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vertex geometry wikipedia 337 captures 30 oct 2007 30 sep 2026 aug sep oct 18 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220918095244 https en wikipedia org wiki vertex_ 28geometry 29 vertex geometry from wikipedia the free encyclopedia jump to navigation jump to search point where two or more curves lines or edges meet for vertices in the geometry of curves see vertex curve for other uses of the word see vertex disambiguation this article needs additional citations for verification please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources vertex geometry news newspapers books scholar jstor july 2022 learn how and when to remove this template message in geometry a vertex in plural form vertices or vertexes often denoted by letters such as s displaystyle s p displaystyle p q displaystyle q r displaystyle r is a point where two or more curves lines or edges meet as a consequence of this definition the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices 1 2 3 contents 1 definition 1 1 of an angle 1 2 of a polytope 1 3 of a plane tiling 2 principal vertex 2 1 ears 2 2 mouths 3 number of vertices of a polyhedron 4 vertices in computer graphics 5 see also 6 references 7 external links definition edit of an angle edit a vertex of an angle is the endpoint where two line or rays come together the vertex of an angle is the point where two rays begin or meet where two line segments join or meet where two lines intersect cross or any appropriate combination of rays segments and lines that result in two straight sides meeting at one place 3 4 of a polytope edit a vertex is a corner point of a polygon polyhedron or other higher dimensional polytope formed by the intersection of edges faces or facets of the object 4 in a polygon a vertex is called convex if the internal angle of the polygon i e the angle formed by the two edges at the vertex with the polygon inside the angle is less than π radians 180 two right angles otherwise it is called concave or reflex 5 more generally a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex and is concave otherwise citation needed polytope vertices are related to vertices of graphs in that the 1 skeleton of a polytope is a graph the vertices of which correspond to the vertices of the polytope 6 and in that a graph can be viewed as a 1 dimensional simplicial complex the vertices of which are the graph s vertices citation needed however in graph theory vertices may have fewer than two incident edges which is usually not allowed for geometric vertices there is also a connection between geometric vertices and the vertices of a curve its points of extreme curvature in some sense the vertices of a polygon are points of infinite curvature and if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex 7 however a smooth curve approximation to a polygon will also have additional vertices at the points where its curvature is minimal citation needed of a plane tiling edit a vertex of a plane tiling or tessellation is a point where three or more tiles meet 8 generally but not always the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles more generally a tessellation can be viewed as a kind of topological cell complex as can the faces of a polyhedron or polytope the vertices of other kinds of complexes such as simplicial complexes are its zero dimensional faces citation needed principal vertex edit vertex b is an ear because the open line segment between c and d is entirely inside the polygon vertex c is a mouth because the open line segment between a and b is entirely outside the polygon a polygon vertex x i of a simple polygon p is a principal polygon vertex if the diagonal x i 1 x i 1 intersects the boundary of p only at x i 1 and x i 1 there are two types of principal vertices ears and mouths 9 ears edit a principal vertex x i of a simple polygon p is called an ear if the diagonal x i 1 x i 1 that bridges x i lies entirely in p see also convex polygon according to the two ears theorem every simple polygon has at least two ears 10 mouths edit a principal vertex x i of a simple polygon p is called a mouth if the diagonal x i 1 x i 1 lies outside the boundary of p number of vertices of a polyhedron edit any convex polyhedron s surface has euler characteristic v e f 2 displaystyle v e f 2 where v is the number of vertices e is the number of edges and f is the number of faces this equation is known as euler s polyhedron formula thus the number of vertices is 2 more than the excess of the number of edges over the number of faces for example since a cube has 12 edges and 6 faces the formula implies that it has eight vertices citation needed vertices in computer graphics edit main article vertex computer graphics in computer graphics objects are often represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but also with other graphical information necessary to render the object correctly such as colors reflectance properties textures and surface normal 11 these properties are used in rendering by a vertex shader part of the vertex pipeline see also edit vertex arrangement vertex figure references edit weisstein eric w vertex mathworld vertices edges and faces www mathsisfun com retrieved 2020 08 16 a b what are vertices in math sciencing retrieved 2020 08 16 a b heath thomas l 1956 the thirteen books of euclid s elements 2nd ed facsimile original publication cambridge university press 1925 ed new york dover publications 3 vols isbn 0 486 60088 2 vol 1 isbn 0 486 60089 0 vol 2 isbn 0 486 60090 4 vol 3 jing lanru stephansson ove 2007 fundamentals of discrete element methods for rock engineering theory and applications elsevier science peter mcmullen egon schulte abstract regular polytopes cambridge university press 2002 isbn 0 521 81496 0 page 29 bobenko alexander i schröder peter sullivan john m ziegler günter m 2008 discrete differential geometry birkhäuser verlag ag isbn 978 3 7643 8620 7 m v jaric ed introduction to the mathematics of quasicrystals aperiodicity and order vol 2 isbn 0 12 040602 0 academic press 1989 devadoss satyan o rourke joseph 2011 discrete and computational geometry princeton university press isbn 978 0 691 14553 2 meisters g h 1975 polygons have ears the american mathematical monthly 82 6 648 651 doi 10 2307 2319703 jstor 2319703 mr 0367792 christen martin clockworkcoders tutorials vertex attributes khronos group archived from the original on 12 april 2019 retrieved 26 january 2009 external links edit weisstein eric w polygon vertex mathworld weisstein eric w polyhedron vertex mathworld weisstein eric w principal vertex mathworld authority control national libraries germany retrieved from https en wikipedia org w index php title vertex_ geometry oldid 1107202594 categories euclidean geometry 3d computer graphics polytopes hidden categories articles with short description short description is different from wikidata articles needing additional references from july 2022 all articles needing additional references all articles with unsourced statements articles with unsourced statements from july 2022 articles with gnd identifiers navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search navigation main page contents current events random article about wikipedia contact us donate contribute help learn to edit community portal recent changes upload file tools what links here related changes upload file special pages permanent link page information cite this page wikidata item print export download as pdf printable version languages العربية asturianu avañe ẽ български català čeština chishona deutsch eesti español esperanto euskara فارسی français 한국어 हिन्दी hrvatski italiano עברית latviešu македонски bahasa melayu nederlands 日本語 norsk bokmål occitan português română русский simple english slovenčina slovenščina ślůnski کوردی suomi svenska தமிழ் taqbaylit తెలుగు türkçe українська 文言 吴语 粵語 中文 edit links this page was last edited on 28 august 2022 at 18 50 utc text is available under the creative commons attribution sharealike license 3 0 additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia mobile view developers statistics cookie statement