"6851758"^^ . "\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u5C16\u70B9\uFF08\u305B\u3093\u3066\u3093\u3001\u82F1: cusp, \u53E4\u304F\u306F\u5C16\u7BC0\u70B9 (spinode)\uFF09\u306F\u3001\u66F2\u7DDA\u306B\u6CBF\u3063\u3066\u8D70\u308B\u52D5\u70B9\u304C\u305D\u3053\u3067\u5411\u304D\u3092\u9006\u8EE2\u3059\u308B\u3088\u3046\u306A\u66F2\u7DDA\u4E0A\u306E\u70B9\u3067\u3042\u308B\u3002\u5C16\u70B9\u306F\u66F2\u7DDA\u306E\u7279\u7570\u70B9\u306E\u4E00\u7A2E\u3068\u3044\u3046\u3053\u3068\u306B\u306A\u308B\u3002 \u89E3\u6790\u7684\u306B\u5A92\u4ECB\u4ED8\u3051\u3089\u308C\u305F\u5E73\u9762\u66F2\u7DDA \u306B\u304A\u3044\u3066\u5C16\u70B9\u306F\u3001f \u304A\u3088\u3073 g \u306E\u5FAE\u5206\u4FC2\u6570\u304C\u3068\u3082\u306B\u6D88\u3048\u3066\u3044\u308B\u3088\u3046\u306A\u70B9\uFF08\u3064\u307E\u308A\u66F2\u7DDA\u306E\u7279\u7570\u70B9\uFF09\u3067\u3042\u3063\u3066\u3001\u305D\u306E\u70B9\u3067\u306E\u63A5\u7DDA\u65B9\u5411\u3078\u306E\u65B9\u5411\u5FAE\u5206\u304C\u7B26\u53F7\u3092\u5909\u3048\u308B\u3082\u306E\u3067\u3042\u308B\uFF08\u3053\u3053\u3067\u300C\u63A5\u7DDA\u65B9\u5411\u300D\u3068\u306F\u3001\u305D\u306E\u8FD1\u508D\u306E\u5404\u70B9\u306B\u304A\u3051\u308B\u50BE\u304D\u306E\u6975\u9650 limg\u2032(t)\u2044f\u2032(t) \u3092\u50BE\u304D\u3068\u3059\u308B\u76F4\u7DDA\u306E\u65B9\u5411\u306E\u610F\uFF09\u3002\u5A92\u4ECB\u5909\u6570 t \u306E\u305F\u3060\u4E00\u3064\u306E\u5024\u306E\u307F\u3067\u6C7A\u307E\u308B\u3068\u3044\u3046\u610F\u5473\u3067\u5C16\u70B9\u306F\u300C\u5C40\u6240\u7684\u306A\u7279\u7570\u70B9\u300D\u3067\u3042\u308B\u3002\u5834\u5408\u306B\u3088\u3063\u3066\u306F\u5C16\u70B9\u306E\u5B9A\u7FA9\u306B\u65B9\u5411\u5FAE\u5206\u306B\u95A2\u3059\u308B\u6761\u4EF6\u3092\u554F\u308F\u306A\u3044\u3053\u3068\u3082\u3042\u308B\u304C\u3001\u305D\u306E\u5834\u5408\u306F\u4E00\u898B\u3059\u308B\u3068\u6B63\u5247\u70B9\u306E\u3088\u3046\u306B\u3082\u898B\u3048\u308B\u7279\u7570\u70B9\u3082\u73FE\u308C\u5F97\u308B\u3053\u3068\u306B\u6CE8\u610F\u3059\u3079\u304D\u3067\u3042\u308B\u3002 \u306A\u3081\u3089\u304B\u306A\u9670\u4F0F\u65B9\u7A0B\u5F0F \u3067\u5B9A\u3081\u3089\u308C\u308B\u66F2\u7DDA\u306B\u304A\u3044\u3066\u5C16\u70B9\u306F\u3001F \u306E\u30C6\u30A4\u30E9\u30FC\u5C55\u958B\u306E\u6700\u4F4E\u6B21\u306E\u9805\u304C\u9069\u5F53\u306A\u4E00\u6B21\u591A\u9805\u5F0F\u306E\u51AA\u3068\u306A\u308B\u70B9\u3068\u306A\u3063\u3066\u3044\u308B\uFF08\u304C\u3001\u3053\u306E\u6027\u8CEA\u3092\u6301\u3064\u70B9\u304C\u5FC5\u305A\u3057\u3082\u5C16\u70B9\u3068\u306A\u308B\u308F\u3051\u3067\u306F\u306A\u3044\u3053\u3068\u306B\u306F\u6CE8\u610F\u3057\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\uFF09\u3002\u8AD6\u304B\u3089\u308F\u304B\u308B\u3053\u3068\u3068\u3057\u3066\u3001F \u304C\u89E3\u6790\u51FD\u6570\uFF08\u305F\u3068\u3048\u3070\u591A\u9805\u5F0F\u51FD\u6570\u306F\u305D\u3046\u3067\u3042\u308B\uFF09\u306A\u3089\u3070\u3001\u5C16\u70B9\u306E\u8FD1\u508D\u306B\u304A\u3044\u3066\u9069\u5F53\u306A\u7DDA\u578B\u5EA7\u6A19\u5909\u63DB\u306B\u3088\u308A\u66F2\u7DDA\u3092 \u3068\u5A92\u4ECB\u8868\u793A\u3067\u304D\u308B\u3053\u3068\u304C\u8A00\u3048\u308B\u3002\u305F\u3060\u3057\u3001a \u306F\u9069\u5F53\u306A\u5B9F\u6570\u3001m \u306F\u6B63\u306E\u5076\u6570\u3067\u3001S(t) \u306F\uFF08\u6700\u3082\u6B21\u6570\u306E\u4F4E\u3044\u975E\u96F6\u9805\u306E\u6B21\u6570\uFF09k \u304C m \u3088\u308A\u5927\u304D\u3044\u51AA\u7D1A\u6570\u3068\u3059\u308B\u3002\u3053\u306E\u3068\u304D\u3001m \u3092\u3053\u306E\u5C16\u70B9\u306E\u4F4D\u6570 (order) \u307E\u305F\u306F\u91CD\u8907\u5EA6 (multiplicity) \u3068\u547C\u3073\u3001\u3053\u308C\u306F F \u306E\u6700\u4F4E\u6B21\u975E\u96F6\u6210\u5206\u306E\u6B21\u6570\u306B\u7B49\u3057\u304F\u306A\u308B\u3002 \u3053\u308C\u3089\u306E\u5B9A\u7FA9\u3092\u3001\u30EB\u30CD\u30FB\u30C8\u30E0\u304A\u3088\u3073\u30A6\u30E9\u30B8\u30FC\u30DF\u30EB\u30FB\u30A2\u30FC\u30CE\u30EB\u30C9\u306F\u3001\u53EF\u5FAE\u5206\u51FD\u6570\u306E\u5B9A\u3081\u308B\u66F2\u7DDA\u306B\u5BFE\u3059\u308B\u3082\u306E\u3078\u4E00\u822C\u5316\u3057\u305F\u3002\u3059\u306A\u308F\u3061\u3001\u66F2\u7DDA\u304C\u3042\u308B\u70B9\u306B\u5C16\u70B9\u3092\u6301\u3064\u3068\u306F\u3001\u5168\u4F53\u7A7A\u9593\u3067\u8003\u3048\u305F\u305D\u306E\u70B9\u306E\u8FD1\u508D\u4E0A\u3067\u5FAE\u5206\u540C\u76F8\u5199\u50CF\u304C\u5B58\u5728\u3057\u3066\u3001\u305D\u306E\u66F2\u7DDA\u3092\u4E0A\u3067\u5B9A\u7FA9\u3055\u308C\u305F\u610F\u5473\u3067\u306E\u5C16\u70B9\u306E\u4E0A\u3078\u5199\u3059\u3053\u3068\u304C\u3067\u304D\u308B\u3068\u304D\u306B\u8A00\u3046\u3002 \u6587\u8108\u306B\u3088\u3063\u3066\u306F\u3001\u5358\u306B\u300C\u5C16\u70B9\u300D\u3068\u8A00\u3048\u3070\u3053\u3053\u3067\u3044\u3046\u4F4D\u6570 m = 2 \u306E\u5C16\u70B9\u306E\u307F\u3092\u7279\u306B\u6307\u3059\u3082\u306E\u3068\u3057\u3066\u5B9A\u3081\u3066\u3044\u308B\u3053\u3068\u3082\u3042\u308B\u3002\u672C\u9805\u3082\u4EE5\u4E0B\u305D\u306E\u3088\u3046\u306A\u5236\u9650\u3055\u308C\u305F\u610F\u5473\u3067\u3053\u308C\u3092\u7528\u3044\u308B\u3053\u3068\u3068\u3059\u308B\u3002\u4F4D\u6570 2 \u306E\u5C16\u70B9\u3092\u6301\u3064\u5E73\u9762\u66F2\u7DDA\u306F\u3001\u9069\u5F53\u306A\u5FAE\u5206\u540C\u76F8\u306B\u3088\u308A\u3001\u9069\u5F53\u306A\u81EA\u7136\u6570 k \u306B\u5BFE\u3059\u308B\u66F2\u7DDA x2 \u2013 y2k+1 = 0 \u306E\u5F62\u306B\u304A\u304F\u3053\u3068\u304C\u3067\u304D\u308B\u3002"@ja . . . . . . "C\u00FAspide (singularidad)"@es . "\u0412 \u043A\u0430\u0441\u043F (\u0430\u043D\u0433\u043B. cusp \u2014 \u0437\u0430\u0433\u043E\u0441\u0442\u0440\u0435\u043D\u043D\u044F) \u0454 \u043E\u0434\u043D\u0438\u043C \u0437 \u0432\u0438\u0434\u0456\u0432 \u043E\u0441\u043E\u0431\u043B\u0438\u0432\u0438\u0445 \u0442\u043E\u0447\u043E\u043A \u043A\u0440\u0438\u0432\u043E\u0457."@uk . "\u041A\u0430\u0441\u043F (\u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430)"@uk . . "En matem\u00E0tiques, en la teoria de la singularitat, una c\u00FAspide \u00E9s un tipus de punt singular d'una corba, on un punt en moviment de la corba ha de comen\u00E7ar a retrocedir. Les c\u00FAspides s\u00F3n singularitats locals que no estan formades per l'autointersecci\u00F3 dels punts de la corba. Per a una corba plana definida per una equaci\u00F3 param\u00E8trica impl\u00EDcita una c\u00FAspide \u00E9s un punt on la derivada de f i g \u00E9s zero, i la derivada direccional, en la direcci\u00F3 de la tangent, canvia signe (la direcci\u00F3 de la tangent \u00E9s la direcci\u00F3 del pendent )."@ca . . . . . . "Point de rebroussement"@fr . . . "\u5C16\u9EDE\uFF08\u82F1\u8A9E\uFF1ACusp\uFF09\u662F\u66F2\u7DDA\u4E2D\u7684\u4E00\u7A2E\u5947\u9EDE\u3002\u66F2\u7DDA\u4E0A\u7684\u52D5\u9EDE\u5728\u79FB\u5230\u5C16\u9EDE\u6642\u6703\u958B\u59CB\u53CD\u5411\u79FB\u52D5\uFF0C\u53F3\u5716\u662F\u4E00\u500B\u5178\u578B\u7684\u4F8B\u5B50\u3002\u7D66\u5B9A\u4E00\u500B\u4EE5\u89E3\u6790\u53C3\u6578\u5F0F\u5B9A\u7FA9\u7684\u5E73\u9762\u66F2\u7DDA\uFF1A \u5C16\u9EDE\u5373\u70BA\u51FD\u6578f\u53CAg\u4E4B\u5C0E\u6578\u70BA\u96F6\u4E4B\u9EDE\uFF0C\u540C\u6642\u65B9\u5411\u5C0E\u6578\u5728\u5207\u7DDA\u65B9\u5411\u6703\u8B8A\u865F\uFF08\u5207\u7DDA\u65B9\u5411\u4E4B\u659C\u7387\u70BA\uFF09\u3002\u5C16\u9EDE\u662F\u5C40\u90E8\u7684\u5947\u9EDE\uFF0C\u53EA\u727D\u6D89\u5230\u53C3\u6578t\u7684\u4E00\u500B\u503C\uFF0C\u4E0D\u50CF\u81EA\u4EA4\u9EDE\u727D\u6D89\u5230t\u7684\u8A31\u591A\u503C\u3002\u5728\u67D0\u4E9B\u6642\u5019\uFF0C\u65B9\u5411\u5C0E\u6578\u8B8A\u865F\u7684\u689D\u4EF6\u6703\u7701\u53BB\uFF0C\u6B64\u6642\u5947\u9EDE\u6709\u53EF\u80FD\u770B\u8D77\u4F86\u50CF\u4E00\u822C\u7684\u9EDE\u3002 \u4EE5\u4E00\u500B\u5149\u6ED1\u96B1\u51FD\u6578\u5B9A\u7FA9\u7684\u66F2\u7DDA\u4F86\u8AAA\uFF0C \u5C07F\u4EE5\u6CF0\u52D2\u7D1A\u6578\u5C55\u958B\uFF0C\u7576\u5176\u6700\u4F4E\u968E\u9805\u53EF\u8868\u70BA\u4E00\u6B21\u591A\u9805\u5F0F\u7684\u6B21\u65B9\u6642\uFF0C\u5373\u70BA\u5C16\u9EDE\u6240\u5728\u8655\u3002\u4F46\u662F\u4E26\u975E\u6240\u6709\u64C1\u6709\u6B64\u6027\u8CEA\u7684\u5947\u9EDE\u90FD\u662F\u5C16\u9EDE\uFF0C\u7531\u76F8\u95DC\u5B9A\u7406\u53EF\u77E5\uFF0C\u82E5F\u662F\u89E3\u6790\u51FD\u6578\uFF0C\u5247\u5728\u5EA7\u6A19\u7DDA\u6027\u8B8A\u63DB\u5F8C\uFF0C\u5728\u5C16\u9EDE\u9644\u8FD1\u53EF\u5C07\u66F2\u7DDA\u53C3\u6578\u5316\u6210\u4EE5\u4E0B\u5F62\u5F0F\uFF1A \u5176\u4E2Da\u662F\u5BE6\u6578\uFF0Cm\u662F\u6B63\u5076\u6578\uFF0CS(t)\u662Fk\u968E\u7684\u51AA\u7D1A\u6578\u4E14k>m\u3002m\u4E5F\u662FF\u6700\u4F4E\u968E\u9805\u4E2D\u975E\u96F6\u90E8\u4EFD\u7684\u968E\u6578\u3002\u9019\u4E9B\u5B9A\u7FA9\u5DF2\u88AB\u52D2\u5185\u00B7\u6258\u59C6\u53CA\u5F17\u62C9\u57FA\u7C73\u723E\u00B7\u963F\u8AFE\u723E\u5FB7\u63A8\u5EE3\u81F3\u4EE5\u53EF\u5FAE\u51FD\u6578\u5B9A\u7FA9\u7684\u66F2\u7DDA\uFF0C\u82E5\u67D0\u9EDE\u9130\u57DF\u5B58\u5728\u5FAE\u5206\u540C\u80DA\uFF0C\u5C07\u66F2\u7DDA\u6620\u81F3\u4EE5\u4E0A\u5B9A\u7FA9\u7684\u5C16\u9EDE\uFF0C\u5247\u8A72\u66F2\u7DDA\u6709\u5C16\u9EDE\u3002\u5728\u67D0\u4E9B\u6642\u5019\uFF0C\u4EE5\u53CA\u4EE5\u4E0B\u6587\u7AE0\uFF0C\u5C16\u9EDE\u88AB\u9650\u5B9A\u70BA\u4E8C\u968E\u5C16\u9EDE\uFF0C\u4E5F\u5C31\u662F\u8AAA{{{1}}}\u3002\u4E00\u500B\u5E73\u9762\u66F2\u7DDA\u7684\u4E8C\u968E\u5C16\u9EDE\u53EF\u88AB\u5FAE\u5206\u540C\u80DA\u8868\u70BAx2 \u2013 y2k+1 = 0\uFF0C\u5176\u4E2Dk\u662F\u6B63\u6574\u6578\u3002"@zh . "Spitze (Singularit\u00E4tentheorie)"@de . "Hrot k\u0159ivky (tak\u00E9 ozna\u010Dovan\u00FD bod vratu nebo bod \u00FAvratu) je v geometrii takov\u00FD bod k\u0159ivky, kde je, neform\u00E1ln\u011B \u0159e\u010Deno, k\u0159ivka \u0161pi\u010Dat\u00E1 \u2013 ozna\u010Den\u00ED \u201Ebod vratu\u201C odpov\u00EDd\u00E1 tomu, \u017Ee pokud by byla k\u0159ivka kreslena perem, tak se pero v dan\u00E9m bod\u011B zastav\u00ED a pak se vyd\u00E1 sm\u011Brem zp\u011Bt. Z form\u00E1ln\u00EDho hlediska se jedn\u00E1 o jeden ze , tedy bod\u016F, kde k\u0159ivka nen\u00ED vyj\u00E1d\u0159iteln\u00E1 , ov\u0161em m\u00E1 v n\u011Bm v tomto p\u0159\u00EDpad\u011B te\u010Dnu (dokonce v ur\u010Dit\u00E9m smyslu dvojn\u00E1sobnou)."@cs . . "Cuspide (matematica)"@it . . . "\u5C16\u9EDE\uFF08\u82F1\u8A9E\uFF1ACusp\uFF09\u662F\u66F2\u7DDA\u4E2D\u7684\u4E00\u7A2E\u5947\u9EDE\u3002\u66F2\u7DDA\u4E0A\u7684\u52D5\u9EDE\u5728\u79FB\u5230\u5C16\u9EDE\u6642\u6703\u958B\u59CB\u53CD\u5411\u79FB\u52D5\uFF0C\u53F3\u5716\u662F\u4E00\u500B\u5178\u578B\u7684\u4F8B\u5B50\u3002\u7D66\u5B9A\u4E00\u500B\u4EE5\u89E3\u6790\u53C3\u6578\u5F0F\u5B9A\u7FA9\u7684\u5E73\u9762\u66F2\u7DDA\uFF1A \u5C16\u9EDE\u5373\u70BA\u51FD\u6578f\u53CAg\u4E4B\u5C0E\u6578\u70BA\u96F6\u4E4B\u9EDE\uFF0C\u540C\u6642\u65B9\u5411\u5C0E\u6578\u5728\u5207\u7DDA\u65B9\u5411\u6703\u8B8A\u865F\uFF08\u5207\u7DDA\u65B9\u5411\u4E4B\u659C\u7387\u70BA\uFF09\u3002\u5C16\u9EDE\u662F\u5C40\u90E8\u7684\u5947\u9EDE\uFF0C\u53EA\u727D\u6D89\u5230\u53C3\u6578t\u7684\u4E00\u500B\u503C\uFF0C\u4E0D\u50CF\u81EA\u4EA4\u9EDE\u727D\u6D89\u5230t\u7684\u8A31\u591A\u503C\u3002\u5728\u67D0\u4E9B\u6642\u5019\uFF0C\u65B9\u5411\u5C0E\u6578\u8B8A\u865F\u7684\u689D\u4EF6\u6703\u7701\u53BB\uFF0C\u6B64\u6642\u5947\u9EDE\u6709\u53EF\u80FD\u770B\u8D77\u4F86\u50CF\u4E00\u822C\u7684\u9EDE\u3002 \u4EE5\u4E00\u500B\u5149\u6ED1\u96B1\u51FD\u6578\u5B9A\u7FA9\u7684\u66F2\u7DDA\u4F86\u8AAA\uFF0C \u5C07F\u4EE5\u6CF0\u52D2\u7D1A\u6578\u5C55\u958B\uFF0C\u7576\u5176\u6700\u4F4E\u968E\u9805\u53EF\u8868\u70BA\u4E00\u6B21\u591A\u9805\u5F0F\u7684\u6B21\u65B9\u6642\uFF0C\u5373\u70BA\u5C16\u9EDE\u6240\u5728\u8655\u3002\u4F46\u662F\u4E26\u975E\u6240\u6709\u64C1\u6709\u6B64\u6027\u8CEA\u7684\u5947\u9EDE\u90FD\u662F\u5C16\u9EDE\uFF0C\u7531\u76F8\u95DC\u5B9A\u7406\u53EF\u77E5\uFF0C\u82E5F\u662F\u89E3\u6790\u51FD\u6578\uFF0C\u5247\u5728\u5EA7\u6A19\u7DDA\u6027\u8B8A\u63DB\u5F8C\uFF0C\u5728\u5C16\u9EDE\u9644\u8FD1\u53EF\u5C07\u66F2\u7DDA\u53C3\u6578\u5316\u6210\u4EE5\u4E0B\u5F62\u5F0F\uFF1A \u5176\u4E2Da\u662F\u5BE6\u6578\uFF0Cm\u662F\u6B63\u5076\u6578\uFF0CS(t)\u662Fk\u968E\u7684\u51AA\u7D1A\u6578\u4E14k>m\u3002m\u4E5F\u662FF\u6700\u4F4E\u968E\u9805\u4E2D\u975E\u96F6\u90E8\u4EFD\u7684\u968E\u6578\u3002\u9019\u4E9B\u5B9A\u7FA9\u5DF2\u88AB\u52D2\u5185\u00B7\u6258\u59C6\u53CA\u5F17\u62C9\u57FA\u7C73\u723E\u00B7\u963F\u8AFE\u723E\u5FB7\u63A8\u5EE3\u81F3\u4EE5\u53EF\u5FAE\u51FD\u6578\u5B9A\u7FA9\u7684\u66F2\u7DDA\uFF0C\u82E5\u67D0\u9EDE\u9130\u57DF\u5B58\u5728\u5FAE\u5206\u540C\u80DA\uFF0C\u5C07\u66F2\u7DDA\u6620\u81F3\u4EE5\u4E0A\u5B9A\u7FA9\u7684\u5C16\u9EDE\uFF0C\u5247\u8A72\u66F2\u7DDA\u6709\u5C16\u9EDE\u3002\u5728\u67D0\u4E9B\u6642\u5019\uFF0C\u4EE5\u53CA\u4EE5\u4E0B\u6587\u7AE0\uFF0C\u5C16\u9EDE\u88AB\u9650\u5B9A\u70BA\u4E8C\u968E\u5C16\u9EDE\uFF0C\u4E5F\u5C31\u662F\u8AAA{{{1}}}\u3002\u4E00\u500B\u5E73\u9762\u66F2\u7DDA\u7684\u4E8C\u968E\u5C16\u9EDE\u53EF\u88AB\u5FAE\u5206\u540C\u80DA\u8868\u70BAx2 \u2013 y2k+1 = 0\uFF0C\u5176\u4E2Dk\u662F\u6B63\u6574\u6578\u3002"@zh . . "\u0639\u0637\u0641\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Cusp)\u200F\u060C \u0644\u063A\u0629\u064B \u0647\u064A \u0632\u0627\u0648\u064A\u0629 \u0628\u0627\u0631\u0632\u0629 \u0623\u0648 \u062A\u0639\u0631\u062C \u0645\u0641\u0627\u062C\u0626 \u0641\u064A \u0634\u064A\u0621. \u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0647\u064A \u0646\u0642\u0637\u0629 \u0641\u064A \u0645\u0646\u062D\u0646\u0649 \u062D\u064A\u062B \u0646\u0642\u0637\u0629 \u0645\u062A\u062D\u0631\u0643\u0629 \u0623\u062E\u0631\u0649 \u0639\u0644\u0649 \u0647\u0630\u0627 \u0627\u0644\u0645\u0646\u062D\u0646\u0649 \u062A\u0636\u0637\u0631 \u0625\u0644\u0649 \u0627\u0644\u0648\u0642\u0648\u0641 \u0644\u0643\u064A \u062A\u0648\u0627\u0635\u0644 \u0637\u0631\u064A\u0642\u0647\u0627. \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0645\u0646\u062D\u0646\u0649 \u0645\u0633\u062A\u0648 \u0645\u0639\u0631\u0641 \u0628\u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0648\u0633\u064A\u0637\u064A\u0629 \u0627\u0644\u062A\u062D\u0644\u064A\u0644\u064A\u0629 \u0627\u0644\u062A\u0627\u0644\u064A\u0629: \u0639\u0637\u0641\u0629\u064C \u0647\u064A \u0646\u0642\u0637\u0629 \u062D\u064A\u062B \u0645\u0634\u062A\u0642 \u0627\u0634\u062A\u0642\u0627\u0642\u0627 \u0627\u0644\u062F\u0627\u0644\u062A\u064A\u0646 f \u0648 g \u064A\u0646\u0639\u062F\u0645\u0627\u0646 \u0648\u063A\u064A\u0631\u0647."