge is oriented from a lower vertex...

example the equator and meridians of a globe); the spherical analog of planar circles, which curve relative to the surface, are called small circles (for...

incompatibility (help) Guang Chen; Fook Fah Yap (2003). An Introduction to Planar Dynamics (3rd ed.). Central Learning Asia/Thomson Learning Asia. p. 34....

In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object...

of two co-ordinate systems, the first being planar motion and the latter being cylindrical motion. In planar motion, the velocity and acceleration components...

universal covering problem, find the smallest convex area that can cover any planar set of unit diameter Bellman's lost-in-a-forest problem, find the shortest...

function. One immediate consequence of Beltrami flow is that it can never be a planar or axisymmetric flow because in those flows, vorticity is always perpendicular...

35/2.8 Distagon = Color-Skoparex 50/1.4 Planar 50/1.8 Planar = Color-Ultron 60/2.8 Macro-Planar 85/1.4 Planar [triangular aperture] 85/2.8 Sonnar = Color-Dynarex...

processing through photolithography. The reason to use either of these planar faces depends on the application for which the silicon wafer will be used...

corrosion. Low planarity on vertical levelers may make this surface finish unsuitable for use with fine pitch components. Improved planarity can be achieved...

In geometry, coaxial means that several three-dimensional linear or planar forms share a common axis. The two-dimensional analogue is concentric. Common...

calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbative...

group. A phenyl group has six carbon atoms bonded together in a hexagonal planar ring, five of which are bonded to individual hydrogen atoms, with the remaining...

corresponded well to a combination of simulated spectra of a sheet-like, quasi-planar global minimum of the B40− anion (Cs symmetry) and its nearly degenerate...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

key-fob designs. Planar Inverted-F Antennas (PIFA) are a form of printed PCB antenna in which the radiating element is implemented as a planar conductive plate...

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...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

1007/978-3-662-69973-7_2. ISBN 978-3-662-69972-0. B. Paul, Kinematics and Dynamics of Planar Machinery, Prentice-Hall, NJ, 1979 L. W. Tsai, Robot Analysis: The mechanics...

Concepts Affine transformation Back-face culling Clipping Collision detection Planar projection Reflection Rendering Beam tracing Cone tracing Checkerboard rendering...

be removed, leaving a planar subgraph. The removed vertex is called the apex. A k-apex graph is a graph that can be made planar by the removal of k vertices...

{\displaystyle c} ; γ {\displaystyle \gamma } is a numerical factor, which in planar problems assumes the value 2 / 2835 {\displaystyle 2/2835} ; g {\displaystyle...

Primary powerpole patent US7153152 B1 2006 1997 Electrical connector with planar contact engaging surface PP75 connector WO2004051681 A2 2004 2003 Finger...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

casual 'Planar Magic' format. The format can be played with two or more players. Each player requires a traditional Magic deck and a 'planar deck' of...

Concepts Affine transformation Back-face culling Clipping Collision detection Planar projection Reflection Rendering Beam tracing Cone tracing Checkerboard rendering...

(Philco-Ford) 1947 1976 Philips 1930 c. 1930 present Pioneer 1985 2010 Planar Systems 1983 present Polaroid - - Polytron 1975 present ProLine - - Now...

Coplanar waveguide is a type of electrical planar transmission line which can be fabricated using printed circuit board technology, and is used to convey...

containing the number 4. It is the sum of 47 + 94 + 141. 282 is the number of planar partitions of 9. This means that 282 is the number of ways to separate 9...

VANGL planar cell polarity protein 2 is a protein that in humans is encoded by the VANGL2 gene. The protein encoded by the VANGL2 gene is a membrane protein...

chlorinated solvents, the tetrahedral isomer converts to the square planar isomer. The square planar form is red and diamagnetic. The phosphine ligands are trans...

hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension...

of the nitrogen in amides is not pyramidal (as in the amines) but planar. This planar restriction prevents rotations about the N linkage and thus has important...

the Contax G1 cost US$2,210 (equivalent to $4,700 in 2025) with the 45 mm Planar lens; the 28 mm Biogon and 90 mm Sonnar lenses were each US$660 (equivalent...

perpendicular axis theorem (or plane figure theorem) states that for a planar lamina the moment of inertia about an axis perpendicular to the plane of...

greater attention to the contrasting qualities of surface (open, closed, planar, broken etc.) while less attention was paid to story-telling and convincing...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

Many d8 complexes of the first row metals exist in tetrahedral or square planar geometry. In some cases these geometries exist in measurable equilibria...

smaller and less pronounced than bedding. Lamination is often regarded as planar structures one centimetre or less in thickness, whereas bedding layers are...

