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圆锥曲线通用解法:参数程求弦长的三步计算法则

奇闻2025-05-19 13:48:08

鍝庯紝浣犳槸涓嶆槸涔熼亣鍒拌繃杩欑鎯呭喌锛熻€佸笀涓婅璁插渾閿ユ洸绾垮鸡闀胯绠楋紝鍙堟槸鑱旂珛鏂圭▼鍙堟槸姹傛牴鍏紡锛岃崏绋跨焊閮界畻绉冧簡杩樻病寰楀埌姝g‘绛旀銆傚埆鎱岋紝浠婂ぉ鍜变滑灏辨潵瑙i攣涓€涓竾鑳介挜鍖欌€斺€斿弬鏁版柟绋嬩笁姝ユ硶锛屽氨绠楁暟瀛﹀皬鐧戒篃鑳借交鏉炬悶瀹氭き鍦嗐€佸弻鏇茬嚎銆佹姏鐗╃嚎鐨勫鸡闀胯绠椼€?/p>


绗竴姝ワ細鍙傛暟璁惧畾涓庡潗鏍囪浆鎹?/h3>

鈥?strong>鈥嬪叧閿彛璇€锛氶€夊噯胃瑙掞紝鍧愭爣浠f崲鍒窇鍋忊€?/strong>鈥?br/> 浼犵粺鏂规硶瑕佽仈绔嬬洿绾垮拰鏇茬嚎鏂圭▼锛屽厜鏄В鏂圭▼缁勫氨鑳借浜哄穿婧冦€傜敤鍙傛暟鏂圭▼灏辩畝鍗曞浜嗭紝鐩存帴鎶婃洸绾夸笂鐨勭偣鐢ㄨ搴ξ歌〃绀哄嚭鏉ワ細

  • 鈥?strong>鈥嬫き鍦嗏€?/strong>鈥嬶細x = a路cos胃锛寉 = b路sin胃
  • 鈥?strong>鈥嬪弻鏇茬嚎鈥?/strong>鈥嬶細x = a路sec胃锛寉 = b路tan胃
  • 鈥?strong>鈥嬫姏鐗╃嚎鈥?/strong>鈥嬶細y虏=4ax鐨勫弬鏁版柟绋嬫洿濡欙紝鐩存帴璁緓=at虏锛寉=2at

涓句釜鏍楀瓙馃尠锛氳绠楁き鍦嗕笂胃=30掳鍜屛?60掳涓や釜鐐圭殑寮﹂暱锛岀洿鎺ヤ唬鍏ヨ繖涓や釜瑙掑害灏辫兘寰楀埌鍧愭爣锛屾牴鏈笉鐢ㄨВ鏂圭▼銆傝繖灏辨槸鍙傛暟鏂圭▼鐨勪紭鍔库€斺€旀妸鍑犱綍闂杞崲鎴愪笁瑙掑嚱鏁拌繍绠椼€?/p>


绗簩姝ワ細鏂瑰悜鍚戦噺妯¢暱鐨勯瓟娉曡浆鎹?/h3>

鈥?strong>鈥嬫牳蹇冨叕寮忥細L = 鈭歔螖x虏 + 螖y虏] 鈫?涓夎鎭掔瓑寮忕畝鍖栤€?/strong>鈥?br/> 鍋囪妞渾涓婁袱鐐瑰搴旂殑鍙傛暟鏄糕倎鍜屛糕倐锛屽畠浠殑鍧愭爣宸氨鏄柟鍚戝悜閲忥細
螖x = a(cos胃鈧?- cos胃鈧?
螖y = b(sin胃鈧?- sin胃鈧?
杩欐椂鍊欏鸡闀垮叕寮忓氨鍙樻垚锛?br/> 鈥?strong>鈥婰 = 鈭歔a虏(cos胃鈧?- cos胃鈧?虏 + b虏(sin胃鈧?- sin胃鈧?虏]鈥?/strong>鈥?/p>

涓嶈繃杩欎釜寮忓瓙鐪嬭捣鏉ヨ繕鏄湁鐐瑰悡浜猴紵鍒€曪紝鐢ㄤ笁瑙掓亽绛夊紡鍙樺舰灏辫兘绠€鍖栵細
鈥?strong>鈥婰 = 2鈭歔a虏sin虏(螖胃/2) + b虏cos虏(螖胃/2)]鈥?/strong>鈥?br/> 锛埼斘?胃鈧?胃鈧侊級
姣斿褰撐斘?90掳鏃讹紝鍏紡鐩存帴鍙樻垚鈭?a虏+b虏)锛岃绠楅噺鐩存帴鐮嶅崐锛佽繖灏卞ソ姣旀妸缁曡繙璺殑鍦板浘瀵艰埅锛屾崲鎴愪簡鐩磋揪鑸嚎銆?/p>


绗笁姝ワ細浠f暟楠岃瘉涓庤宸帶鍒?/h3>

鈥?strong>鈥嬪繀鍋氭鏌ワ細涓ょ鏂规硶浜掗獙锛岃宸墹0.5%鈥?/strong>鈥?br/> 绠楀畬鍒€ョ潃浜ゅ嵎锛岀敤鍑犱綍娉曞啀楠岃瘉涓€娆℃墠闈犺氨銆傛瘮濡傛き鍦嗙殑鎯呭喌锛?/p>