@ar . . "En math\u00E9matiques, on appelle point de rebroussement, fronce (selon Ren\u00E9 Thom) ou parfois cusp, selon la terminologie anglaise, un type particulier de point singulier sur une courbe.Dans le cas d'une courbe admettant une \u00E9quation , les points de rebroussement ont les propri\u00E9t\u00E9s : 1. \n* ; 2. \n* ; 3. \n* La matrice hessienne (la matrice des d\u00E9riv\u00E9es secondes) a un d\u00E9terminant nul. L'\u00E9tude de la g\u00E9om\u00E9trie d'une courbe, alg\u00E9brique ou analytique, au voisinage d'un tel point, repose notamment sur la notion d'\u00E9clatement."@fr . . . . . . . . . . "En matem\u00E0tiques, en la teoria de la singularitat, una c\u00FAspide \u00E9s un tipus de punt singular d'una corba, on un punt en moviment de la corba ha de comen\u00E7ar a retrocedir. Les c\u00FAspides s\u00F3n singularitats locals que no estan formades per l'autointersecci\u00F3 dels punts de la corba. Per a una corba plana definida per una equaci\u00F3 param\u00E8trica impl\u00EDcita una c\u00FAspide \u00E9s un punt on la derivada de f i g \u00E9s zero, i la derivada direccional, en la direcci\u00F3 de la tangent, canvia signe (la direcci\u00F3 de la tangent \u00E9s la direcci\u00F3 del pendent ). La c\u00FAspide \u00E9s una singularitat local en el sentit que impliquen nom\u00E9s un valor del par\u00E0metre t, a difer\u00E8ncia dels punts d'intersecci\u00F3 lliures que impliquen m\u00E9s d'un valor. En alguns contexts, es pot ometre la condici\u00F3 de la derivada direccional, encara que, en aquest cas, la singularitat pot semblar un punt normal. Per a una corba definida per una equaci\u00F3 impl\u00EDcita cont\u00EDnuament diferenciable les c\u00FAspides s\u00F3n punts on els termes de menor grau de la s\u00E8rie de Taylor de F s\u00F3n una pot\u00E8ncia d'un polinomi lineal; no obstant aix\u00F2, no tots els punts singulars que tenen aquesta propietat s\u00F3n c\u00FAspides. La teoria de la implica que, si F \u00E9s una funci\u00F3 anal\u00EDtica (per exemple, un polinomi), un canvi lineal de coordenades permet parametritzar la corba, en un ve\u00EFnat de la c\u00FAspide, com on a \u00E9s un nombre real, m \u00E9s un enter positiu, i S(t) \u00E9s una s\u00E8rie de pot\u00E8ncies d'ordre k (grau del terme no-zero del grau m\u00E9s baix) m\u00E9s gran que m. El nombre m es denomina de vegades ordre o multiplicitat de la c\u00FAspide, i \u00E9s igual al grau de la part no-zero del grau m\u00E9s baix de F. Aquestes definicions s'han generalitzat a les corbes definides per funcions diferenciables per Ren\u00E9 Thom i Vladimir Arnold de la seg\u00FCent manera: Una corba t\u00E9 una c\u00FAspide en un punt si hi ha un difeomorfisme d'un ve\u00EFnat del punt en l'espai ambient, que assigna la corba a una de les c\u00FAspides abans esmentades. En alguns contexts i en la resta d'aquest article, la definici\u00F3 d'una c\u00FAspide est\u00E0 restringida al cas de c\u00FAspides d'ordre dos, \u00E9s a dir, el cas on m = 2. Les c\u00FAspides de corbes planes (d'ordre dos) es poden posar per difeomorfisme al pla amb la forma x\u00B2 \u2212 y2k+1 = 0, on k \u00E9s un nombre enter positiu (k\u2265 1)."@ca . . . . . . "\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u5C16\u70B9\uFF08\u305B\u3093\u3066\u3093\u3001\u82F1: cusp, \u53E4\u304F\u306F\u5C16\u7BC0\u70B9 (spinode)\uFF09\u306F\u3001\u66F2\u7DDA\u306B\u6CBF\u3063\u3066\u8D70\u308B\u52D5\u70B9\u304C\u305D\u3053\u3067\u5411\u304D\u3092\u9006\u8EE2\u3059\u308B\u3088\u3046\u306A\u66F2\u7DDA\u4E0A\u306E\u70B9\u3067\u3042\u308B\u3002\u5C16\u70B9\u306F\u66F2\u7DDA\u306E\u7279\u7570\u70B9\u306E\u4E00\u7A2E\u3068\u3044\u3046\u3053\u3068\u306B\u306A\u308B\u3002 \u89E3\u6790\u7684\u306B\u5A92\u4ECB\u4ED8\u3051\u3089\u308C\u305F\u5E73\u9762\u66F2\u7DDA \u306B\u304A\u3044\u3066\u5C16\u70B9\u306F\u3001f \u304A\u3088\u3073 g \u306E\u5FAE\u5206\u4FC2\u6570\u304C\u3068\u3082\u306B\u6D88\u3048\u3066\u3044\u308B\u3088\u3046\u306A\u70B9\uFF08\u3064\u307E\u308A\u66F2\u7DDA\u306E\u7279\u7570\u70B9\uFF09\u3067\u3042\u3063\u3066\u3001\u305D\u306E\u70B9\u3067\u306E\u63A5\u7DDA\u65B9\u5411\u3078\u306E\u65B9\u5411\u5FAE\u5206\u304C\u7B26\u53F7\u3092\u5909\u3048\u308B\u3082\u306E\u3067\u3042\u308B\uFF08\u3053\u3053\u3067\u300C\u63A5\u7DDA\u65B9\u5411\u300D\u3068\u306F\u3001\u305D\u306E\u8FD1\u508D\u306E\u5404\u70B9\u306B\u304A\u3051\u308B\u50BE\u304D\u306E\u6975\u9650 limg\u2032(t)\u2044f\u2032(t) \u3092\u50BE\u304D\u3068\u3059\u308B\u76F4\u7DDA\u306E\u65B9\u5411\u306E\u610F\uFF09\u3002\u5A92\u4ECB\u5909\u6570 t \u306E\u305F\u3060\u4E00\u3064\u306E\u5024\u306E\u307F\u3067\u6C7A\u307E\u308B\u3068\u3044\u3046\u610F\u5473\u3067\u5C16\u70B9\u306F\u300C\u5C40\u6240\u7684\u306A\u7279\u7570\u70B9\u300D\u3067\u3042\u308B\u3002\u5834\u5408\u306B\u3088\u3063\u3066\u306F\u5C16\u70B9\u306E\u5B9A\u7FA9\u306B\u65B9\u5411\u5FAE\u5206\u306B\u95A2\u3059\u308B\u6761\u4EF6\u3092\u554F\u308F\u306A\u3044\u3053\u3068\u3082\u3042\u308B\u304C\u3001\u305D\u306E\u5834\u5408\u306F\u4E00\u898B\u3059\u308B\u3068\u6B63\u5247\u70B9\u306E\u3088\u3046\u306B\u3082\u898B\u3048\u308B\u7279\u7570\u70B9\u3082\u73FE\u308C\u5F97\u308B\u3053\u3068\u306B\u6CE8\u610F\u3059\u3079\u304D\u3067\u3042\u308B\u3002 \u306A\u3081\u3089\u304B\u306A\u9670\u4F0F\u65B9\u7A0B\u5F0F \u3053\u308C\u3089\u306E\u5B9A\u7FA9\u3092\u3001\u30EB\u30CD\u30FB\u30C8\u30E0\u304A\u3088\u3073\u30A6\u30E9\u30B8\u30FC\u30DF\u30EB\u30FB\u30A2\u30FC\u30CE\u30EB\u30C9\u306F\u3001\u53EF\u5FAE\u5206\u51FD\u6570\u306E\u5B9A\u3081\u308B\u66F2\u7DDA\u306B\u5BFE\u3059\u308B\u3082\u306E\u3078\u4E00\u822C\u5316\u3057\u305F\u3002\u3059\u306A\u308F\u3061\u3001\u66F2\u7DDA\u304C\u3042\u308B\u70B9\u306B\u5C16\u70B9\u3092\u6301\u3064\u3068\u306F\u3001\u5168\u4F53\u7A7A\u9593\u3067\u8003\u3048\u305F\u305D\u306E\u70B9\u306E\u8FD1\u508D\u4E0A\u3067\u5FAE\u5206\u540C\u76F8\u5199\u50CF\u304C\u5B58\u5728\u3057\u3066\u3001\u305D\u306E\u66F2\u7DDA\u3092\u4E0A\u3067\u5B9A\u7FA9\u3055\u308C\u305F\u610F\u5473\u3067\u306E\u5C16\u70B9\u306E\u4E0A\u3078\u5199\u3059\u3053\u3068\u304C\u3067\u304D\u308B\u3068\u304D\u306B\u8A00\u3046\u3002"@ja . "\u041A\u0430\u0441\u043F (\u043E\u0442 \u0430\u043D\u0433\u043B. cusp \u2014 \u0437\u0430\u043E\u0441\u0442\u0440\u0435\u043D\u0438\u0435, \u043F\u0438\u043A), \u0438\u043B\u0438 \u0442\u043E\u0447\u043A\u0430 \u0432\u043E\u0437\u0432\u0440\u0430\u0442\u0430, \u2014 \u043E\u0441\u043E\u0431\u0430\u044F \u0442\u043E\u0447\u043A\u0430, \u0432 \u043A\u043E\u0442\u043E\u0440\u043E\u0439 \u043A\u0440\u0438\u0432\u0430\u044F \u043B\u0438\u043D\u0438\u044F \u0440\u0430\u0437\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u043D\u0430 \u0434\u0432\u0435 (\u0438\u043B\u0438 \u0431\u043E\u043B\u0435\u0435) \u0432\u0435\u0442\u0432\u0438, \u0438\u043C\u0435\u044E\u0449\u0438\u0435 \u0432 \u044D\u0442\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043E\u0434\u0438\u043D\u0430\u043A\u043E\u0432\u044B\u0439 \u043D\u0430\u043F\u0440\u0430\u0432\u043B\u044F\u044E\u0449\u0438\u0439 \u0432\u0435\u043A\u0442\u043E\u0440. \u0422\u043E \u0435\u0441\u0442\u044C \u0432\u0435\u0442\u0432\u0438 \u0432 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u0438\u043C\u0435\u044E\u0442 \u043E\u0431\u0449\u0443\u044E \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u0443\u044E, \u0438 \u0434\u0432\u0438\u0436\u0435\u043D\u0438\u0435 \u0432\u0434\u043E\u043B\u044C \u043D\u0438\u0445 \u0438\u0437 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0438 \u0438\u0437\u043D\u0430\u0447\u0430\u043B\u044C\u043D\u043E \u043F\u0440\u043E\u0438\u0441\u0445\u043E\u0434\u0438\u0442 \u0432 \u043E\u0434\u043D\u043E\u043C \u0438 \u0442\u043E\u043C \u0436\u0435 \u043D\u0430\u043F\u0440\u0430\u0432\u043B\u0435\u043D\u0438\u0438. \u0418\u043D\u043E\u0433\u0434\u0430 \u043A\u0430\u0441\u043F \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u0432 \u0431\u043E\u043B\u0435\u0435 \u0443\u0437\u043A\u043E\u043C \u0441\u043C\u044B\u0441\u043B\u0435 \u2014 \u043A\u0430\u043A \u043E\u0441\u043E\u0431\u0430\u044F \u0442\u043E\u0447\u043A\u0430 \u0441\u043F\u0435\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0442\u0438\u043F\u0430 \u043D\u0430 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u0440\u0438\u0432\u043E\u0439. \u0410 \u0438\u043C\u0435\u043D\u043D\u043E: \u043E\u0441\u043E\u0431\u0430\u044F \u0442\u043E\u0447\u043A\u0430 