笛卡尔心形曲线公式 心形函数图像，写成f（x）的形式
\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u91cc\u7684\u6843\u5fc3\u516c\u5f0f\u662f\u4ec0\u4e48\u5fc3\u5f62\u7ebf\u7684\u5e73\u9762\u76f4\u89d2\u5750\u6807\u7cfb\u65b9\u7a0b\u8868\u8fbe\u5f0f\u5206\u522b\u4e3a x^2+y^2+a*x=a*sqrt(x^2+y^2) \u548c x^2+y^2-a*x=a*sqrt(x^2+y^2\uff09\u3002
\u6781\u5750\u6807\u65b9\u7a0b\uff1a
\u6c34\u5e73\u65b9\u5411\uff1a \u03c1=a(1-cos\u03b8) \u6216 \u03c1=a(1+cos\u03b8) (a>0)\u3002
\u5782\u76f4\u65b9\u5411\uff1a \u03c1=a(1-sin\u03b8) \u6216 \u03c1=a(1+sin\u03b8) (a>0)\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u5fc3\u5f62\u7ebf\u7684\u7531\u6765\uff1a
\u7b1b\u5361\u5c14\u5728\u7ed9\u514b\u91cc\u65af\u6c40\u5bc4\u51fa\u7b2c\u5341\u4e09\u5c01\u4fe1\u540e\u5c31\u6c14\u7edd\u8eab\u4ea1\u4e86\uff0c\u8fd9\u7b2c\u5341\u4e09\u5c01\u4fe1\u5185\u5bb9\u53ea\u6709\u77ed\u77ed\u7684\u4e00\u4e2a\u516c\u5f0f\uff1ar=a(1-sin\u03b8\uff09\u3002\u56fd\u738b\u770b\u4e0d\u61c2\uff0c\u89c9\u5f97\u4ed6\u4eec\u4fe9\u4e4b\u95f4\u5e76\u4e0d\u662f\u603b\u662f\u8bf4\u60c5\u8bdd\u7684\uff0c\u5c06\u5168\u57ce\u7684\u6570\u5b66\u5bb6\u53ec\u96c6\u5230\u7687\u5bab\uff0c\u4f46\u6ca1\u6709\u4e00\u4e2a\u4eba\u80fd\u89e3\u5f00\uff0c\u4ed6\u4e0d\u5fcd\u5fc3\u770b\u7740\u5fc3\u7231\u7684\u5973\u513f\u6574\u65e5\u95f7\u95f7\u4e0d\u4e50\uff0c\u5c31\u628a\u8fd9\u5c01\u4fe1\u4ea4\u7ed9\u4e00\u76f4\u95f7\u95f7\u4e0d\u4e50\u7684\u514b\u91cc\u65af\u6c40\u3002
\u516c\u4e3b\u770b\u5230\u540e\uff0c\u7acb\u5373\u660e\u4e86\u604b\u4eba\u7684\u610f\u56fe\uff0c\u5979\u9a6c\u4e0a\u7740\u624b\u628a\u65b9\u7a0b\u7684\u56fe\u5f62\u753b\u51fa\u6765\uff0c\u770b\u5230\u56fe\u5f62\uff0c\u5979\u5f00\u5fc3\u6781\u4e86\uff0c\u5979\u77e5\u9053\u604b\u4eba\u4ecd\u7136\u7231\u7740\u5979\uff0c\u539f\u6765\u65b9\u7a0b\u7684\u56fe\u5f62\u662f\u4e00\u9897\u5fc3\u7684\u5f62\u72b6\u3002\u8fd9\u4e5f\u5c31\u662f\u8457\u540d\u7684\u201c\u5fc3\u5f62\u7ebf\u201d\u3002
\u5fc3\u5f62\u7ebf\u7684\u9762\u79ef\uff1a
-pi<=t<=pi \u6216 0<=t<=2*pi\u3002
x=a*(2*cos(t)-cos(2*t))\u3002
y=a*(2*sin(t)-sin(2*t))\u3002
