@prefix rdf: . @prefix dbr: . @prefix yago: . dbr:Cross-ratio rdf:type yago:MagnitudeRelation113815152 , yago:Relation100031921 . @prefix owl: . dbr:Cross-ratio rdf:type owl:Thing , yago:Abstraction100002137 , yago:WikicatRatios , yago:Ratio113819207 . @prefix rdfs: . dbr:Cross-ratio rdfs:label "\u8907\u6BD4"@ja , "Doppelverh\u00E4ltnis"@de , "\uBE44\uC870\uD654\uBE44"@ko , "Birapport"@fr , "\u0414\u0432\u043E\u0439\u043D\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435"@ru , "Dubbelverhouding"@nl , "Raz\u00E3o anarm\u00F4nica"@pt , "\u0646\u0633\u0628\u0629 \u062A\u0628\u0627\u062F\u0644\u064A\u0629"@ar , "Ra\u00F3 doble"@ca , "Cross-ratio"@en , "\u041F\u043E\u0434\u0432\u0456\u0439\u043D\u0435 \u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u043D\u044F"@uk , "Raz\u00F3n anarm\u00F3nica"@es , "Dwustosunek"@pl , "\u4EA4\u6BD4"@zh , "Birapporto"@it ; rdfs:comment "Das Doppelverh\u00E4ltnis ist in der Geometrie im einfachsten Fall das Verh\u00E4ltnis zweier Teilverh\u00E4ltnisse. Wird zum Beispiel die Strecke sowohl durch einen Punkt als auch durch einen Punkt in jeweils zwei Teilstrecken und bzw. und (s. erstes Beispiel) geteilt, so ist das Verh\u00E4ltnis das (affine) Doppelverh\u00E4ltnis, in dem die Teilpunkte die gegebene Strecke teilen. Die gro\u00DFe Bedeutung erh\u00E4lt das Doppelverh\u00E4ltnis als Invariante bei Zentralprojektionen, denn das anschaulichere Teilverh\u00E4ltnis ist zwar invariant unter Parallelprojektionen, aber nicht unter Zentralprojektionen.Eine Verallgemeinerung f\u00FChrt zur Definition des Doppelverh\u00E4ltnisses f\u00FCr Punkte einer projektiven Gerade (das hei\u00DFt, einer affinen Geraden, der ein Fernpunkt hinzugef\u00FCgt wird)."@de , "La ra\u00F3 doble, tamb\u00E9 anomenada ra\u00F3 anharm\u00F2nica, \u00E9s una poderosa eina en geometria, especialment en geometria projectiva. El nom de ra\u00F3 anharm\u00F2nica va ser creat per Michel Chasles, per\u00F2 la noci\u00F3 es remunta a Pappos d'Alexandria."@ca , "\u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629\u060C \u0627\u0644\u0646\u0633\u0628\u0629 \u0627\u0644\u062A\u0628\u0627\u062F\u0644\u064A\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Cross-ratio)\u200F \u0647\u064A \u0646\u0633\u0628\u0629\u064C \u0645\u064F\u0631\u062A\u0628\u0637\u0629\u064C \u0628\u0623\u0631\u0628\u0639\u0650 \u0646\u0642\u0627\u0637\u064D \u0645\u064F\u062A\u0633\u0627\u0645\u062A\u0629. \u0625\u0630\u0627 \u0643\u0627\u0646\u062A \u0627\u0644\u0646\u0642\u0627\u0637 \u0639\u0644\u0649 \u0627\u0633\u062A\u0642\u0627\u0645\u0629\u064D \u0648\u0627\u062D\u062F\u0629\u064D\u060C \u0641\u0625\u0646\u064E\u0651 \u0646\u0633\u0628\u062A\u0647\u0645 \u0627\u0644\u062A\u0628\u0627\u062F\u0644\u064A\u0629 \u062A\u064F\u0639\u0631\u0651\u0641 \u0643\u0627\u0644\u0622\u062A\u064A: \u062D\u064A\u062B \u0623\u0646\u064E\u0651 \u0627\u0644\u0646\u0651\u0633\u0628 \u0646\u0633\u0628\u064C \u0645\u064F\u0648\u062C\u0651\u0647\u0629\u064C. \u0625\u0630\u0627 \u0643\u0627\u0646\u062A \u0648\u0627\u062D\u062F\u0629 \u0645\u0646 \u0627\u0644\u0646\u0642\u0627\u0637 \u0627\u0644\u0623\u0631\u0628\u0639 \u0646\u0642\u0637\u0629\u064B \u0641\u064A \u0627\u0644\u0644\u0627\u0646\u0647\u0627\u064A\u0629\u060C \u0641\u0625\u0646\u064E\u0651 \u0627\u0644\u0645\u0633\u0627\u0641\u062A\u064A\u0646 \u0627\u0644\u0648\u0627\u0635\u0644\u062A\u064A\u0646 \u0628\u0647\u0630\u0647 \u0627\u0644\u0646\u0642\u0637\u0629 \u062A\u064F\u062D\u0630\u0641 \u0645\u0646 \u0627\u0644\u0635\u064A\u063A\u0629. \u062A\u064F\u0639\u0631\u0651\u0641\u064F \u0627\u0644\u0646\u0642\u0637\u0629 D \u0639\u0644\u0649 \u0623\u0646\u0651\u0647\u0627 \u0627\u0644\u0645\u0631\u0627\u0641\u0642 \u0627\u0644\u062A\u0648\u0627\u0641\u0642\u064A \u0644\u0644\u0646\u0642\u0637\u0629 C \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0640A \u0648 B."@ar , "In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry."@en , "La raz\u00F3n anarm\u00F3nica o raz\u00F3n doble es una poderosa herramienta en geometr\u00EDa, especialmente en geometr\u00EDa proyectiva. El nombre de raz\u00F3n anarm\u00F3nica fue creado por Michel Chasles, pero la noci\u00F3n se remonta a Papo de Alejandr\u00EDa."@es , "In de meetkunde is de dubbelverhouding van vier collineaire punten gedefinieerd als de verhouding van twee deelverhoudingen. De dubbelverhouding is invariant onder centrale projectie."@nl , "\u0414\u0432\u043E\u0439\u043D\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435 (\u0438\u043B\u0438 \u0441\u043B\u043E\u0436\u043D\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435 \u0438\u043B\u0438 \u0443\u0441\u0442\u0430\u0440\u0435\u0432\u0448\u0435\u0435 \u0430\u043D\u0433\u0430\u0440\u043C\u043E\u043D\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435) \u0447\u0435\u0442\u0432\u0451\u0440\u043A\u0438 \u0447\u0438\u0441\u0435\u043B , , , (\u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0445 \u0438\u043B\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0445) \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u043A\u0430\u043A \u0422\u0430\u043A\u0436\u0435 \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044E\u0442\u0441\u044F \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0435\u043D\u0438\u044F \u0438 ."@ru , "Dwustosunek (stosunek anharmoniczny) czterech wsp\u00F3\u0142liniowych punkt\u00F3w \u2013 funkcja postaci: gdzie punkty A, B, C, D spe\u0142niaj\u0105 oraz jest wsp\u00F3\u0142rz\u0119dn\u0105 punktu X w uk\u0142adzie wsp\u00F3\u0142rz\u0119dnych na danej prostej. Jest to podstawowe poj\u0119cie geometrii rzutowej. Jak wida\u0107, powy\u017Csza definicja zak\u0142ada istnienie uk\u0142adu wsp\u00F3\u0142rz\u0119dnych na rozpatrywanej prostej. Wyb\u00F3r uk\u0142adu wsp\u00F3\u0142rz\u0119dnych z wielu mo\u017Cliwych nie wp\u0142ywa na warto\u015B\u0107 dwustosunku."@pl , "Le birapport, ou rapport anharmonique selon la d\u00E9nomination de Michel Chasles est un outil puissant de la g\u00E9om\u00E9trie, en particulier la g\u00E9om\u00E9trie projective. La notion remonte \u00E0 Pappus d'Alexandrie, mais son \u00E9tude syst\u00E9matique est r\u00E9alis\u00E9e en 1827 par M\u00F6bius."@fr , "\uC0AC\uC601\uAE30\uD558\uD559\uC5D0\uC11C, \uBE44\uC870\uD654\uBE44(\u975E\u8ABF\u548C\u6BD4, \uC601\uC5B4: anharmonic ratio) \uB610\uB294 \uBCF5\uBE44(\u8907\u6BD4, \uC601\uC5B4: double ratio)\uB294 \uAC19\uC740 \uC9C1\uC120 \uC704\uC5D0 \uC788\uB294 \uB124 \uC810\uC758 \uC720\uC77C\uD55C \uC0AC\uC601 \uBD88\uBCC0\uB7C9\uC774\uB2E4."@ko , "\u041F\u043E\u0434\u0432\u0456\u0301\u0439\u043D\u0435 \u0432\u0456\u0434\u043D\u043E\u0301\u0448\u0435\u043D\u043D\u044F (\u0430\u0431\u043E \u0441\u043A\u043B\u0430\u0434\u043D\u0435\u0301 \u0432\u0456\u0434\u043D\u043E\u0301\u0448\u0435\u043D\u043D\u044F \u0430\u0431\u043E \u0437\u0430\u0441\u0442\u0430\u0440\u0456\u043B\u0435 \u0430\u043D\u0433\u0430\u0440\u043C\u043E\u043D\u0456\u0447\u043D\u0435 \u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u043D\u044F) \u0447\u0435\u0442\u0432\u0456\u0440\u043A\u0438 \u0447\u0438\u0441\u0435\u043B , , , (\u0434\u0456\u0439\u0441\u043D\u0438\u0445 \u0447\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u0445) \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u044F\u043A"@uk , "\u6570\u5B66\u4E0A\uFF0C\u8907\u5E73\u9762\u4E0A\u56DB\u70B9\u7684\u4EA4\u6BD4\u662F \u3002 \u8FD9\u4E2A\u5B9A\u4E49\u53EF\u4EE5\u8FDE\u7EED\u5EF6\u62D3\u81F3\u6574\u4E2A\u9ECE\u66FC\u7403\u9762\uFF0C\u5373\u8907\u5E73\u9762\u52A0\u4E0A\u65E0\u7A77\u8FDC\u70B9\u3002 \u4E00\u822C\u6765\u8BF4\uFF0C\u4EA4\u6BD4\u53EF\u4EE5\u5B9A\u4E49\u5728\uFF08\u9ECE\u66FC\u7403\u9762\u5C31\u662F\u8907\u5C04\u5F71\u76F4\u7DDA\uFF09\u3002\u5728\u4EFB\u4F55\u4E2D\uFF0C\u4EA4\u6BD4\u7531\u4E0A\u5F0F\u7ED9\u51FA\u3002\u4EA4\u6BD4\u662F\u5C04\u5F71\u51E0\u4F55\u7684\u4E0D\u53D8\u91CF\uFF0C\u5C31\u662F\u8BF4\u4FDD\u6301\u4EA4\u6BD4\u4E0D\u53D8\u3002\u4ECE\u524D\u4EBA\u4EEC\u6CE8\u610F\u5230\u5982\u679C\u56DB\u6761\u76F4\u7EBF\u7A7F\u8FC7\u4E00\u70B9P\uFF0C\u7B2C\u4E94\u6761\u76F4\u7EBFL\u4E0D\u7A7F\u8FC7P\uFF0C\u5206\u522B\u4E0E\u56DB\u6761\u76F4\u7EBF\u4EA4\u4E8E\u56DB\u70B9\uFF0C\u90A3\u4E48\u5728L\u4E0A\u6309\u5E8F\u53D6\u56DB\u70B9\u7684\u6709\u5411\u957F\u5EA6\uFF0C\u6240\u7B97\u51FA\u7684\u4EA4\u6BD4\u662F\u72EC\u7ACB\u4E8EL\u3002\u5B83\u662F\u8FD9\u56DB\u76F4\u7EBF\u7CFB\u7684\u4E0D\u53D8\u91CF\u3002 \u56DB\u4E2A\u8907\u6570\u7684\u4EA4\u6BD4\u4E3A\u5B9E\u6570\u5F53\u4E14\u552F\u5F53\u56DB\u70B9\u5171\u7EBF\u6216\u3002"@zh , "Il birapporto \u00E8 una grandezza associata a una quaterna di punti di una retta. Si tratta di uno strumento importante in geometria proiettiva: risulta infatti definito anche se uno dei quattro punti \u00E8 all'infinito (la retta in questione \u00E8 quindi una retta proiettiva) ed \u00E8 invariante tramite trasformazioni proiettive. La retta su cui giacciono i punti pu\u00F2 essere definita su un campo diverso dai numeri reali. Ad esempio, se definita sui numeri complessi, la retta \u00E8 in realt\u00E0 la sfera di Riemann, ovvero il piano complesso a cui va aggiunto un punto all'infinito."@it , "\u8907\u6BD4\uFF08\u3075\u304F\u3072\u3001\u82F1: double ratio\uFF09\u306F\u3001\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u6982\u5FF5\u306E1\u3064\u3067\u3001\u4EA4\u5DEE\u6BD4\uFF08\u3053\u3046\u3055\u3072\u3001\u82F1: cross-ratio\uFF09\u304A\u3088\u3073\u975E\u8ABF\u548C\u6BD4\uFF08\u3072\u3061\u3087\u3046\u308F\u3072\u3001\u82F1: anharmonic ratio\uFF09\u3068\u3082\u547C\u3070\u308C\u30014\u3064\u306E\u5171\u7DDA\u4E0A\u306E\u70B9\u3001\u7279\u306B\u5C04\u5F71\u76F4\u7DDA\u4E0A\u306E\u70B9\u306E\u96C6\u5408\u306B\u95A2\u9023\u4ED8\u3051\u3089\u308C\u305F\u6570\u5024\u3067\u3042\u308B\u3002\u76F4\u7DDA\u4E0A\u306E4\u3064\u306E\u70B9 A, B, C, D \u304C\u4E0E\u3048\u3089\u308C\u308B\u3068\u3001\u305D\u308C\u3089\u306E\u8907\u6BD4\u306F\u6B21\u306E\u3088\u3046\u306B\u5B9A\u7FA9\u3055\u308C\u308B\u3002 \u3053\u3053\u3067\u3001\u5404\u8DDD\u96E2\u306E\u7B26\u53F7\u306F\u7DDA\u306E\u5411\u304D\u306B\u3088\u3063\u3066\u6C7A\u307E\u308A\u3001\u8DDD\u96E2\u306F\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u7A7A\u9593\u306B\u5C04\u5F71\u3055\u308C\u3066\u6E2C\u5B9A\u3055\u308C\u308B\u3002\uFF084\u3064\u306E\u70B9\u306E1\u3064\u304C\u76F4\u7DDA\u306E\u7121\u9650\u9060\u70B9\u3067\u3042\u308B\u5834\u5408\u3001\u305D\u306E\u70B9\u3092\u542B\u30802\u3064\u306E\u8DDD\u96E2\u306F\u5F0F\u304B\u3089\u524A\u9664\u3055\u308C\u308B\u3002\uFF09\u8907\u6BD4\u304C\u6B63\u78BA\u306B-1\u306E\u5834\u5408\u3001\u70B9D\u306FA\u3068B\u306B\u5BFE\u3059\u308BC\u306E\u3067\u3042\u308A\u3001\u8ABF\u548C\u6BD4\u3068\u547C\u3070\u308C\u308B\u3002\u3057\u305F\u304C\u3063\u3066\u3001\u8907\u6BD4\u306F\u30014\u3064\u7D44\u306E\u8ABF\u548C\u6BD4\u304B\u3089\u306E\u504F\u5DEE\u3092\u6E2C\u5B9A\u3059\u308B\u3082\u306E\u3068\u307F\u306A\u305B\u308B\u3002\u305D\u306E\u305F\u3081\u975E\u8ABF\u548C\u6BD4\u3068\u3082\u547C\u3070\u308C\u308B\u3002 \u8907\u6BD4\u306F\u7DDA\u5F62\u5206\u6570\u5909\u63DB\u306E\u4E0B\u3067\u4E0D\u5909\u3067\u3042\u308B\u3002\u3053\u308C\u306F\u672C\u8CEA\u7684\u306B4\u3064\u306E\u540C\u4E00\u7DDA\u4E0A\u306E\u70B9\u306E\u552F\u4E00\u306E\u5C04\u5F71\u4E0D\u5909\u91CF\u3067\u3042\u308B\u3002\u3053\u306E\u3053\u3068\u306F\u5C04\u5F71\u5E7E\u4F55\u5B66\u306E\u6839\u5E95\u306B\u3042\u308B\u91CD\u8981\u306A\u6027\u8CEA\u3067\u3042\u308B\u3002 \u8907\u6BD4\u306F\u3001\u53E4\u4EE3\u3088\u308A\u304A\u305D\u3089\u304F\u306F\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u306B\u3088\u3063\u3066\u5B9A\u7FA9\u3055\u308C\u3001\u30D1\u30C3\u30D7\u30B9\u306B\u3088\u3063\u3066\u305D\u306E\u91CD\u8981\u306A\u666E\u904D\u6027\u7279\u6027\u306B\u6CE8\u76EE\u3057\u305F\u8003\u5BDF\u304C\u306A\u3055\u308C\u305F\u300219\u4E16\u7D00\u306B\u306F\u5E83\u304F\u7814\u7A76\u3055\u308C\u308B\u3088\u3046\u306B\u306A\u3063\u305F\u3002"@ja , "Em geometria projetiva, dist\u00E2ncias e \u00E2ngulos n\u00E3o s\u00E3o preservados. O conceito m\u00E9trico que \u00E9 preservado pelas \u00E9 a raz\u00E3o anarm\u00F4nica. A raz\u00E3o anarm\u00F4nica de quatro pontos colineares \u00E9 definida por: em que os segmentos de reta devem ser interpretados como segmentos orientados. Os quatro pontos est\u00E3o na raz\u00E3o harm\u00F4nica quando a raz\u00E3o anarm\u00F4nica entre eles vale -1."@pt ; owl:differentFrom dbr:Odds_ratio . @prefix foaf: . dbr:Cross-ratio foaf:depiction , . @prefix dcterms: . @prefix dbc: . dbr:Cross-ratio dcterms:subject dbc:Ratios , dbc:Projective_geometry . @prefix dbo: . dbr:Cross-ratio dbo:wikiPageID 401767 ; dbo:wikiPageRevisionID 1104327723 ; dbo:wikiPageWikiLink dbr:Euclid , dbr:Klein_four-group , dbr:Quotient_group , , dbr:Carl_von_Staudt , dbr:Homogeneous_coordinate , dbr:Pappus_of_Alexandria , , , dbr:Dihedral_group_of_order_6 , dbr:Riemann_surfaces , dbr:Mathematical_Intelligencer , dbr:Homography , dbr:Projective_harmonic_conjugate , , dbr:Projective_line_over_a_ring , dbr:Complex_plane , , dbr:Theta_functions , dbr:Generating_set_of_a_group , , dbr:Complex_number , , dbr:Unit_circle , , dbr:Concyclic , dbr:Euclidean_space , dbr:Multiplicative_inverse , dbr:Transitive_action , dbr:Non-Euclidean_geometry , dbr:Geometry , , dbr:Group_action , dbr:Dirk_Struik , , dbr:Elliptic_transform , dbr:Five_points_determine_a_conic , , , , dbr:Symmetric_group , dbr:Projective_geometry , , dbr:Projective_plane , dbr:Collinear , dbr:Michel_Chasles , dbr:Effective_group_action , dbr:General_position , , , dbr:Generalized_circle , , dbr:Stabilizer_subgroup , dbr:Projectively_extended_real_line , , , dbr:Projective_transformation , dbr:Conic_section , dbr:Homothetic_transformation , dbr:Felix_Klein , , dbr:Real_number , dbr:Linear_fractional_transformation , dbr:Point_at_infinity , dbc:Ratios , dbr:Addison-Wesley , dbr:Cross-ratio , dbr:Projective_connection , dbr:Factorial , dbr:Karl_von_Staudt , dbr:Projective_line , dbr:Projective_linear_group , dbr:Projective_group , dbr:Isaac_Newton , dbr:Permutation , dbr:Brady_Haran , dbr:Complex_projective_line , dbr:Cycle_notation , dbr:Igor_Shafarevich , dbr:Arthur_Cayley , dbr:Hilbert_metric , dbr:Riemann_sphere , , dbr:Gaussian_curvature , , dbr:Special_conformal_transformation , dbr:McGraw-Hill , dbr:Hyperbolic_geometry , dbr:Affine_transformation , dbr:Schwarzian_derivative , dbr:Lazare_Carnot , dbr:Simply_transitive , , dbr:Euclidean_geometry , dbr:Cambridge_University_Press , dbr:Robert_Simson , dbr:Cut-the-knot , dbr:William_Kingdon_Clifford , , dbr:Lars_Ahlfors , dbc:Projective_geometry ; dbo:wikiPageExternalLink , , , , , . @prefix dbpedia-nn: . dbr:Cross-ratio owl:sameAs dbpedia-nn:Dobbeltforhold , . @prefix dbpedia-pl: . dbr:Cross-ratio owl:sameAs dbpedia-pl:Dwustosunek , , , , . @prefix dbpedia-it: . dbr:Cross-ratio owl:sameAs dbpedia-it:Birapporto , , , . @prefix dbpedia-fr: . dbr:Cross-ratio owl:sameAs dbpedia-fr:Birapport , , , . @prefix dbpedia-nl: . dbr:Cross-ratio owl:sameAs dbpedia-nl:Dubbelverhouding . @prefix dbpedia-sl: . dbr:Cross-ratio owl:sameAs dbpedia-sl:Dvorazmerje , . @prefix dbpedia-no: . dbr:Cross-ratio owl:sameAs dbpedia-no:Dobbeltforhold , , . @prefix dbpedia-fi: . dbr:Cross-ratio owl:sameAs dbpedia-fi:Kaksoissuhde . @prefix wikidata: . dbr:Cross-ratio owl:sameAs wikidata:Q899539 , , . @prefix yago-res: . dbr:Cross-ratio owl:sameAs yago-res:Cross-ratio , . @prefix dbpedia-ro: . dbr:Cross-ratio owl:sameAs dbpedia-ro:Raport_anarmonic . @prefix dbp: . @prefix dbt: . dbr:Cross-ratio dbp:wikiPageUsesTemplate dbt:Cbignore , dbt:Mathworld , dbt:Further , dbt:ISBN , dbt:Citation_needed , dbt:Null , dbt:Clear , dbt:Short_description , dbt:Cite_web , dbt:Main , dbt:Reflist , dbt:Distinguish , , dbt:Visible_anchor , ; dbo:thumbnail ; dbo:abstract "\u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629\u060C \u0627\u0644\u0646\u0633\u0628\u0629 \u0627\u0644\u062A\u0628\u0627\u062F\u0644\u064A\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Cross-ratio)\u200F \u0647\u064A \u0646\u0633\u0628\u0629\u064C \u0645\u064F\u0631\u062A\u0628\u0637\u0629\u064C \u0628\u0623\u0631\u0628\u0639\u0650 \u0646\u0642\u0627\u0637\u064D \u0645\u064F\u062A\u0633\u0627\u0645\u062A\u0629. \u0625\u0630\u0627 \u0643\u0627\u0646\u062A \u0627\u0644\u0646\u0642\u0627\u0637 \u0639\u0644\u0649 \u0627\u0633\u062A\u0642\u0627\u0645\u0629\u064D \u0648\u0627\u062D\u062F\u0629\u064D\u060C \u0641\u0625\u0646\u064E\u0651 \u0646\u0633\u0628\u062A\u0647\u0645 \u0627\u0644\u062A\u0628\u0627\u062F\u0644\u064A\u0629 \u062A\u064F\u0639\u0631\u0651\u0641 \u0643\u0627\u0644\u0622\u062A\u064A: \u062D\u064A\u062B \u0623\u0646\u064E\u0651 \u0627\u0644\u0646\u0651\u0633\u0628 \u0646\u0633\u0628\u064C \u0645\u064F\u0648\u062C\u0651\u0647\u0629\u064C. \u0625\u0630\u0627 \u0643\u0627\u0646\u062A \u0648\u0627\u062D\u062F\u0629 \u0645\u0646 \u0627\u0644\u0646\u0642\u0627\u0637 \u0627\u0644\u0623\u0631\u0628\u0639 \u0646\u0642\u0637\u0629\u064B \u0641\u064A \u0627\u0644\u0644\u0627\u0646\u0647\u0627\u064A\u0629\u060C \u0641\u0625\u0646\u064E\u0651 \u0627\u0644\u0645\u0633\u0627\u0641\u062A\u064A\u0646 \u0627\u0644\u0648\u0627\u0635\u0644\u062A\u064A\u0646 \u0628\u0647\u0630\u0647 \u0627\u0644\u0646\u0642\u0637\u0629 \u062A\u064F\u062D\u0630\u0641 \u0645\u0646 \u0627\u0644\u0635\u064A\u063A\u0629. \u062A\u064F\u0639\u0631\u0651\u0641\u064F \u0627\u0644\u0646\u0642\u0637\u0629 D \u0639\u0644\u0649 \u0623\u0646\u0651\u0647\u0627 \u0627\u0644\u0645\u0631\u0627\u0641\u0642 \u0627\u0644\u062A\u0648\u0627\u0641\u0642\u064A \u0644\u0644\u0646\u0642\u0637\u0629 C \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0640A \u0648 B."@ar , "La raz\u00F3n anarm\u00F3nica o raz\u00F3n doble es una poderosa herramienta en geometr\u00EDa, especialmente en geometr\u00EDa proyectiva. El nombre de raz\u00F3n anarm\u00F3nica fue creado por Michel Chasles, pero la noci\u00F3n se remonta a Papo de Alejandr\u00EDa."@es , "\u8907\u6BD4\uFF08\u3075\u304F\u3072\u3001\u82F1: double ratio\uFF09\u306F\u3001\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u6982\u5FF5\u306E1\u3064\u3067\u3001\u4EA4\u5DEE\u6BD4\uFF08\u3053\u3046\u3055\u3072\u3001\u82F1: cross-ratio\uFF09\u304A\u3088\u3073\u975E\u8ABF\u548C\u6BD4\uFF08\u3072\u3061\u3087\u3046\u308F\u3072\u3001\u82F1: anharmonic ratio\uFF09\u3068\u3082\u547C\u3070\u308C\u30014\u3064\u306E\u5171\u7DDA\u4E0A\u306E\u70B9\u3001\u7279\u306B\u5C04\u5F71\u76F4\u7DDA\u4E0A\u306E\u70B9\u306E\u96C6\u5408\u306B\u95A2\u9023\u4ED8\u3051\u3089\u308C\u305F\u6570\u5024\u3067\u3042\u308B\u3002\u76F4\u7DDA\u4E0A\u306E4\u3064\u306E\u70B9 A, B, C, D \u304C\u4E0E\u3048\u3089\u308C\u308B\u3068\u3001\u305D\u308C\u3089\u306E\u8907\u6BD4\u306F\u6B21\u306E\u3088\u3046\u306B\u5B9A\u7FA9\u3055\u308C\u308B\u3002 \u3053\u3053\u3067\u3001\u5404\u8DDD\u96E2\u306E\u7B26\u53F7\u306F\u7DDA\u306E\u5411\u304D\u306B\u3088\u3063\u3066\u6C7A\u307E\u308A\u3001\u8DDD\u96E2\u306F\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u7A7A\u9593\u306B\u5C04\u5F71\u3055\u308C\u3066\u6E2C\u5B9A\u3055\u308C\u308B\u3002\uFF084\u3064\u306E\u70B9\u306E1\u3064\u304C\u76F4\u7DDA\u306E\u7121\u9650\u9060\u70B9\u3067\u3042\u308B\u5834\u5408\u3001\u305D\u306E\u70B9\u3092\u542B\u30802\u3064\u306E\u8DDD\u96E2\u306F\u5F0F\u304B\u3089\u524A\u9664\u3055\u308C\u308B\u3002\uFF09\u8907\u6BD4\u304C\u6B63\u78BA\u306B-1\u306E\u5834\u5408\u3001\u70B9D\u306FA\u3068B\u306B\u5BFE\u3059\u308BC\u306E\u3067\u3042\u308A\u3001\u8ABF\u548C\u6BD4\u3068\u547C\u3070\u308C\u308B\u3002\u3057\u305F\u304C\u3063\u3066\u3001\u8907\u6BD4\u306F\u30014\u3064\u7D44\u306E\u8ABF\u548C\u6BD4\u304B\u3089\u306E\u504F\u5DEE\u3092\u6E2C\u5B9A\u3059\u308B\u3082\u306E\u3068\u307F\u306A\u305B\u308B\u3002\u305D\u306E\u305F\u3081\u975E\u8ABF\u548C\u6BD4\u3068\u3082\u547C\u3070\u308C\u308B\u3002 \u8907\u6BD4\u306F\u7DDA\u5F62\u5206\u6570\u5909\u63DB\u306E\u4E0B\u3067\u4E0D\u5909\u3067\u3042\u308B\u3002\u3053\u308C\u306F\u672C\u8CEA\u7684\u306B4\u3064\u306E\u540C\u4E00\u7DDA\u4E0A\u306E\u70B9\u306E\u552F\u4E00\u306E\u5C04\u5F71\u4E0D\u5909\u91CF\u3067\u3042\u308B\u3002\u3053\u306E\u3053\u3068\u306F\u5C04\u5F71\u5E7E\u4F55\u5B66\u306E\u6839\u5E95\u306B\u3042\u308B\u91CD\u8981\u306A\u6027\u8CEA\u3067\u3042\u308B\u3002 \u8907\u6BD4\u306F\u3001\u53E4\u4EE3\u3088\u308A\u304A\u305D\u3089\u304F\u306F\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u306B\u3088\u3063\u3066\u5B9A\u7FA9\u3055\u308C\u3001\u30D1\u30C3\u30D7\u30B9\u306B\u3088\u3063\u3066\u305D\u306E\u91CD\u8981\u306A\u666E\u904D\u6027\u7279\u6027\u306B\u6CE8\u76EE\u3057\u305F\u8003\u5BDF\u304C\u306A\u3055\u308C\u305F\u300219\u4E16\u7D00\u306B\u306F\u5E83\u304F\u7814\u7A76\u3055\u308C\u308B\u3088\u3046\u306B\u306A\u3063\u305F\u3002 \u5C04\u5F71\u5E73\u9762\u4E0A\u30671\u70B9\u3067\u4EA4\u308F\u308B4\u7DDA\uFF08\u82F1: concurrent lines\uFF09\u3084\u3001\u30EA\u30FC\u30DE\u30F3\u7403\u9762\u4E0A\u306E4\u70B9\u306B\u3064\u3044\u3066\u306E\u6D3E\u751F\u3057\u305F\u6982\u5FF5\u3082\u5B58\u5728\u3059\u308B\u3002\u53CC\u66F2\u5E7E\u4F55\u5B66\u306E\u3067\u306F\u3001\u7279\u5B9A\u306E\u8907\u6BD4\u306B\u3088\u308A\u70B9\u9593\u306E\u8DDD\u96E2\u304C\u8868\u3055\u308C\u308B\u3002"@ja , "\uC0AC\uC601\uAE30\uD558\uD559\uC5D0\uC11C, \uBE44\uC870\uD654\uBE44(\u975E\u8ABF\u548C\u6BD4, \uC601\uC5B4: anharmonic ratio) \uB610\uB294 \uBCF5\uBE44(\u8907\u6BD4, \uC601\uC5B4: double ratio)\uB294 \uAC19\uC740 \uC9C1\uC120 \uC704\uC5D0 \uC788\uB294 \uB124 \uC810\uC758 \uC720\uC77C\uD55C \uC0AC\uC601 \uBD88\uBCC0\uB7C9\uC774\uB2E4."@ko , "\u0414\u0432\u043E\u0439\u043D\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435 (\u0438\u043B\u0438 \u0441\u043B\u043E\u0436\u043D\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435 \u0438\u043B\u0438 \u0443\u0441\u0442\u0430\u0440\u0435\u0432\u0448\u0435\u0435 \u0430\u043D\u0433\u0430\u0440\u043C\u043E\u043D\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u043E\u0442\u043D\u043E\u0448\u0435\u043D\u0438\u0435) \u0447\u0435\u0442\u0432\u0451\u0440\u043A\u0438 \u0447\u0438\u0441\u0435\u043B , , , (\u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0445 \u0438\u043B\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0445) \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u043A\u0430\u043A \u0422\u0430\u043A\u0436\u0435 \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044E\u0442\u0441\u044F \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0435\u043D\u0438\u044F \u0438 ."@ru , "Dwustosunek (stosunek anharmoniczny) czterech wsp\u00F3\u0142liniowych punkt\u00F3w \u2013 funkcja postaci: gdzie punkty A, B, C, D spe\u0142niaj\u0105 oraz jest wsp\u00F3\u0142rz\u0119dn\u0105 punktu X w uk\u0142adzie wsp\u00F3\u0142rz\u0119dnych na danej prostej. Jest to podstawowe poj\u0119cie geometrii rzutowej. Jak wida\u0107, powy\u017Csza definicja zak\u0142ada istnienie uk\u0142adu wsp\u00F3\u0142rz\u0119dnych na rozpatrywanej prostej. Je\u015Bli dwustosunek stosujemy na p\u0142aszczy\u017Anie euklidesowej to wystarczy zbudowa\u0107 dowolny kartezja\u0144ski uk\u0142ad wsp\u00F3\u0142rz\u0119dnych wykorzystuj\u0105c relacj\u0119 przystawania i relacj\u0119 prostopad\u0142o\u015Bci. Je\u015Bli stosujemy go na p\u0142aszczy\u017Anie rzutowej to trzeba zbudowa\u0107 jaki\u015B wykorzystuj\u0105c relacj\u0119 harmoniczno\u015Bci punkt\u00F3w rzutowych Wyb\u00F3r uk\u0142adu wsp\u00F3\u0142rz\u0119dnych z wielu mo\u017Cliwych nie wp\u0142ywa na warto\u015B\u0107 dwustosunku."@pl , "Em geometria projetiva, dist\u00E2ncias e \u00E2ngulos n\u00E3o s\u00E3o preservados. O conceito m\u00E9trico que \u00E9 preservado pelas \u00E9 a raz\u00E3o anarm\u00F4nica. A raz\u00E3o anarm\u00F4nica de quatro pontos colineares \u00E9 definida por: em que os segmentos de reta devem ser interpretados como segmentos orientados. Os quatro pontos est\u00E3o na raz\u00E3o harm\u00F4nica quando a raz\u00E3o anarm\u00F4nica entre eles vale -1."@pt , "Le birapport, ou rapport anharmonique selon la d\u00E9nomination de Michel Chasles est un outil puissant de la g\u00E9om\u00E9trie, en particulier la g\u00E9om\u00E9trie projective. La notion remonte \u00E0 Pappus d'Alexandrie, mais son \u00E9tude syst\u00E9matique est r\u00E9alis\u00E9e en 1827 par M\u00F6bius."@fr , "La ra\u00F3 doble, tamb\u00E9 anomenada ra\u00F3 anharm\u00F2nica, \u00E9s una poderosa eina en geometria, especialment en geometria projectiva. El nom de ra\u00F3 anharm\u00F2nica va ser creat per Michel Chasles, per\u00F2 la noci\u00F3 es remunta a Pappos d'Alexandria."@ca , "\u041F\u043E\u0434\u0432\u0456\u0301\u0439\u043D\u0435 \u0432\u0456\u0434\u043D\u043E\u0301\u0448\u0435\u043D\u043D\u044F (\u0430\u0431\u043E \u0441\u043A\u043B\u0430\u0434\u043D\u0435\u0301 \u0432\u0456\u0434\u043D\u043E\u0301\u0448\u0435\u043D\u043D\u044F \u0430\u0431\u043E \u0437\u0430\u0441\u0442\u0430\u0440\u0456\u043B\u0435 \u0430\u043D\u0433\u0430\u0440\u043C\u043E\u043D\u0456\u0447\u043D\u0435 \u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u043D\u044F) \u0447\u0435\u0442\u0432\u0456\u0440\u043A\u0438 \u0447\u0438\u0441\u0435\u043B , , , (\u0434\u0456\u0439\u0441\u043D\u0438\u0445 \u0447\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u0445) \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u044F\u043A"@uk , "In de meetkunde is de dubbelverhouding van vier collineaire punten gedefinieerd als de verhouding van twee deelverhoudingen. De dubbelverhouding is invariant onder centrale projectie."@nl , "\u6570\u5B66\u4E0A\uFF0C\u8907\u5E73\u9762\u4E0A\u56DB\u70B9\u7684\u4EA4\u6BD4\u662F \u3002 \u8FD9\u4E2A\u5B9A\u4E49\u53EF\u4EE5\u8FDE\u7EED\u5EF6\u62D3\u81F3\u6574\u4E2A\u9ECE\u66FC\u7403\u9762\uFF0C\u5373\u8907\u5E73\u9762\u52A0\u4E0A\u65E0\u7A77\u8FDC\u70B9\u3002 \u4E00\u822C\u6765\u8BF4\uFF0C\u4EA4\u6BD4\u53EF\u4EE5\u5B9A\u4E49\u5728\uFF08\u9ECE\u66FC\u7403\u9762\u5C31\u662F\u8907\u5C04\u5F71\u76F4\u7DDA\uFF09\u3002\u5728\u4EFB\u4F55\u4E2D\uFF0C\u4EA4\u6BD4\u7531\u4E0A\u5F0F\u7ED9\u51FA\u3002\u4EA4\u6BD4\u662F\u5C04\u5F71\u51E0\u4F55\u7684\u4E0D\u53D8\u91CF\uFF0C\u5C31\u662F\u8BF4\u4FDD\u6301\u4EA4\u6BD4\u4E0D\u53D8\u3002\u4ECE\u524D\u4EBA\u4EEC\u6CE8\u610F\u5230\u5982\u679C\u56DB\u6761\u76F4\u7EBF\u7A7F\u8FC7\u4E00\u70B9P\uFF0C\u7B2C\u4E94\u6761\u76F4\u7EBFL\u4E0D\u7A7F\u8FC7P\uFF0C\u5206\u522B\u4E0E\u56DB\u6761\u76F4\u7EBF\u4EA4\u4E8E\u56DB\u70B9\uFF0C\u90A3\u4E48\u5728L\u4E0A\u6309\u5E8F\u53D6\u56DB\u70B9\u7684\u6709\u5411\u957F\u5EA6\uFF0C\u6240\u7B97\u51FA\u7684\u4EA4\u6BD4\u662F\u72EC\u7ACB\u4E8EL\u3002\u5B83\u662F\u8FD9\u56DB\u76F4\u7EBF\u7CFB\u7684\u4E0D\u53D8\u91CF\u3002 \u56DB\u4E2A\u8907\u6570\u7684\u4EA4\u6BD4\u4E3A\u5B9E\u6570\u5F53\u4E14\u552F\u5F53\u56DB\u70B9\u5171\u7EBF\u6216\u3002"@zh , "Das Doppelverh\u00E4ltnis ist in der Geometrie im einfachsten Fall das Verh\u00E4ltnis zweier Teilverh\u00E4ltnisse. Wird zum Beispiel die Strecke sowohl durch einen Punkt als auch durch einen Punkt in jeweils zwei Teilstrecken und bzw. und (s. erstes Beispiel) geteilt, so ist das Verh\u00E4ltnis das (affine) Doppelverh\u00E4ltnis, in dem die Teilpunkte die gegebene Strecke teilen. Die gro\u00DFe Bedeutung erh\u00E4lt das Doppelverh\u00E4ltnis als Invariante bei Zentralprojektionen, denn das anschaulichere Teilverh\u00E4ltnis ist zwar invariant unter Parallelprojektionen, aber nicht unter Zentralprojektionen.Eine Verallgemeinerung f\u00FChrt zur Definition des Doppelverh\u00E4ltnisses f\u00FCr Punkte einer projektiven Gerade (das hei\u00DFt, einer affinen Geraden, der ein Fernpunkt hinzugef\u00FCgt wird). Ein besonderer Fall liegt vor, wenn das Doppelverh\u00E4ltnis den Wert \u22121 annimmt. In diesem Fall spricht man von einer harmonischen Teilung der Strecke durch das Punktepaar und sagt, liegen harmonisch. W\u00E4hrend man das Teilverh\u00E4ltnis dreier Punkte noch gut an der Lage der Punkte absch\u00E4tzen kann, ist dies f\u00FCr das Doppelverh\u00E4ltnis fast unm\u00F6glich. Das Doppelverh\u00E4ltnis hat in der analytischen und projektiven Geometrie haupts\u00E4chlich theoretische Bedeutung (Invariante bei projektiven Kollineationen). In der Darstellenden Geometrie allerdings wird es (ohne Rechnung) zur Rekonstruktion ebener Figuren verwendet."@de , "Il birapporto \u00E8 una grandezza associata a una quaterna di punti di una retta. Si tratta di uno strumento importante in geometria proiettiva: risulta infatti definito anche se uno dei quattro punti \u00E8 all'infinito (la retta in questione \u00E8 quindi una retta proiettiva) ed \u00E8 invariante tramite trasformazioni proiettive. La retta su cui giacciono i punti pu\u00F2 essere definita su un campo diverso dai numeri reali. Ad esempio, se definita sui numeri complessi, la retta \u00E8 in realt\u00E0 la sfera di Riemann, ovvero il piano complesso a cui va aggiunto un punto all'infinito. Il birapporto ha nella geometria proiettiva un ruolo vagamente simile a quello della distanza tra punti in geometria euclidea. Il birapporto viene chiamato anche rapporto anarmonico, termine coniato da Michel Chasles per una nozione nota prima delle sue ricerche geometriche."@it , "In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.)The point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is \u22121, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.In the Cayley\u2013Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio."@en . @prefix prov: . dbr:Cross-ratio prov:wasDerivedFrom . @prefix xsd: . dbr:Cross-ratio dbo:wikiPageLength "26066"^^xsd:nonNegativeInteger . @prefix wikipedia-en: . dbr:Cross-ratio foaf:isPrimaryTopicOf wikipedia-en:Cross-ratio .