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u0440\u0438\u0432\u043E\u0439 \u043D\u0430\u0434 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u0438 \u0437\u0430\u043C\u043A\u043D\u0443\u0442\u044B\u043C \u043F\u043E\u043B\u0435\u043C \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043A\u0430\u0441\u043F\u043E\u043C, \u0435\u0441\u043B\u0438 \u043F\u043E\u043F\u043E\u043B\u043D\u0435\u043D\u0438\u0435 \u0435\u0451 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u043A\u043E\u043B\u044C\u0446\u0430 \u0438\u0437\u043E\u043C\u043E\u0440\u0444\u043D\u043E \u043F\u043E\u043F\u043E\u043B\u043D\u0435\u043D\u0438\u044E \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u043A\u043E\u043B\u044C\u0446\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u0440\u0438\u0432\u043E\u0439 (\u043F\u043E\u043B\u0443\u043A\u0443\u0431\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043F\u0430\u0440\u0430\u0431\u043E\u043B\u044B) \u0432 \u043D\u0430\u0447\u0430\u043B\u0435 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442. \u0412 \u044D\u0442\u043E\u043C \u0441\u043B\u0443\u0447\u0430\u0435 \u043A\u0430\u0441\u043F \u0435\u0449\u0451 \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442 \u043E\u0431\u044B\u043A\u043D\u043E\u0432\u0435\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u043E\u0439 \u0432\u043E\u0437\u0432\u0440\u0430\u0442\u0430."@ru . . "\u5C16\u9EDE"@zh . . "1072351906"^^ . . . "In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation a cusp is a point where both derivatives of f and g are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope ). Cusps are local singularities in the sense that they involve only one value of the parameter t, in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an implicit equation which is smooth, cusps are points where the terms of lowest degree of the Taylor expansion of F are a power of a linear polynomial; however, not all singular points that have this property are cusps. The theory of Puiseux series implies that, if F is an analytic function (for example a polynomial), a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as where a is a real number, m is a positive even integer, and S(t) is a power series of order k (degree of the nonzero term of the lowest degree) larger than m. The number m is sometimes called the order or the multiplicity of the cusp, and is equal to the degree of the nonzero part of lowest degree of F. In some contexts, the definition of a cusp is restricted to the case of cusps of order two\u2014that is, the case where m = 2. The definitions for plane curves and implicitly-defined curves have been generalized by Ren\u00E9 Thom and Vladimir Arnold to curves defined by differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps."@en . . . . . . . . . . . "\u041A\u0430\u0441\u043F (\u043E\u0442 \u0430\u043D\u0433\u043B. cusp \u2014 \u0437\u0430\u043E\u0441\u0442\u0440\u0435\u043D\u0438\u0435, \u043F\u0438\u043A), \u0438\u043B\u0438 \u0442\u043E\u0447\u043A\u0430 \u0432\u043E\u0437\u0432\u0440\u0430\u0442\u0430, \u2014 \u043E\u0441\u043E\u0431\u0430\u044F \u0442\u043E\u0447\u043A\u0430, \u0432 \u043A\u043E\u0442\u043E\u0440\u043E\u0439 \u043A\u0440\u0438\u0432\u0430\u044F \u043B\u0438\u043D\u0438\u044F \u0440\u0430\u0437\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u043D\u0430 \u0434\u0432\u0435 (\u0438\u043B\u0438 \u0431\u043E\u043B\u0435\u0435) \u0432\u0435\u0442\u0432\u0438, \u0438\u043C\u0435\u044E\u0449\u0438\u0435 \u0432 \u044D\u0442\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043E\u0434\u0438\u043D\u0430\u043A\u043E\u0432\u044B\u0439 \u043D\u0430\u043F\u0440\u0430\u0432\u043B\u044F\u044E\u0449\u0438\u0439 \u0432\u0435\u043A\u0442\u043E\u0440. \u0422\u043E \u0435\u0441\u0442\u044C \u0432\u0435\u0442\u0432\u0438 \u0432 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u0438\u043C\u0435\u044E\u0442 \u043E\u0431\u0449\u0443\u044E \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u0443\u044E, \u0438 \u0434\u0432\u0438\u0436\u0435\u043D\u0438\u0435 \u0432\u0434\u043E\u043B\u044C \u043D\u0438\u0445 \u0438\u0437 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0438 \u0438\u0437\u043D\u0430\u0447\u0430\u043B\u044C\u043D\u043E \u043F\u0440\u043E\u0438\u0441\u0445\u043E\u0434\u0438\u0442 \u0432 \u043E\u0434\u043D\u043E\u043C \u0438 \u0442\u043E\u043C \u0436\u0435 \u043D\u0430\u043F\u0440\u0430\u0432\u043B\u0435\u043D\u0438\u0438."@ru . . "In der Mathematik sind Spitzen (auch Kuspen, engl.: cusps) ein Typ von Singularit\u00E4ten von Kurven. Ein sich auf der Kurve bewegender Punkt m\u00FCsste an der Spitze seine Richtung abrupt \u00E4ndern."@de . . . . . "C\u00FAspide (matem\u00E0tiques)"@ca . "En math\u00E9matiques, on appelle point de rebroussement, fronce (selon Ren\u00E9 Thom) ou parfois cusp, selon la terminologie anglaise, un type particulier de point singulier sur une courbe.Dans le cas d'une courbe admettant une \u00E9quation , les points de rebroussement ont les propri\u00E9t\u00E9s : 1. \n* ; 2. \n* ; 3. \n* La matrice hessienne (la matrice des d\u00E9riv\u00E9es secondes) a un d\u00E9terminant nul. L'\u00E9tude de la g\u00E9om\u00E9trie d'une courbe, alg\u00E9brique ou analytique, au voisinage d'un tel point, repose notamment sur la notion d'\u00E9clatement."@fr . . "10635"^^ . "En matem\u00E1ticas, una c\u00FAspide es un punto de una curva donde un punto m\u00F3vil que recorra la curva debe comenzar a retroceder. Un ejemplo t\u00EDpico se da en la figura adjunta. Una c\u00FAspide es, por lo tanto, un tipo de punto singular de una curva. Para una curva plana definida por una ecuaci\u00F3n param\u00E9trica anal\u00EDtica\u200B Para una curva definida por una ecuaci\u00F3n impl\u00EDcita suave (continuamente diferenciable) En algunos contextos, y en el resto de este art\u00EDculo, la definici\u00F3n de una c\u00FAspide se limita al caso de las c\u00FAspides de orden dos, es decir, el caso donde m = 2."@es . . . . . . . . "Cusp (singularity)"@en . "In der Mathematik sind Spitzen (auch Kuspen, engl.: cusps) ein Typ von Singularit\u00E4ten von Kurven. Ein sich auf der Kurve bewegender Punkt m\u00FCsste an der Spitze seine Richtung abrupt \u00E4ndern."@de . . . . . . "In analisi matematica, si dice che una funzione di variabile reale continua in un punto del dominio, ha una cuspide in se si verifica la seguente condizione ossia i limiti destro e sinistro del rapporto incrementale in sono divergenti (tendenti a ) con segno opposto. Geometricamente, si pu\u00F2 osservare come le semitangenti destra e sinistra siano verticali e formino un angolo nullo."@it . . . . "\u5C16\u70B9"@ja . . "In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation For a curve defined by an implicit equation"@en . . . . . . . . . . . . "In analisi matematica, si dice che una funzione di variabile reale continua in un punto del dominio, ha una cuspide in se si verifica la seguente condizione ossia i limiti destro e sinistro del rapporto incrementale in sono divergenti (tendenti a ) con segno opposto. Geometricamente, si pu\u00F2 osservare come le semitangenti destra e sinistra siano verticali e formino un angolo nullo."@it . . . . . . . "Hrot k\u0159ivky"@cs . . . "\u041A\u0430\u0441\u043F"@ru . . . . . . "En kuspo estas singulara punkto de kurbo. Por kurbo difinita kiel la nula aro de funkcio de du variabloj f(x, y)=0, la kuspoj sur la kurbo estas punktoj (x, y) kiuj havas samtemple \u0109iujn jenajn propra\u0135ojn: \n* f(x, y)=0 \n* \n* La matrico de Hessian de la duaj deriva\u0135oj havas nulan determinanton. Klasika ekzemplo de kuspo estas punkto (0, 0) sur kurbo x3-y2=0 \u0108i tiu kurbo povas esti esprimita parametre kiel x=t2, y=t3 Kuspoj estas ofte trovitaj en optiko kiel formo de . Ili estas anka\u016D trovataj en projekcioj de profilo de surfaco."@eo . . . "Hrot k\u0159ivky (tak\u00E9 ozna\u010Dovan\u00FD bod vratu nebo bod \u00FAvratu) je v geometrii takov\u00FD bod k\u0159ivky, kde je, neform\u00E1ln\u011B \u0159e\u010Deno, k\u0159ivka \u0161pi\u010Dat\u00E1 \u2013 ozna\u010Den\u00ED \u201Ebod vratu\u201C odpov\u00EDd\u00E1 tomu, \u017Ee pokud by byla k\u0159ivka kreslena perem, tak se pero v dan\u00E9m bod\u011B zastav\u00ED a pak se vyd\u00E1 sm\u011Brem zp\u011Bt. Z form\u00E1ln\u00EDho hlediska se jedn\u00E1 o jeden ze , tedy bod\u016F, kde k\u0159ivka nen\u00ED vyj\u00E1d\u0159iteln\u00E1 , ov\u0161em m\u00E1 v n\u011Bm v tomto p\u0159\u00EDpad\u011B te\u010Dnu (dokonce v ur\u010Dit\u00E9m smyslu dvojn\u00E1sobnou)."@cs . "En kuspo estas singulara punkto de kurbo. Por kurbo difinita kiel la nula aro de funkcio de du variabloj f(x, y)=0, la kuspoj sur la kurbo estas punktoj (x, y) kiuj havas samtemple \u0109iujn jenajn propra\u0135ojn: \n* f(x, y)=0 \n* \n* La matrico de Hessian de la duaj deriva\u0135oj havas nulan determinanton. Klasika ekzemplo de kuspo estas punkto (0, 0) sur kurbo x3-y2=0 \u0108i tiu kurbo povas esti esprimita parametre kiel x=t2, y=t3 Kuspoj estas ofte trovitaj en optiko kiel formo de . Ili estas anka\u016D trovataj en projekcioj de profilo de surfaco."@eo . . . . . . "En matem\u00E1ticas, una c\u00FAspide es un punto de una curva donde un punto m\u00F3vil que recorra la curva debe comenzar a retroceder. Un ejemplo t\u00EDpico se da en la figura adjunta. Una c\u00FAspide es, por lo tanto, un tipo de punto singular de una curva. Para una curva plana definida por una ecuaci\u00F3n param\u00E9trica anal\u00EDtica\u200B una c\u00FAspide es un punto donde las derivadas de f y g son simult\u00E1neamente cero, y la derivada direccional, en la direcci\u00F3n de la tangente, cambia de signo (la direcci\u00F3n de la tangente es la direcci\u00F3n de la pendiente ) Las c\u00FAspides son singularidades locales en el sentido de que involucran solo un valor del par\u00E1metro t, en contraste con los puntos de auto-intersecci\u00F3n que involucran m\u00E1s de un valor. En algunos contextos, la condici\u00F3n en la derivada direccional puede omitirse, aunque, en este caso, la singularidad puede parecer un punto regular. Para una curva definida por una ecuaci\u00F3n impl\u00EDcita suave (continuamente diferenciable) las c\u00FAspides son puntos donde los t\u00E9rminos del grado m\u00E1s bajo de la expansi\u00F3n en serie de Taylor de F son una potencia de un polinomio lineal; sin embargo, no todos los puntos singulares que tienen esta propiedad son c\u00FAspides. La teor\u00EDa de la implica que, si F es una funci\u00F3n anal\u00EDtica (por ejemplo, un polinomio), un cambio lineal de coordenadas permite que la curva se parametrice, en una vecindad de la c\u00FAspide, como donde a es un n\u00FAmero real, m es un entero par positivo y S(t) es una serie de potencias de orden k (grado del t\u00E9rmino distinto de cero del grado m\u00E1s bajo) mayor que m. El n\u00FAmero m se llama el orden o la multiplicidad de la c\u00FAspide, y es igual al grado de la parte distinta de cero del grado m\u00E1s bajo de F. Estas definiciones han sido generalizadas a las curvas definidas por funciones diferenciables por Ren\u00E9 Thom y Vladimir Arnold, de la siguiente manera. Una curva tiene una c\u00FAspide en un punto si hay un difeomorfismo de una vecindad del punto en el espacio del entorno, que aplica la curva en una de las c\u00FAspides definidas anteriormente. En algunos contextos, y en el resto de este art\u00EDculo, la definici\u00F3n de una c\u00FAspide se limita al caso de las c\u00FAspides de orden dos, es decir, el caso donde m = 2. Una c\u00FAspide de una curva plana (de orden dos) se puede poner en la siguiente forma mediante un difeomorfismo del plano: x2 \u2013 y2k+1 = 0, donde k es un n\u00FAmero entero positivo.[cita requerida]"@es . "\u0639\u0637\u0641\u0629 (\u0631\u064A\u0627\u0636\u064A\u0627\u062A)"@ar . . . . "\u0639\u0637\u0641\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Cusp)\u200F\u060C \u0644\u063A\u0629\u064B \u0647\u064A \u0632\u0627\u0648\u064A\u0629 \u0628\u0627\u0631\u0632\u0629 \u0623\u0648 \u062A\u0639\u0631\u062C \u0645\u0641\u0627\u062C\u0626 \u0641\u064A \u0634\u064A\u0621. \u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0647\u064A \u0646\u0642\u0637\u0629 \u0641\u064A \u0645\u0646\u062D\u0646\u0649 \u062D\u064A\u062B \u0646\u0642\u0637\u0629 \u0645\u062A\u062D\u0631\u0643\u0629 \u0623\u062E\u0631\u0649 \u0639\u0644\u0649 \u0647\u0630\u0627 \u0627\u0644\u0645\u0646\u062D\u0646\u0649 \u062A\u0636\u0637\u0631 \u0625\u0644\u0649 \u0627\u0644\u0648\u0642\u0648\u0641 \u0644\u0643\u064A \u062A\u0648\u0627\u0635\u0644 \u0637\u0631\u064A\u0642\u0647\u0627. \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0645\u0646\u062D\u0646\u0649 \u0645\u0633\u062A\u0648 \u0645\u0639\u0631\u0641 \u0628\u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0648\u0633\u064A\u0637\u064A\u0629 \u0627\u0644\u062A\u062D\u0644\u064A\u0644\u064A\u0629 \u0627\u0644\u062A\u0627\u0644\u064A\u0629: \u0639\u0637\u0641\u0629\u064C \u0647\u064A \u0646\u0642\u0637\u0629 \u062D\u064A\u062B \u0645\u0634\u062A\u0642 \u0627\u0634\u062A\u0642\u0627\u0642\u0627 \u0627\u0644\u062F\u0627\u0644\u062A\u064A\u0646 f \u0648 g \u064A\u0646\u0639\u062F\u0645\u0627\u0646 \u0648\u063A\u064A\u0631\u0647."@ar . "\u0412 \u043A\u0430\u0441\u043F (\u0430\u043D\u0433\u043B. cusp \u2014 \u0437\u0430\u0433\u043E\u0441\u0442\u0440\u0435\u043D\u043D\u044F) \u0454 \u043E\u0434\u043D\u0438\u043C \u0437 \u0432\u0438\u0434\u0456\u0432 \u043E\u0441\u043E\u0431\u043B\u0438\u0432\u0438\u0445 \u0442\u043E\u0447\u043E\u043A \u043A\u0440\u0438\u0432\u043E\u0457."@uk . "Kuspo (speciala\u0135o)"@eo . . . . .