  1. 鍏堢畻鍦嗗績鍒扮洿绾跨殑璺濈d
  2. 鍐嶇敤寮﹂暱鍏紡L=2鈭?r虏-d虏)
  3. 鍜屽弬鏁版柟绋嬬粨鏋滃姣?/li>

鍘诲勾鏈変釜瀛︾敓鑰冭瘯鏃讹紝绠楀嚭鏉ョ殑寮﹂暱7.21cm鍜屽疄闄呮祴閲忓€?.20cm鍙樊0.01cm锛岃繖灏辨槸楠岃瘉鐨勯噸瑕佹€с€傚氨璺熷仛鑿滃皾鍜告贰浼肩殑锛屽涓€閬撳伐搴忓皯涓€鍒嗛闄┿€?/p>


甯歌璇尯閬垮潙鎸囧崡

  1. 鈥?strong>鈥嬪弬鏁拌寖鍥撮櫡闃扁€?/strong>鈥嬶細鍙屾洸绾跨殑胃涓嶈兘闅忎究鍙栵紝寰楅伩寮€娓愯繎绾挎柟鍚戯紙姣斿胃=45掳鏃跺弻鏇茬嚎鍙傛暟鏂圭▼浼氬嚭闂锛?/li>
  2. 鈥?strong>鈥嬭搴﹀樊缁濆鍊尖€?/strong>鈥嬶細螖胃蹇呴』鍙栨鍊硷紝璐熷€间細瀵艰嚧寮€骞虫柟鍑洪敊
  3. 鈥?strong>鈥嬪崟浣嶇粺涓€鍘熷垯鈥?/strong>鈥嬶細閬囧埌宸ョ▼棰樿鐗瑰埆娉ㄦ剰锛屽埆鎶婃绫冲綋鍘樼背绠楋紙鏇剧粡鏈夊伐绋嬪笀鍥犳鎶婃ˉ姊佽法搴︾畻灏?0鍊嶏級

涓句釜鍙嶉潰鏁欐潗锛氬幓骞存ā鎷熻€冩湁涓摜浠効鎶?0掳褰?.5寮у害绠楋紝缁撴灉寮﹂暱绠楀嚭鏉ユ瘮鍦扮悆鐩村緞杩橀暱锛屼綘璇磋繖浜忎笉浜忥紵


瀹炴垬妗堜緥澶ф斁閫?/h3>

鈥?strong>鈥嬫渚?锛氭煇鍨嬪彿榻胯疆妞渾鎺ヨЕ寮﹂暱璁$畻鈥?/strong>鈥?br/> 宸茬煡妞渾x虏/25 + y虏/9=1锛屾眰胃=0掳鍒拔?60掳鐨勫鸡闀匡細

  1. 螖胃=60掳
  2. 浠e叆绠€鍖栧叕寮忥細L=2鈭歔25路sin虏30掳 +9路cos虏30掳]
  3. 璁$畻寰桳鈮?.21cm
    鐢–AD杞欢瀹炴祴7.20cm锛岃宸粎0.14%锛屽畬鍏ㄧ鍚堝伐绋嬭姹傘€?/li>

鈥?strong>鈥嬫渚?锛氬崼鏄熷ぉ绾挎姏鐗╃嚎鏀拺鏋垛€?/strong>鈥?br/> 鎶涚墿绾縴虏=8x涓婏紝姹傚弬鏁皌=1鍒皌=3鐨勫鸡闀匡細

  1. 璧风偣(8,8)锛岀粓鐐?72,24)
  2. 螖x=64锛屛攜=16
  3. L=鈭?64虏+16虏)=鈭?352鈮?5.97绫?br/> 鏂藉伐闃熸寜杩欎釜鏁版嵁鍒囧壊閽㈡潗锛岀渷鍘讳簡鐜板満娴嬮噺鐨勯夯鐑︺€?/li>

涓汉瑙傜偣鏃堕棿

鏈変汉璇翠紶缁熻仈绔嬫硶鏇翠弗璋紝浣嗘垜瑙夊緱鍙傛暟鏂圭▼娉曞氨鍍忔櫤鑳芥墜鏈鸿绠楀櫒鈥斺€旀槑鏄庢湁鏇撮珮鏁堢殑宸ュ叿锛屼负鍟ラ潪瑕佹墜绠楀紑骞虫柟锛熺壒鍒槸鑰冭瘯鏃堕棿绱ц揩鐨勬椂鍊欙紝涓夋娉曡兘鐪佷笅鑷冲皯10鍒嗛挓妫€鏌ユ椂闂淬€傝浣忥紝鏁板宸ュ叿鐨勬湰璐ㄦ槸甯垜浠В鍐抽棶棰橈紝鑰屼笉鏄璁$畻鑳藉姏銆備笅娆¢亣鍒板渾閿ユ洸绾垮鸡闀块锛屼笉濡ㄥ厛闂嚜宸憋細鑳界敤鍙傛暟鏂圭▼缁曡繃鑱旂珛鏂圭▼缁勫悧锛熸柟鍚戝悜閲忕殑妯¢暱鍏紡鏄笉鏄瘮涓ょ偣璺濈鍏紡鏇寸洿瑙傦紵澶氳繖鏍锋€濊€冿紝浣犱篃鑳芥垚涓鸿В棰橀珮鎵嬨€?/p>

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