3D NAND is comparable with scaled down (32 nm or less) planar NAND flash. However, with planar NAND scaling stopping at 16 nm, the cost per bit reduction...

k-quasi-planar if it has no k pairwise crossing edges. Using this terminology, if a topological graph is 2-quasi-planar, then it is a planar graph. It...

search for CP violation in Xi hyperon decays and a study of surface-treated planar microstrip gas chambers. purdue.edu (PhD thesis). Purdue University. OCLC 1194793889...

hydrocarbons (PAHs) in which the fused aromatic molecules deviate from the usual planarity. The comparison of structures of graphene and fullerene can help comprehend...

shape that is elliptical in all sections through the long axis. elliptical Planar, shaped like a flattened circle, symmetrical about both the long and the...

are valid for most other geometries. The exception is square-planar because square-planar complexes typically abide by the 16-electron rule. Assuming ligands...

David Schwartz, Mark Seifter, Jeffery Swank, and Linda Zayas-Palmer [27] Planar Adventures June 27, 2018 192 ISBN 978-1-64078-044-6 Hardcover PZO1141 James...

In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges. It consists of a complete graph...

mask design, is the representation of an integrated circuit in terms of planar geometric shapes which correspond to the patterns of metal, oxide, or semiconductor...

shutter Portrait lenses 120 f/2–16 Planar (HFT) SL66 7 5 0.4 m (14 in) 90×87 mm (3.5×3.4 in) 810 g (29 oz) 86 f/5.6–45 S-Planar (HFT) SL66 6 4 0.4 m (14 in)...

NINA-W132 PIFA Belongs to the u-blox NINA-W13 series of Wi-Fi modules. On board planar inverted-F antenna (PIFA) is shaped (cut & bent) metal, not a PCB trace...

anions [XeF5]−. The [XeF5]− ion was the first example of a pentagonal planar molecular geometry AX5E2 species. It was prepared by the reaction of [N(CH3)4]F...

aromatics, are aromatic organic compounds that consist only of a conjugated planar ring system.[citation needed] Many simple aromatic rings have trivial names...

polynomial-time algorithms for Max-Cut in general graphs are known. However, in planar graphs, the maximum cut problem is dual to the route inspection problem...

sunset or sunrise. Determining the angle of reflection with respect to a planar surface is trivial, but the computation for almost any other surface is...

measurements was made at ten frequencies (0.25 to 8-MHz). There were two planar electron traps mounted on opposite sides of the spacecraft. The trap consisted...

electronic-photonic unit cells that can be tiled together to form large-scale planar apertures (up to 10 centimeters in diameter) that can run at 100 watts of...

group is known as the double group, D4′. With species such as the square-planar complex of the silver(II) ion [AgF4]2− the relevant double group is also...

lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical...

technique using gamma rays. It is very similar to conventional nuclear medicine planar imaging using a gamma camera (that is, scintigraphy), but is able to provide...

the Kantorovich potential field entirely solves the problem. Unlike the planar case, a non-intersecting family of line congruences cannot in general be...

wheel graphs. A maximal planar graph is a planar graph in which all faces, even the outer face, are triangles. A maximal planar graph is node-pancyclic...

between two planar semi-infinite walls. This approximation is widely used to estimate forces between colloidal particles, as forces between two planar bodies...

planar magnetic or electrostatically-driven loudspeakers, and should not be confused with planar or true ribbon loudspeakers. In contrast to a planar...

forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K 5 {\displaystyle...

algorithm Kirkpatrick, David G.; Seidel, Raimund (1986). "The ultimate planar convex hull algorithm?". SIAM Journal on Computing. 15 (1): 287–299. doi:10...

application is still to add shadows to 3D applications. It is also used for planar reflections. Other rendering techniques, such as portal rendering, use the...

compute locations using multilateration. Projected coordinate system (or planar, grid) A standardized cartesian coordinate system that models the surface...

compound of boron and iodine with chemical formula BI3. It has a trigonal planar molecular geometry. Boron triiodide can be prepared by the reaction of boron...

3.4 GHz CPU). It is based on a fast Hough-transform voting strategy for planar regions, inspired by the Kernel-based Hough transform (KHT). This 3D kernel-based...

V-shaped, U-shaped, or semi-circular defect deliberately introduced into a planar material. In structural components, a notch causes a stress concentration...

estimates as well as exact numbers for the number of rectangulations of planar point sets. That means, we fix a point set and consider the number of rectangulations...

slope number. As Keszegh, Pach & Pálvölgyi (2011) showed, every planar graph has a planar straight-line drawing in which the number of distinct slopes is...

Pseudoconical Bonne Bottomley Polyconic American Chinese Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert...

(illustration, right) the central arch and side arches are raised on four massive planar piers[clarification needed]. Donato Bramante's original plan for St Peter's...

&{\text{for planar and wedge failure}}\\1,&{\text{for toppling failure}}\end{cases}}} where parameter B is the discontinuity dip in degrees for planar failure...

using a planar coil rather than a toroidal coil. In order to reject the influence of conductors outside the sensor's measurement region, these planar Rogowski...

Example of a MST: The minimum spanning tree of a planar graph. Each edge is labeled with its weight, which here is roughly proportional to its length....

planar waves instead of a ray tracing model. This reduces computation and lowers requirements for optics knowledge. The waves are considered planar between...

for conjugation is orbital overlap. Thus, the conjugated system must be planar (or nearly so). As a consequence, lone pairs which do participate in conjugated...