\u6240\u56f4\u9762\u79ef\u4e3a3/2*PI*a^2\uff0c\u5f62\u6210\u7684\u5f27\u957f\u4e3a8a\u3002
\u6240\u56f4\u9762\u79ef\u7684\u6c42\u6cd5\uff1a\u4ee5\u03c1=a(1+cos\u03b8)\u4e3a\u4f8b\u3002
\u4ee4\u9762\u79ef\u5143\u4e3adA\uff0c\u5219\uff1a
dA=1/2*a\u22272*(1+cos\u03b8)\u22272*d\u03b8\u3002
\u8fd0\u7528\u79ef\u5206\u6cd5\u4e0a\u534a\u8f74\u7684\u9762\u79ef\u5f97\uff1a
A=\u222b(\u03c0\u21920)1/2*a\u22272*(1+cos\u03b8)\u22272*d\u03b8\u3002
=3/4*a\u22272*\u03c0
\u6240\u4ee5\u6574\u4e2a\u5fc3\u5f62\u7ebf\u6240\u56f4\u6210\u7684\u9762\u79efS=2A=3/2*a\u22272*\u03c0\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5fc3\u5f62\u7ebf
水平方向： r=a(1-cosθ) 或 r=a(1+cosθ) (a>0)
垂直方向： r=a(1-sinθ) 或 r=a(1+sinθ) (a>0)
直角坐标方程
心形线的平面直角坐标系方程表达式分别为 x^2+y^2+a*x=a*sqrt(x^2+y^2) 和 x^2+y^2-a*x=a*sqrt(x^2+y^2）
参数方程
x=a*(2*cos(t)-cos(2*t))
y=a*(2*sin(t)-sin(2*t))
r = a(1 - sinθ)
绛旓細绗涘崱灏鐨蹇冨舰绾匡紙Cardioid锛夋槸涓绉嶆瀬鍧愭爣鏂圭▼锛屽彲浠ョ敤绗涘崱灏斿潗鏍囩郴琛ㄧず涓猴細x = a * (2 * cos(t) - cos(2t)) y = a * (2 * sin(t) - sin(2t))鍏朵腑锛(x, y)鏄績褰㈢嚎涓婄殑鐐圭殑绗涘崱灏斿潗鏍囷紝t鏄弬鏁帮紝a鏄竴涓父鏁帮紝鐢ㄦ潵鎺у埗蹇冨舰绾跨殑澶у皬銆傚績褰㈢嚎鏄竴涓棴鍚鏇茬嚎锛屽舰鐘剁被浼间簬蹇冭剰锛屽畠鐢...
绛旓細绗涘崱灏斿績褰 鏋佸潗鏍囨柟绋 姘村钩鏂瑰悜锛 r=a(1-cos胃) 鎴 r=a(1+cos胃) (a>0)鍨傜洿鏂瑰悜锛 r=a(1-sin胃) 鎴 r=a(1+sin胃) (a>0)鐩磋鍧愭爣鏂圭▼ 蹇冨舰绾跨殑骞抽潰鐩磋鍧愭爣绯绘柟绋嬭〃杈惧紡鍒嗗埆涓 x^2+y^2+a*x=a*sqrt(x^2+y^2) 鍜 x^2+y^2-a*x=a*sqrt(x^2+y^2锛夊弬鏁版柟绋 x=a*(...
绛旓細绗涘崱灏浜岀淮鍧愭爣绯婚噷鐨勬蹇鍏紡锛歳=a(1-sin胃锛夋瀬鍧愭爣鏂圭▼锛氭按骞虫柟鍚戯細 蟻=a(1-cos胃) 鎴 蟻=a(1+cos胃) (a>0)鍨傜洿鏂瑰悜锛 蟻=a(1-sin胃) 鎴 蟻=a(1+sin胃) (a>0)鐩磋鍧愭爣鏂圭▼锛蹇冨舰绾跨殑骞抽潰鐩磋鍧愭爣绯绘柟绋嬭〃杈惧紡鍒嗗埆涓 x^2+y^2+a*x=a*sqrt(x^2+y^2) 鍜 x^2+y^2-a*...
绛旓細鎵嶅厤鍏朵竴姝伙紝绗涘崱灏琚揩鍥炲埌娉曞浗銆傚湪娉曞浗鐨勭瑳鍗″皵涓鐩寸粰鍏嬮噷鏂眬鍐欎俊锛屼絾鎵鏈変俊浠跺潎琚浗鐜嬫墸鐣欍傜瑳鍗″皵鍐欎簡鏈鍚庝竴灏佷俊鍚庯紝姘旂粷韬骸銆傝繖灏佷俊鍙湁涓涓暟瀛鍏紡r=a(1-sin胃)銆傚浗鐜嬩笉瑙ｅ叾鎰忥紝鍏富鐪嬭繃鍏紡鍚庯紝椹笂鐢诲嚭涓涓編涓界殑蹇冨舰鏇茬嚎銆備絾閬楁喚鐨勬槸锛岃繖涓〃杈剧埍鎰忕殑缇庝附蹇冨舰锛岀珶鐒舵垚涓虹瑳鍗″皵鐨勭埍鎯呯粷绗斻
绛旓細浠栨棩澶滄濆康鍏富锛屾瘡澶╁潥鎸佺粰濂瑰啓淇★紝鐩兼湜鐫濂圭殑鍥為煶銆傚湪绗涘崱灏缁欏厠閲屾柉姹瀵勫嚭绗崄涓夊皝淇″悗锛屼粬姘歌繙鍦扮寮浜嗚繖涓笘鐣岋紝杩欐渶鍚庣殑涓灏佷俊涓婃病鏈夊啓涓鍙ヨ瘽锛屽彧鏈変竴涓柟绋嬪紡锛歳=a(1-sin胃)锛岃繖鏉鏇茬嚎灏辨槸钁楀悕鐨勨蹇冨舰绾库濄傝繖灏佷韩瑾変笘鐣岀殑鍙︾被鎯呬功锛岃嚦浠婅繕淇濆瓨鍦ㄦ娲茬瑳鍗″皵绾康棣嗛噷锛岀邯蹇电潃杩欐鍞編鐨勭埍鎯呫
绛旓細绗涘崱灏斿績褰绾匡紝涔熺О涓虹瑳鍗″皵蹇冨舰鎴栧績褰㈢嚎锛屾槸涓涓粡鍏哥殑浠ｆ暟鏇茬嚎锛屽畠鐨勬暟瀛﹁〃杈惧紡濡備笅锛歺² + y² = a² * (x² - y²)鍏朵腑锛- x 鍜 y 鏄瑳鍗″皵鍧愭爣绯讳腑鐨勫彉閲忥紝琛ㄧず骞抽潰涓婄殑鐐圭殑鍧愭爣銆- a 鏄績褰㈢嚎鐨勫弬鏁帮紝鎺у埗鐫蹇冨舰绾跨殑澶у皬鍜屽舰鐘躲傚績褰㈢嚎鐨勫悕瀛楁潵婧愪簬瀹冪殑...
绛旓細褰揳=1鏃讹紝蹇冪嚎鍛ㄩ暱涓8锛屽皝闂潰绉负3/2銆2.蹇冭剰绾夸篃鏄幆绾跨殑涓绉嶃3.鏇煎痉灏斿竷缃楀竷鏅腑闂寸殑鍥惧舰鏄竴鏉″績鑴忕嚎銆4.蹇冭剰绾库蹇冨舰鈥濈殑鑻辨枃鍚嶇О鐢眃e Castillon浜1741骞淬奝hilosophical Transactions of the Royal Society銆嬪勾鍑虹増 绗涘崱灏鐖辨儏鍧愭爣鍏紡鐨勭敾娉 棣栧厛锛屽畠涓嶆槸涓骞呯敾锛岃屾槸涓涓叕寮忋傜瑳鍗″皵鍧愭爣绯讳笅...
绛旓細鏁板鍙蹭笂娴极鏁板鍏紡鏄痳=a(1-sin胃锛夈傝В鏋愯繃绋嬶細r锛漚锛1-sin胃锛夎繖涓嚱鏁版湁涓や釜鍙橀噺锛屽彲瀵筧璧嬪硷紝鐒跺悗杩涜姹傝В銆傚嚱鏁板浘鍍忔槸蹇冨舰绾裤傝繖涓柟绋嬪張琚О涓衡绗涘崱灏鐨勭埍鎯呭潗鏍囧叕寮忊濄傚鍥炬墍绀猴紝鍒嗗埆鏄痑=1銆乤=2銆乤=3鏃剁殑鍥惧儚銆傚叕寮忔潵婧 绗涘崱灏斾簲鍗佸宀佺殑鏃跺欙紝娆ф床澶ч檰鐖嗗彂浜嗗彲鎬曠殑榛戞鐥咃紝浠栬杩祦娴...
绛旓細1銆佺洿瑙掑潗鏍囨柟绋 蹇冨舰绾跨殑骞抽潰鐩磋鍧愭爣绯绘柟绋嬭〃杈惧紡鍒嗗埆涓 x^2+y^2+a*x=a*sqrt(x^2+y^2) 鍜 x^2+y^2-a*x=a*sqrt(x^2+y^2锛夈2銆佹瀬鍧愭爣鏂圭▼ 姘村钩鏂瑰悜锛 蟻=a(1-cos胃) 鎴 蟻=a(1+cos胃) (a>0)鍨傜洿鏂瑰悜锛 蟻=a(1-sin胃) 鎴 蟻=a(1+sin胃) (a>0)鏋佸潗鏍囩郴涓嬬粯鍒...
绛旓細绗涘崱灏鐨蹇冨舰绾匡紙Cartesian Heart Curve锛夋槸涓绉嶇壒娈婄殑鏁板鏇茬嚎锛屽畠鐨勬柟绋嬪彲浠ョ敤绗涘崱灏斿潗鏍囩郴琛ㄧず涓猴細(x^2 + y^2 - 1)^3 - x^2 * y^3 = 0 鍏朵腑锛寈鍜寉鍒嗗埆浠ｈ〃绗涘崱灏斿潗鏍囩郴涓殑妯潗鏍囧拰绾靛潗鏍囥傝繖涓柟绋嬫弿杩颁簡涓涓績褰㈢殑鏇茬嚎锛屽叾褰㈢姸绫讳技浜庝袱涓浉浜ょ殑鍦嗗姬锛屼腑闂存湁涓涓皷灏栫殑鍑归櫡閮ㄥ垎锛屼娇鍏...
