The red dot represents the point at which the two lines intersect.
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or does not exist (if the lines are parallel). Other types of geometric intersection include:
For two non-parallel line segments and there is not necessarily an intersection point (see diagram), because the intersection point of the corresponding lines need not to be contained in the line segments. In order to check the situation one uses parametric representations of the lines:
The line segments intersect only in a common point of the corresponding lines if the corresponding parameters fulfill the condition .
The parameters are the solution of the linear system
It can be solved for s and t using Cramer's rule (see above). If the condition is fulfilled one inserts or into the corresponding parametric representation and gets the intersection point .
Example: For the line segments and one gets the linear system
and . That means: the lines intersect at point .
Remark: Considering lines, instead of segments, determined by pairs of points, each condition can be dropped and the method yields the intersection point of the lines (see above).
one solves the line equation for x or y and substitutes it into the equation of the circle and gets for the solution (using the formula of a quadratic equation) with
if If this condition holds with strict inequality, there are two intersection points; in this case the line is called a secant line of the circle, and the line segment connecting the intersection points is called a chord of the circle.
If holds, there exists only one intersection point and the line is tangent to the circle. If the weak inequality does not hold, the line does not intersect the circle.
If the circle's midpoint is not the origin, see.[1] The intersection of a line and a parabola or hyperbola may be treated analogously.
The determination of the intersection points of two circles
can be reduced to the previous case of intersecting a line and a circle. By subtraction of the two given equations one gets the line equation:
This special line is the radical line of the two circles.
Intersection of two circles with centers on the x-axis, their radical line is dark red
Special case :
In this case the origin is the center of the first circle and the second center lies on the x-axis (s. diagram). The equation of the radical line simplifies to and the points of intersection can be written as with
In case of the circles have no points in common.
In case of the circles have one point in common and the radical line is a common tangent.
Any general case as written above can be transformed by a shift and a rotation into the special case.
The intersection of two disks (the interiors of the two circles) forms a shape called a lens.
circle–ellipse intersection
Two conic sections
The problem of intersection of an ellipse/hyperbola/parabola with another conic section leads to a system of quadratic equations, which can be solved in special cases easily by elimination of one coordinate. Special properties of conic sections may be used to obtain a solution. In general the intersection points can be determined by solving the equation by a Newton iteration. If a) both conics are given implicitly (by an equation) a 2-dimensional Newton iteration b) one implicitly and the other parametrically given a 1-dimensional Newton iteration is necessary. See next section.
Two smooth curves
A transversal intersection of two curvestouching intersection (left), touching (right)
Two curves in (two-dimensional space), which are continuously differentiable (i.e. there is no sharp bend),
have an intersection point, if they have a point of the plane in common and have at this point (see diagram):
a) different tangent lines (transversal intersection, after transversality), or
b) the tangent line in common and they are crossing each other (touching intersection, after tangency).
If both the curves have a point S and the tangent line there in common but do not cross each other, they are just touching at point S.
Because touching intersections appear rarely and are difficult to deal with, the following considerations omit this case. In any case below all necessary differential conditions are presupposed. The determination of intersection points always leads to one or two non-linear equations which can be solved by Newton iteration. A list of the appearing cases follows:
intersection of a parametric curve and an implicit curveintersection of two implicit curves
If both curves are explicitly given: , equating them yields the equation
If both curves are parametrically given:
Equating them yields two equations in two variables:
If one curve is parametrically and the other implicitly given:
This is the simplest case besides the explicit case. One has to insert the parametric representation of into the equation of curve and one gets the equation:
If both curves are implicitly given:
Here, an intersection point is a solution of the system
Any Newton iteration needs convenient starting values, which can be derived by a visualization of both the curves. A parametrically or explicitly given curve can easily be visualized, because to any parameter t or x respectively it is easy to calculate the corresponding point. For implicitly given curves this task is not as easy. In this case one has to determine a curve point with help of starting values and an iteration. See
.[2]
Examples:
1: and circle (see diagram).
The Newton iteration for function
has to be done. As start values one can choose −1 and 1.5.
The intersection points are: (−1.1073, −1.3578), (1.6011, 4.1046)
2:
(see diagram).
The Newton iteration
has to be performed, where is the solution of the linear system
at point . As starting values one can choose(−0.5, 1) and (1, −0.5).
The linear system can be solved by Cramer's rule.
The intersection points are (−0.3686, 0.9953) and (0.9953, −0.3686).