Harary, who proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph. The same graph had already been given as an example of a non-Hamiltonian...

4 to 7) arranged in a specific geometric pattern, such as a circular or planar array. Each element receives signals independently. Signal Processing Unit:...

did still remain in production until the 1960s, by which time the silicon planar transistor was dominating the market. While point-contact transistors usually...

readily than smooth compound curves, the Lockheed design used a set of planar facets. Overholser died on April 28, 2026, at the age of 86. Overholser...

(December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits". Science China Physics, Mechanics & Astronomy...

invariant of planar graphs that measures the number of bounded faces of the graph, as a fraction of the possible number of faces for other planar graphs with...

the slight negative charge of the aromatic system on another molecule. Planar monocyclic molecules containing 4n π electrons are called antiaromatic and...

atom is substituted for the nitrogen atom. Whereas the pyrrole molecule is planar, the arsole molecule is not, and the hydrogen atom bonded to arsenic extends...

known as a space crank, unlike the Peaucellier–Lipkin linkage which is a planar mechanism. One of its main advantages is that it can be used to lift the...

College, University of London Thesis The influence of the lipid state on the planar distribution of membrane proteins in Acholeplasma laidlawii  (1977) Website...

Pyridine rings (here shown using the parent compound as an example) are planar and aromatic...

theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar, undirected graph with 2n + 1 vertices and 3n edges. The friendship graph...

Spiric sections as planar sections of a torus...

not cover the surface of a circle finitely many times. In addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons...

graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph whose vertices and edges can be drawn in a plane such that...

chirality, a stereogenic axis (axial chirality) and a stereogenic plane (planar chirality). Finally, the inherent curvature of a molecule can also give...

Awards Prix Versailles 2019 LEAF Award (2009, 2011, 2013, 2014, 2015, 2017, 2018) Practice Studio MK27 Buildings Gama-Issa House Jungle House Planar House...

indicatrix is to relate how distances on the sphere change when mapped to a planar surface. Specifically, the desired relation is the transform T {\displaystyle...

this principle is Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory...

to hydrothermal circulation. Veins are classically thought of as being planar fractures in rocks, with the crystal growth occurring normal to the walls...

J. Robert King, (ISBN 0-7869-0501-8) Planar Powers (August 1997), by J. Robert King, (ISBN 0-7869-0532-8) Planar Powers won the Origins Awards for Best...

sand, silt, or gravel, similar to shoals and river islands. polar aspect A planar map projection with its origin located at either the North or the South...

interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia...

spectroscopy and X-ray crystallography in 1963. The structure is square planar, as has been confirmed by neutron diffraction studies. According to VSEPR...

Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K5 (top) or K3,3 (bottom) subgraphs...

trigonal planar 145–149 BO3− 3, CO2− 3 La6(BN3)O6 B 2N8− 4 planar 147–150 177–182 C 2O2− 4 La3B2N4, ((La3+)3(B 2N8− 4)(e−)) B 3N9− 6 planar or chair form...

Planar analytic geometry uses the coordinate space R2. Depicted: description of a line as the solution set in x → {\displaystyle {\vec {x}}} of the vector...

Institutions Monash University Thesis Stetige planare Funktionen [Continuous planar functions]  (1993) Doctoral advisor Karl Strambach YouTube information Channel...

Harvard in 1962. In their extensive evaluation of Maxwell's equations at the planar interface between a linear and nonlinear medium, several rules for the interaction...

efficient approximation algorithms for restricted classes of graphs. In planar graphs, the maximum independent set may be approximated to within any approximation...

Concepts Affine transformation Back-face culling Clipping Collision detection Planar projection Reflection Rendering Beam tracing Cone tracing Checkerboard rendering...

can be constructed as a strong product of path graphs. Decompositions of planar graphs and related graph classes into strong products have been used as...

glass or microporous polymer membranes with appropriate surface chemistry. Planar shallow electroosmotic pumps are made of parallel shallow microchannels...

the plane of focus (PoF) is parallel to the lens and image planes. If a planar subject (such as the side of a building) is also parallel to the image plane...

about "random planar metrics".[B] Benjamini was a long time collaborator of Oded Schramm. Their joint works included papers on limits of planar graphs,[BS]...

Thiophene is a heterocyclic compound with the formula C4H4S. Consisting of a planar five-membered ring, it is aromatic as indicated by its extensive substitution...

it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow. In the Couette flow, a fluid is trapped between two infinitely...

AN/SPY-6(V)1 was selected in October 2025. Each ship will be equipped with four planar arrays, providing a 360° situational awareness. Each of those arrays are...

degree Δ or Δ+1. For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always Δ, and for multigraphs, the number...

the same hybridization due to the different d-orbitals involved. A square planar complex has one unoccupied p-orbital and hence has 16 valence electrons...

In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes...