. "Une surface d'Enneper est une surface minimale, param\u00E9tris\u00E9e en 1863 par le math\u00E9maticien allemand Alfred Enneper. On peut la d\u00E9crire par un param\u00E9trage cart\u00E9sien : Cette surface repr\u00E9sente un film de savon \u00AB fant\u00F4me \u00BB, c\u2019est-\u00E0-dire un \u00E9quilibre instable de l\u2019\u00E9nergie potentielle. On peut imaginer une surface d'Enneper comme s'appuyant sur un contour comme celui trac\u00E9 sur une balle de tennis. Sur ce contour, deux films de savon r\u00E9els peuvent s'accrocher : un pour chaque moiti\u00E9 de la surface de la balle de tennis. La surface d'Enneper repr\u00E9sente alors l'\u00E9quilibre instable de la surface minimale entre ces deux surfaces stables."@fr . . "1074400810"^^ . . "In de differentiaalmeetkunde en de algebra\u00EFsche meetkunde, deelgebieden van de wiskunde, is het Enneper-oppervlak een oppervlak dat zichzelf snijdt en parametrisch als volgt kan worden beschreven Het Enneper-oppervlak werd in 1864 ge\u00EFntroduceerd door Alfred Enneper in verband met zijn theorie over minimaaloppervlakken. Met de impliciteringsmethoden van de algebra\u00EFsche meetkunde wordt gevonden dat de punten in het hierboven gegeven Enneper-oppervlak voldoen aan de volgende vergelijking van de graad 9: Duaal is het raakvlak op het punt met gegeven parameters gelijk aan , waarin"@nl . "\u041F\u043E\u0432\u0435\u0440\u0445\u043D\u044F \u0415\u043D\u043D\u0435\u043F\u0435\u0440\u0430"@uk . . "p/e035710"@en . . . . . "\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762"@ja . "Une surface d'Enneper est une surface minimale, param\u00E9tris\u00E9e en 1863 par le math\u00E9maticien allemand Alfred Enneper. On peut la d\u00E9crire par un param\u00E9trage cart\u00E9sien : Cette surface repr\u00E9sente un film de savon \u00AB fant\u00F4me \u00BB, c\u2019est-\u00E0-dire un \u00E9quilibre instable de l\u2019\u00E9nergie potentielle. On peut imaginer une surface d'Enneper comme s'appuyant sur un contour comme celui trac\u00E9 sur une balle de tennis. Sur ce contour, deux films de savon r\u00E9els peuvent s'accrocher : un pour chaque moiti\u00E9 de la surface de la balle de tennis. La surface d'Enneper repr\u00E9sente alors l'\u00E9quilibre instable de la surface minimale entre ces deux surfaces stables."@fr . . . "Surface d'Enneper"@fr . "In matematica, nel campo della geometria differenziale e in geometria algebrica, la superficie di Enneper \u00E8 una superficie che pu\u00F2 essere descritta in forma parametrica da: \u00C8 stata introdotta da Alfred Enneper in connessione con la Teoria delle superfici minime. I metodi di implicitizzazione della geometria algebrica possono essere utilizzati per dimostrare che i punti appartenenti alla superficie di Enneper soddisfano la seguente equazione polinomiale di nono grado Dualmente, il piano tangente nel punto con parametri dati \u00E8 dove: I suoi coefficienti soddisfano l'equazione polinomiale implicita di 6\u00BA grado:Lo jacobiano, la Curvatura gaussiana e la Curvatura media sono date da: La curvatura totale \u00E8 . Osserman dimostr\u00F2 che una superficie minima completa in con curvatura totale \u00E8 una catenoide oppure una superficie di Enneper. Un'altra propriet\u00E0 \u00E8 che tutte le superfici di B\u00E9zier bicubiche minimali sono, a meno di trasformazioni affini, pezzi della superficie di Enneper. Usando la parametrizzazione di Weierstrass-Enneper , per un intero , si pu\u00F2 generalizzare la superficie di Enneper ad ordini maggiori di simmetrie rotazionali. Inoltre, si pu\u00F2 generalizzare la superficie in dimensioni maggiori. Si \u00E8 dimostrata l'esistenza di superfici di Enneper in per ."@it . . . "4688830"^^ . . . . . . "In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory. The Weierstrass\u2013Enneper parameterization is very simple, , and the real parametric form can easily be calculated from it. The surface is conjugate to itself. Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation The Jacobian, Gaussian curvature and mean curvature are"@en . "En matem\u00E0tiques, en els camps de la geometria diferencial i geometria algebraica, la superf\u00EDcie d'Enneper \u00E9s una superf\u00EDcie que s'autointersecciona i que pot ser descrita param\u00E8tricament per: Va ser introdu\u00EFda el 1864 per Alfred Enneper en connexi\u00F3 amb la teoria de la . La \u00E9s molt simple, , i la forma param\u00E8trica real es pot calcular a partir d'aquesta. La superf\u00EDcie est\u00E0 conjugada amb si mateixa. Es poden usar m\u00E8todes d'implicitaci\u00F3 de geometria algebraica per a trobar els punts de la superf\u00EDcie d'Enneper que satisfacen l'equaci\u00F3 polin\u00F2mica de grau 9: on:"@ca . . . . "4334"^^ . . . . . . "\u6069\u5185\u4F69\u5C14\u66F2\u9762"@zh . . . "Enneper-oppervlak"@nl . . . "Superficie de Enneper"@es . "Superf\u00EDcie d'Enneper"@ca . . "Enneper surface"@en . . . . . . . . . . "\u041F\u043E\u0432\u0435\u0440\u0445\u043D\u043E\u0441\u0442\u044C \u042D\u043D\u043D\u0435\u043F\u0435\u0440\u0430 \u2014 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0451\u043D\u043D\u044B\u0439 \u0442\u0438\u043F \u0441\u0430\u043C\u043E\u043F\u0435\u0440\u0435\u0441\u0435\u043A\u0430\u044E\u0449\u0435\u0439\u0441\u044F \u043C\u0438\u043D\u0438\u043C\u0430\u043B\u044C\u043D\u043E\u0439 \u043F\u043E\u0432\u0435\u0440\u0445\u043D\u043E\u0441\u0442\u0438. \u0420\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u043B\u0430\u0441\u044C \u0410\u043B\u044C\u0444\u0440\u0435\u0434\u043E\u043C \u042D\u043D\u043D\u0435\u043F\u0435\u0440\u043E\u043C \u0432 1864 \u0433\u043E\u0434\u0443."@ru . "\u041F\u043E\u0432\u0435\u0440\u0445\u043D\u043E\u0441\u0442\u044C \u042D\u043D\u043D\u0435\u043F\u0435\u0440\u0430"@ru . . . . "En matem\u00E1ticas, en los campos de la geometr\u00EDa diferencial y geometr\u00EDa algebraica, la superficie de Enneper es una superficie que se auto-intersecciona y que puede ser descrita param\u00E9tricamente por: Fue introducida en 1864 por en conexi\u00F3n con la teor\u00EDa de la superficie minimal.\u200B\u200B\u200B\u200B La es muy simple, , y la forma param\u00E9trica real se puede calcular de ella. La superficie est\u00E1 consigo misma. Se pueden usar m\u00E9todos de implicitaci\u00F3n de geometr\u00EDa algebraica para encontrar los puntos de la superficie de Enneper dados arriba que satisfagan la ecuaci\u00F3n polin\u00F3mica de grado 9:"@es . "\u6069\u5185\u4F69\u5C14\u66F2\u9762\uFF08\u82F1\u8A9E\uFF1AEnneper surface\uFF09\u662F\u4E00\u79CD\u6781\u5C0F\u66F2\u9762\uFF0C\u7531\u5FB7\u56FD\u6570\u5B66\u5BB6\u4E8E1864\u5E74\u63D0\u51FA\u3002\u6069\u5185\u4F69\u5C14\u66F2\u9762\u7684\u53C2\u6570\u65B9\u7A0B\u4E3A \u5728\u9B4F\u5C14\u65AF\u7279\u62C9\u65AF\uFF0D\u6069\u5185\u4F69\u5C14\uFF08Weierstrass\u2013Enneper\uFF09\u8868\u793A\u4E2D\uFF0C\u4EE4\uFF0C\u4FBF\u80FD\u5F97\u5230\u6069\u5185\u4F69\u5C14\u66F2\u9762\u3002"@zh . "Superficie di Enneper"@it . . . "\u6570\u5B66\u306E\u5206\u79D1\u3001\u5FAE\u5206\u5E7E\u4F55\u5B66\u3068\u4EE3\u6570\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\uFF08\u30A8\u30F3\u30CD\u30D1\u30FC\u304D\u3087\u304F\u3081\u3093\u3001\u82F1: Enneper surface\uFF09\u3068\u306F\u3001\u6B21\u306E\u5A92\u4ECB\u5909\u6570\u8868\u793A\u3067\u66F8\u3051\u308B\u3001\u81EA\u5DF1\u4EA4\u5DEE\u6027\u3092\u6301\u3064\u66F2\u9762\u3067\u3042\u308B\u3002 \u3053\u306E\u66F2\u9762\u306F1864\u5E74\u3001\u306B\u3088\u3063\u3066\u7406\u8AD6\u3068\u306E\u95A2\u308F\u308A\u304B\u3089\u5C0E\u5165\u3055\u308C\u305F\u3002 \u306F\u975E\u5E38\u306B\u7C21\u5358\u3067\u3001 \u3068\u306A\u308B\u3002\u5B9F\u5909\u6570\u3067\u306E\u5A92\u4ECB\u5909\u6570\u8868\u793A\u306F\u3053\u306E\u5F0F\u304B\u3089\u5BB9\u6613\u306B\u8A08\u7B97\u3067\u304D\u308B\u3002\u3053\u306E\u66F2\u9762\u306F\u5171\u5F79\u6975\u5C0F\u66F2\u9762\uFF08conjugate minimal surface\uFF09\u304C\u81EA\u5206\u81EA\u8EAB\u3068\u4E00\u81F4\u3059\u308B\uFF08\u3092\u53C2\u7167\uFF09\u3002 \u4EE3\u6570\u5E7E\u4F55\u306E\u9670\u95A2\u6570\u8868\u793A\u3067\u306F\u3001\u4E0A\u5F0F\u3067\u4E0E\u3048\u305F\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306E\u5404\u70B9\u306F\u6B21\u306E9\u6B21\u591A\u9805\u5F0F\u3092\u6E80\u305F\u3059\u3002 \u53CC\u5BFE\u7684\u306B\u3001\u5A92\u4ECB\u5909\u6570\u3067\u4E0E\u3048\u3089\u308C\u305F\u3042\u308B\u70B9\u3067\u306E\u63A5\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306F \u3001\u3053\u3053\u3067 \u3068\u66F8\u3051\u308B\u3002\u3053\u306E\u4FC2\u6570\u306F\u6B21\u306E6\u6B21\u591A\u9805\u5F0F\u3092\u6E80\u305F\u3059\u3002 \u30E4\u30B3\u30D3\u884C\u5217\u5F0F\u3001\u30AC\u30A6\u30B9\u66F2\u7387\u3001\u306F\u305D\u308C\u305E\u308C \u3068\u306A\u308B\u3002\u306F \u3067\u3042\u308B\u3002\u306F\u3001\u5168\u66F2\u7387\u304C \u3067\u3042\u308B\u3088\u3046\u306A \u306B\u304A\u3051\u308B\u5B8C\u5099\u306A\u6975\u5C0F\u66F2\u9762\u306F\u304B\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306E\u3044\u305A\u308C\u304B\u3067\u3042\u308B\u3053\u3068\u3092\u8A3C\u660E\u3057\u305F\u3002 \u4ED6\u306E\u6027\u8CEA\u3068\u3057\u3066\u3001\u5168\u3066\u306E\u53CC3\u6B21\u306A\uFF08\u30D0\u30A4\u30AD\u30E5\u30FC\u30D3\u30C3\u30AF\u306A, bicubical\uFF09\u6975\u5C0F\u306F\u3001\u30A2\u30D5\u30A3\u30F3\u5909\u63DB\u306B\u3088\u308B\u5DEE\u3092\u9664\u3051\u3070\u3001\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306E\u4E00\u90E8\u306B\u306A\u308B\u3002"@ja . . "In matematica, nel campo della geometria differenziale e in geometria algebrica, la superficie di Enneper \u00E8 una superficie che pu\u00F2 essere descritta in forma parametrica da: \u00C8 stata introdotta da Alfred Enneper in connessione con la Teoria delle superfici minime. I metodi di implicitizzazione della geometria algebrica possono essere utilizzati per dimostrare che i punti appartenenti alla superficie di Enneper soddisfano la seguente equazione polinomiale di nono grado Dualmente, il piano tangente nel punto con parametri dati \u00E8 dove:"@it . . . "Enneper surface"@en . . "En matem\u00E1ticas, en los campos de la geometr\u00EDa diferencial y geometr\u00EDa algebraica, la superficie de Enneper es una superficie que se auto-intersecciona y que puede ser descrita param\u00E9tricamente por: Fue introducida en 1864 por en conexi\u00F3n con la teor\u00EDa de la superficie minimal.\u200B\u200B\u200B\u200B La es muy simple, , y la forma param\u00E9trica real se puede calcular de ella. La superficie est\u00E1 consigo misma. Se pueden usar m\u00E9todos de implicitaci\u00F3n de geometr\u00EDa algebraica para encontrar los puntos de la superficie de Enneper dados arriba que satisfagan la ecuaci\u00F3n polin\u00F3mica de grado 9: Dualmente, el plano tangente en el punto con los par\u00E1metros dados es donde: Sus coeficientes satisfacen la ecuaci\u00F3n polin\u00F3mica de grado seis impl\u00EDcita: El jacobiano, la curvatura de Gauss y la curvatura media son: La curvatura total es . Osserman prob\u00F3 que una superficie minimal completa en con una curvatura total de es o bien el catenoide o la superficie de Enneper.\u200B Otra propiedad es que todas las minimales bic\u00FAbicas, hasta una transformaci\u00F3n af\u00EDn, son trozos de esta superficie.\u200B Se puede generalizar a \u00F3rdenes de simetr\u00EDa rotacional mayores usando la parametrizaci\u00F3n de Weierstra\u00DF\u2013Enneper para enteros k>1.\u200B Puede ser generalizada para mayores dimensiones; en (hasta n 7) se conocen superficies similares a la superficie de Enneper.\u200B"@es . "\u6570\u5B66\u306E\u5206\u79D1\u3001\u5FAE\u5206\u5E7E\u4F55\u5B66\u3068\u4EE3\u6570\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\uFF08\u30A8\u30F3\u30CD\u30D1\u30FC\u304D\u3087\u304F\u3081\u3093\u3001\u82F1: Enneper surface\uFF09\u3068\u306F\u3001\u6B21\u306E\u5A92\u4ECB\u5909\u6570\u8868\u793A\u3067\u66F8\u3051\u308B\u3001\u81EA\u5DF1\u4EA4\u5DEE\u6027\u3092\u6301\u3064\u66F2\u9762\u3067\u3042\u308B\u3002 \u3053\u306E\u66F2\u9762\u306F1864\u5E74\u3001\u306B\u3088\u3063\u3066\u7406\u8AD6\u3068\u306E\u95A2\u308F\u308A\u304B\u3089\u5C0E\u5165\u3055\u308C\u305F\u3002 \u306F\u975E\u5E38\u306B\u7C21\u5358\u3067\u3001 \u3068\u306A\u308B\u3002\u5B9F\u5909\u6570\u3067\u306E\u5A92\u4ECB\u5909\u6570\u8868\u793A\u306F\u3053\u306E\u5F0F\u304B\u3089\u5BB9\u6613\u306B\u8A08\u7B97\u3067\u304D\u308B\u3002\u3053\u306E\u66F2\u9762\u306F\u5171\u5F79\u6975\u5C0F\u66F2\u9762\uFF08conjugate minimal surface\uFF09\u304C\u81EA\u5206\u81EA\u8EAB\u3068\u4E00\u81F4\u3059\u308B\uFF08\u3092\u53C2\u7167\uFF09\u3002 \u4EE3\u6570\u5E7E\u4F55\u306E\u9670\u95A2\u6570\u8868\u793A\u3067\u306F\u3001\u4E0A\u5F0F\u3067\u4E0E\u3048\u305F\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306E\u5404\u70B9\u306F\u6B21\u306E9\u6B21\u591A\u9805\u5F0F\u3092\u6E80\u305F\u3059\u3002 \u53CC\u5BFE\u7684\u306B\u3001\u5A92\u4ECB\u5909\u6570\u3067\u4E0E\u3048\u3089\u308C\u305F\u3042\u308B\u70B9\u3067\u306E\u63A5\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306F \u3001\u3053\u3053\u3067 \u3068\u66F8\u3051\u308B\u3002\u3053\u306E\u4FC2\u6570\u306F\u6B21\u306E6\u6B21\u591A\u9805\u5F0F\u3092\u6E80\u305F\u3059\u3002 \u30E4\u30B3\u30D3\u884C\u5217\u5F0F\u3001\u30AC\u30A6\u30B9\u66F2\u7387\u3001\u306F\u305D\u308C\u305E\u308C \u3068\u306A\u308B\u3002\u306F \u3067\u3042\u308B\u3002\u306F\u3001\u5168\u66F2\u7387\u304C \u3067\u3042\u308B\u3088\u3046\u306A \u306B\u304A\u3051\u308B\u5B8C\u5099\u306A\u6975\u5C0F\u66F2\u9762\u306F\u304B\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306E\u3044\u305A\u308C\u304B\u3067\u3042\u308B\u3053\u3068\u3092\u8A3C\u660E\u3057\u305F\u3002 \u4ED6\u306E\u6027\u8CEA\u3068\u3057\u3066\u3001\u5168\u3066\u306E\u53CC3\u6B21\u306A\uFF08\u30D0\u30A4\u30AD\u30E5\u30FC\u30D3\u30C3\u30AF\u306A, bicubical\uFF09\u6975\u5C0F\u306F\u3001\u30A2\u30D5\u30A3\u30F3\u5909\u63DB\u306B\u3088\u308B\u5DEE\u3092\u9664\u3051\u3070\u3001\u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306E\u4E00\u90E8\u306B\u306A\u308B\u3002 \u30A8\u30F3\u30CD\u30D1\u30FC\u66F2\u9762\u306F\u3001\u30EF\u30A4\u30A8\u30EB\u30B7\u30E5\u30C8\u30E9\u30B9\u2013\u30A8\u30F3\u30CD\u30D1\u30FC\u306E\u5A92\u4ECB\u5909\u6570\u8868\u793A\u3067 \uFF08k>1 \u306F\u6574\u6570\uFF09\u3068\u3059\u308B\u3053\u3068\u3067\u3088\u308A\u9AD8\u6B21\u306E\u5BFE\u79F0\u5F0F\u306B\u3088\u308B\u66F2\u9762\u3078\u3068\u4E00\u822C\u5316\u3059\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u4E00\u65B9\u3001\u3088\u308A\u9AD8\u6B21\u306E\u7A7A\u9593\u3078\u3068\u4E00\u822C\u5316\u3059\u308B\u3053\u3068\u3082\u3067\u304D\u308B\u30027\u307E\u3067\u306En\u306B\u3064\u3044\u3066\u7A7A\u9593 \u306B\u304A\u3051\u308B\u30A8\u30F3\u30CD\u30D1\u30FC\u69D8\uFF08Enneper-like\uFF09\u8D85\u66F2\u9762\u306E\u5B58\u5728\u304C\u77E5\u3089\u308C\u3066\u3044\u308B\u3002"@ja . "In de differentiaalmeetkunde en de algebra\u00EFsche meetkunde, deelgebieden van de wiskunde, is het Enneper-oppervlak een oppervlak dat zichzelf snijdt en parametrisch als volgt kan worden beschreven Het Enneper-oppervlak werd in 1864 ge\u00EFntroduceerd door Alfred Enneper in verband met zijn theorie over minimaaloppervlakken. Met de impliciteringsmethoden van de algebra\u00EFsche meetkunde wordt gevonden dat de punten in het hierboven gegeven Enneper-oppervlak voldoen aan de volgende vergelijking van de graad 9: Duaal is het raakvlak op het punt met gegeven parameters gelijk aan , waarin Haar co\u00EBffici\u00EBnten voldoen aan de impliciete vergelijking Het Enneper-oppervlak is een minimaaloppervlak. De Jacobiaan, de Gaussiaanse kromming en de gemiddelde kromming zijn:"@nl . . . . . "En matem\u00E0tiques, en els camps de la geometria diferencial i geometria algebraica, la superf\u00EDcie d'Enneper \u00E9s una superf\u00EDcie que s'autointersecciona i que pot ser descrita param\u00E8tricament per: Va ser introdu\u00EFda el 1864 per Alfred Enneper en connexi\u00F3 amb la teoria de la . La \u00E9s molt simple, , i la forma param\u00E8trica real es pot calcular a partir d'aquesta. La superf\u00EDcie est\u00E0 conjugada amb si mateixa. Es poden usar m\u00E8todes d'implicitaci\u00F3 de geometria algebraica per a trobar els punts de la superf\u00EDcie d'Enneper que satisfacen l'equaci\u00F3 polin\u00F2mica de grau 9: Dualment, el pla tangent en el punt amb els par\u00E0metres donats \u00E9s on: Els seus coeficients satisfan l'equaci\u00F3 polin\u00F2mica de grau sis impl\u00EDcita: El jacobi\u00E0, la curvatura de Gauss i la s\u00F3n: La curvatura total \u00E9s . Osserman va demostrar que una superf\u00EDcie minimal completa en amb una curvatura total de \u00E9s o b\u00E9 el o la superf\u00EDcie d'Enneper. Una altra propietat n'\u00E9s que totes les minimals bic\u00FAbiques, fins a una transformaci\u00F3 af\u00ED, s\u00F3n trossos d'aquesta superf\u00EDcie. Es pot generalitzar a ordres de majors usant la parametritzaci\u00F3 de Weierstra\u00DF\u2013Enneper per a sencers k>1. Pot ser generalitzada per a majors dimensions; es coneixen superf\u00EDcies semblants a la superf\u00EDcie d'Enneper en fins a n igual a 7."@ca . . . . "Enneperfl\u00E4che"@de . "\u6069\u5185\u4F69\u5C14\u66F2\u9762\uFF08\u82F1\u8A9E\uFF1AEnneper surface\uFF09\u662F\u4E00\u79CD\u6781\u5C0F\u66F2\u9762\uFF0C\u7531\u5FB7\u56FD\u6570\u5B66\u5BB6\u4E8E1864\u5E74\u63D0\u51FA\u3002\u6069\u5185\u4F69\u5C14\u66F2\u9762\u7684\u53C2\u6570\u65B9\u7A0B\u4E3A \u5728\u9B4F\u5C14\u65AF\u7279\u62C9\u65AF\uFF0D\u6069\u5185\u4F69\u5C14\uFF08Weierstrass\u2013Enneper\uFF09\u8868\u793A\u4E2D\uFF0C\u4EE4\uFF0C\u4FBF\u80FD\u5F97\u5230\u6069\u5185\u4F69\u5C14\u66F2\u9762\u3002"@zh . "\u041F\u043E\u0432\u0435\u0440\u0445\u043D\u043E\u0441\u0442\u044C \u042D\u043D\u043D\u0435\u043F\u0435\u0440\u0430 \u2014 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0451\u043D\u043D\u044B\u0439 \u0442\u0438\u043F \u0441\u0430\u043C\u043E\u043F\u0435\u0440\u0435\u0441\u0435\u043A\u0430\u044E\u0449\u0435\u0439\u0441\u044F \u043C\u0438\u043D\u0438\u043C\u0430\u043B\u044C\u043D\u043E\u0439 \u043F\u043E\u0432\u0435\u0440\u0445\u043D\u043E\u0441\u0442\u0438. \u0420\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u043B\u0430\u0441\u044C \u0410\u043B\u044C\u0444\u0440\u0435\u0434\u043E\u043C \u042D\u043D\u043D\u0435\u043F\u0435\u0440\u043E\u043C \u0432 1864 \u0433\u043E\u0434\u0443."@ru . "In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory. The Weierstrass\u2013Enneper parameterization is very simple, , and the real parametric form can easily be calculated from it. The surface is conjugate to itself. Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation Dually, the tangent plane at the point with given parameters is where Its coefficients satisfy the implicit degree-6 polynomial equation The Jacobian, Gaussian curvature and mean curvature are The total curvature is . Osserman proved that a complete minimal surface in with total curvature is either the catenoid or the Enneper surface. Another property is that all bicubical minimal B\u00E9zier surfaces are, up to an affine transformation, pieces of the surface. It can be generalized to higher order rotational symmetries by using the Weierstrass\u2013Enneper parameterization for integer k>1. It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in for n up to 7."@en .