Two polygons
intersection of two polygons: window test
If one wants to determine the intersection points of two polygons, one can check the intersection of any pair of line segments of the polygons (see above). For polygons with many segments this method is rather time-consuming. In practice one accelerates the intersection algorithm by using window tests. In this case one divides the polygons into small sub-polygons and determines the smallest window (rectangle with sides parallel to the coordinate axes) for any sub-polygon. Before starting the time-consuming determination of the intersection point of two line segments any pair of windows is tested for common points. See.[3]
In 3-dimensional space there are intersection points (common points) between curves and surfaces. In the following sections we consider transversal intersection only.
The intersection of a line and a plane in general position in three dimensions is a point.
Commonly a line in space is represented parametrically and a plane by an equation . Inserting the parameter representation into the equation yields the linear equation
for parameter of the intersection point .
If the linear equation has no solution, the line either lies on the plane or is parallel to it.
Three planes
If a line is defined by two intersecting planes and should be intersected by a third plane , the common intersection point of the three planes has to be evaluated.
Three planes with linear independent normal vectors have the intersection point
For the proof one should establish using the rules of a scalar triple product. If the scalar triple product equals to 0, then planes either do not have the triple intersection or it is a line (or a plane, if all three planes are the same).
A curve and a surface
intersection of curve with surface
Analogously to the plane case the following cases lead to non-linear systems, which can be solved using a 1- or 3-dimensional Newton iteration.[4]
parametric curve and
parametric surface
parametric curve and
implicit surface
Example:
parametric curve and
implicit surface (s. picture).
The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332).
Like the case of a line and a plane, the intersection of a curve and a surface in general position consists of discrete points, but a curve may be partly or totally contained in a surface.
When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:
Let S be a sphere with center O, P a plane which intersects S. Draw OE perpendicular to P and meeting P at E. Let A and B be any two different points in the intersection. Then AOE and BOE are right triangles with a common side, OE, and hypotenuses AO and BO equal. Therefore, the remaining sides AE and BE are equal. This proves that all points in the intersection are the same distance from the point E in the plane P, in other words all points in the intersection lie on a circle C with center E.[5] This proves that the intersection of P and S is contained in C. Note that OE is the axis of the circle.
Now consider a point D of the circle C. Since C lies in P, so does D. On the other hand, the triangles AOE and DOE are right triangles with a common side, OE, and legs EA and ED equal. Therefore, the hypotenuses AO and DO are equal, and equal to the radius of S, so that D lies in S. This proves that C is contained in the intersection of P and S.
As a corollary, on a sphere there is exactly one circle that can be drawn through three given points.[6]
The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.[7]
To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius ) is centered at the origin. Points on this sphere satisfy
Also without loss of generality, assume that the second sphere, with radius , is centered at a point on the positive x-axis, at distance from the origin. Its points satisfy
The intersection of the spheres is the set of points satisfying both equations. Subtracting the equations gives
In the singular case , the spheres are concentric. There are two possibilities: if , the spheres coincide, and the intersection is the entire sphere; if , the spheres are disjoint and the intersection is empty.
When a is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.
The result follows from the previous proof for sphere-plane intersections.
Haines, Eric (June 6, 2021). "Intersections (Ray Tracing Resources Page)". Real-Time Rendering. Retrieved December 14, 2023. a grid of intersection routines for various popular objects, pointing to resources in books and on the web.
Nicholas M. Patrikalakis and Takashi Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2002, ISBN3540424547, 9783540424543, pp. 408. [1]
Sykes, M.; Comstock, C.E. (1922). Solid Geometry. Rand McNally. pp. 81 ff.
My Father and My Son Babam ve OğlumSutradaraÇağan IrmakDitulis olehÇağan IrmakPemeranÇetin TekindorFikret KuşkanHümeyraŞerif SezerEge TanmanÖzge ÖzberkTuba BüyüküstünTanggal rilisDurasi108 menitBahasaTurki My Father and My Son (bahasa Turki: Babam ve Oğlum) adalah sebuah film drama Turki 2005 yang ditulis dan disutradarai oleh Çağan Irmak tentang sebuah keluarga yang terpisah karena kudeta Turki 1980. Film tersebut yang dirilis di seluruh bagian negara pada 18 November 2005…
The following table shows the world record progression in the men's and women's triple jump, officially ratified by the IAAF. Men Ratified Not ratified Ratified but later rescinded Pending ratification The first world record in the men's triple jump was recognised by the International Association of Athletics Federations in 1912. That inaugural record was the 15.52 m performance by Dan Ahearn in 1911.[1] As of June 21, 2009, 27 world records have been ratified by the IAAF in the event.…
Kazamaura-mura 風間浦村 La mairie de Kazamaura. Drapeau Administration Pays Japon Région Tōhoku Préfecture Aomori Code postal 039-4500 Démographie Population 2 550 hab. (30 avril 2014) Densité 37 hab./km2 Géographie Coordonnées 41° 29′ 15″ nord, 140° 59′ 45″ est Superficie 6 960 ha = 69,60 km2 Localisation Géolocalisation sur la carte : Japon Kazamaura-mura Géolocalisation sur la carte : Japon …
Artikel ini bukan mengenai Air berat. Air semiberat Model pengisian ruang dari air Nama Nama IUPAC Air(O-2H1) Nama lain Deuterium hidrogen monoksidaDeuterium hidrogen oksidaAir-d1Air-d Penanda Nomor CAS 14940-63-7 Model 3D (JSmol) Gambar interaktif 3DMet {{{3DMet}}} ChEBI CHEBI:33806 ChemSpider 123344 Nomor EC Referensi Gmelin 115 PubChem CID 139859 Nomor RTECS {{{value}}} CompTox Dashboard (EPA) DTXSID50164262 InChI InChI=1S/H2O/h1H2/i/hDKey: XLYOFNOQVPJJNP-DYCDLGHISA-N SMILES [2H]O Sifat …
Biro Kereta Api WuhanJenisBadan usaha milik negara TiongkokIndustriTransportasi kereta apiPendahuluAdministrasi Kereta Api WuhanDidirikan19 November 2017KantorpusatJalan Bayi No.2, Wuchang, Wuhan, TiongkokWilayah operasiHubeiHenan selatanPemilikPemerintah TiongkokIndukKereta api TiongkokSitus webOfficial Website Biro Kereta Api Wuhan Kantor Biro Kereta Api Wuhan, dekat Alun-alun Hongshan. Hanzi sederhana: 中国铁路武汉局集团 Hanzi tradisional: 中國鐵路武漢局集團 Alih aksara Mand…
Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Aktris Terbaik dalam sebuah Peran Pendukung Penghargaan Robert – berita · surat kabar · buku · cendekiawan · JSTOR Penerima 1980an 1984: Mette Munk Plum – Thunderbirds (film 1983) 1985: Aase Hansen …
كيرلس الثالث معلومات شخصية الميلاد سنة 1235[1] مصر الوفاة مارس 10, 1243مصر الإقامة الكنيسة المعلقة مواطنة مصر اللغة الأم القبطية اللغات القبطية تعديل مصدري - تعديل كيرلس الثالث بابا الإسكندرية وبطريرك الكنيسة القبطية رقم 75 (1235 - 1243) في عهد تملك الملك العادل.…
American college football season 1912 Florida Gators footballThe 1912 Gators practicing on the UF campusBacardi Bowl, W 28–0 vs. Vedado Tennis ClubConferenceSouthern Intercollegiate Athletic AssociationRecord5–2–1 (0–2–1 SIAA)Head coachGeorge E. Pyle (4th season)CaptainSam BuieHome stadiumUniversity Athletic FieldSeasons← 19111913 → 1912 Southern Intercollegiate Athletic Association football standings vte Conf Overall Team W L T W L …
Uruguayan footballer (born 1983) In this Spanish name, the first or paternal surname is Conde and the second or maternal family name is Quintana. Esteban Conde Conde playing for Nacional in 2016Personal informationFull name Néstor Esteban Conde QuintanaDate of birth (1983-03-04) 4 March 1983 (age 41)Place of birth Young, UruguayHeight 1.86 m (6 ft 1 in)Position(s) GoalkeeperYouth career2001–2003 RentistasSenior career*Years Team Apps (Gls)2003–2004 Rentistas 46 (…
Kala Cinta MenggodaGenreSinetronPembuatTripar Multivision PlusPemeran Angel Karamoy Nicky Tirta Bunga Citra Lestari Harlan Chaniago Yana Zein Aty Cancer Egy Fedly Bunga Zainal Ronaldo Barky Arie Dwi Andhika Lia Ananta Penggubah lagu temaDidi AGPLagu pembukaKala Cinta Menggoda, ChrisyeLagu penutupKala Cinta Menggoda, ChrisyePenata musikJoseph SaryufNegara asalIndonesiaBahasa asliIndonesiaJawaJmlh. episode10 ([episode])ProduksiProduserRaam PunjabiLokasi produksiJakarta, IndonesiaDurasi60 men…
Europe Écologie Logotype d'Europe Écologie. Élections concernées par l'alliance Européennes 2009, Régionales 2010 Organisations politiques concernées Les Verts,Régions et peuples solidaires (RPS) Présidents au Parlement européen Groupe des Verts/Alliance libre européenne Idéologie Politique écologiqueFédéralismeEuropéisme Couleurs Vert modifier Europe Écologie est le nom donné aux listes du rassemblement de la mouvance écologiste en France lancé par Les Verts pour les …
Earthquake in Indiana, United States 2010 Indiana earthquakeMap of the maximum intensity in the surrounding counties with a star as the epicenter.UTC time2010-12-30 12:55:21ISC eventn/a USGS-ANSSComCatLocal dateDecember 30, 2010 (2010-12-30)Local time7:55 a.m.[1]Magnitude3.8 Mw Depth3 mi (4.83 km)Epicenter40°25′37″N 85°53′17″W / 40.427°N 85.888°W / 40.427; -85.888Areas affectedUnited StatesIndianaMax. …
Intense but shallow attraction For other uses, see Infatuation (disambiguation). This article is written like a personal reflection, personal essay, or argumentative essay that states a Wikipedia editor's personal feelings or presents an original argument about a topic. Please help improve it by rewriting it in an encyclopedic style. (September 2018) (Learn how and when to remove this template message) An illustrated depiction of Mahābhārata character Ulupi becoming infatuated with Arjuna Infa…
Église orthodoxe serbe(sr) Српска православна црква Fondateur(s) Saint Sava (1er archevêque) Autocéphalie ou autonomie déclarée 1219–14591557–1776Depuis 1832 Reconnaissance 1922 (par le Patriarcat œcuménique) Primat actuel Porphyre Siège Bâtiment du patriarcat, Belgrade (Serbie) Territoire primaire Europe du Sud-Est : Serbie, Bosnie-Herzégovine, Monténégro, Croatie, Kosovo et Slovénie Extension territoriale Archevêché orthodoxe d'Ohrid Rite Byzantin …
National anthem of Jamaica Jamaica, Land We LoveNational anthem of JamaicaLyricsHugh Sherlock, July 1962[1]MusicRobert Lightbourne (arranged by Mapletoft Poulle), July 1962[1]Adopted19 July 1962 (1962-07-19)Audio sampleU.S. Navy Band instrumental version (one verse)filehelp Jamaica, Land We Love is the national anthem of Jamaica, officially adopted in July 1962. It was chosen after a competition from September 1961 to 31 March 1962, in which the lyrics of the natio…
مسجد ومدرسة وتكية محمد بك أبو الذهب إحداثيات 30°02′45″N 31°15′43″E / 30.045933°N 31.261841°E / 30.045933; 31.261841 معلومات عامة القرية أو المدينة القاهرة الدولة مصر سنة التأسيس 1774 تاريخ بدء البناء منذ عام 1187هـ/1773م وحتى عام 1188هـ/1774م التصميم والإنشاء النمط المعماري المماليك مع…
British physicist For other people named Paul Davies, see Paul Davies (disambiguation). Paul DaviesAMDavies in 2016BornPaul Charles William Davies (1946-04-22) 22 April 1946 (age 77)London, United KingdomAlma materUniversity College LondonKnown forFulling–Davies–Unruh effectBunch–Davies vacuum stateMoving MirrorsAwardsTempleton Prize (1995)Kelvin Medal (2001)Faraday Prize (2002)Klumpke-Roberts Award (2011)Scientific careerFieldsPhysicsInstitutionsArizona State UniversityUniv…
Eurasian Patent ConventionParties (purple), former parties (green) and signatories that did not ratify (blue)Signed9 September 1994 (1994-09-09)[1]LocationMoscow, RussiaEffective12 August 1995[2]ConditionRatification by three States[1]Signatories10Parties8 (Armenia, Azerbaijan, Belarus, Russia, Kyrgyzstan, Kazakhstan, Tajikistan, Turkmenistan)[2][3]DepositaryDirector-General of WIPO[1]LanguageRussian[1][3] The Eurasia…
This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles; try the Find link tool for suggestions. (August 2023) School in Gangche, Pandua, West Bengal, IndiaKhanpur Gangche High SchoolLocationGangche, Pandua, West Bengal, 721232IndiaCoordinates22°39′40″N 87°35′30″E / 22.6611453°N 87.5915912°E / 22.6611453; 87.5915912InformationEstablished1959 (1959)School districtPaschim Medinipur Khanpur Gangch…