心脏线的参数方程 心脏线方程
\u5fc3\u810f\u7ebf\u7684\u65b9\u7a0b\u5fc3\u810f\u7ebf\u7684\u65b9\u7a0b\u4e3a\uff1a
1\u3001\u6781\u5750\u6807\u65b9\u7a0b
\u6c34\u5e73\u65b9\u5411\uff1a \u03c1=a(1-cos\u03b8) \u6216 \u03c1=a(1+cos\u03b8) (a>0)
\u5782\u76f4\u65b9\u5411\uff1a \u03c1=a(1-sin\u03b8) \u6216 \u03c1=a(1+sin\u03b8) (a>0)
2\u3001\u76f4\u89d2\u5750\u6807\u65b9\u7a0b
\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
3\u3001\u53c2\u6570\u65b9\u7a0b
x=a*(2*cos(t)-cos(2*t))y=a*(2*sin(t)-sin(2*t))
\u6269\u5c55\u8d44\u6599\uff1a
\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\u5219dA=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
=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
\u57fa\u672c\u6027\u8d28\u3000\u3000\u5fc3\u810f\u7ebf\u662f\u5916\u6446\u7ebf\u7684\u4e00\u79cd\uff0c\u5176 n \u4e3a 2\u3002\u5b83\u4ea6\u53ef\u4ee5\u6781\u5750\u6807\u7684\u5f62\u5f0f\u8868\u793a\uff1a
\u3000\u3000r = 1 + cos \u03b8
\u3000\u3000\u8fd9\u6837\u7684\u5fc3\u810f\u7ebf\u7684\u5468\u957f\u4e3a 8\uff0c\u56f4\u5f97\u7684\u9762\u79ef\u4e3a3\u03c0/2\u3002
\u3000\u3000\u5fc3\u810f\u7ebf\u4ea6\u4e3a\u86b6\u7ebf\u7684\u4e00\u79cd\u3002
\u3000\u3000\u5728 Mandelbrot set \u6b63\u4e2d\u95f4\u7684\u56fe\u5f62\u4fbf\u662f\u4e00\u4e2a\u5fc3\u810f\u7ebf\u3002
\u3000\u3000\u5fc3\u810f\u7ebf\u7684\u82f1\u6587\u540d\u79f0\u201cCardioid\u201d\u662f de Castillon \u5728 1741\u5e74 \u7684\u300aPhilosophical Transactions of the Royal Society\u300b\u53d1\u8868\u7684\uff1b\u610f\u4e3a\u201c\u50cf\u5fc3\u810f\u7684\u201d\u3002\u65b9\u7a0b\u3000\u3000\u5728\u7b1b\u5361\u513f\u5750\u6807\u7cfb\u4e2d\uff0c\u5fc3\u810f\u7ebf\u7684\u53c2\u6570\u65b9\u7a0b\u4e3a\uff1a
\u3000\u3000x(t)=2r(cost-cos2t/2)
\u3000\u3000y(t)=2r(sint-sin2t/2)
\u3000\u3000\u5176\u4e2dr\u662f\u5706\u7684\u534a\u5f84\u3002\u66f2\u7ebf\u7684\u5c16\u70b9\u4f4d\u4e8e\uff08r\uff0c0\uff09\u3002
\u3000\u3000\u5728\u6781\u5750\u6807\u7cfb\u4e2d\u7684\u65b9\u7a0b\u4e3a\uff1a
\u3000\u3000\u03c1(\u03b8)=2r(1-cos\u03b8)
绛旓細鍙傛暟鏂圭▼锛歺=a*(2*cos(t)-cos(2*t)),y=a*(2*sin(t)-sin(2*t))鐩磋鍧愭爣鏂圭▼锛(x^2+y^2-2*a*x)^2=4*a^2*(x^2+y^2)
绛旓細蹇冭剰绾夸害涓鸿毝绾跨殑涓绉嶃傚湪 Mandelbrot set 姝ｄ腑闂寸殑鍥惧舰渚挎槸涓涓績鑴忕嚎銆傚績鑴忕嚎鐨勮嫳鏂囧悕绉扳淐ardioid鈥濇槸 de Castillon 鍦 1741骞 鐨勩奝hilosophical Transactions of the Royal Society銆嬪彂琛ㄧ殑锛涙剰涓衡滃儚蹇冭剰鐨勨濄備笁銆佹柟绋 鍦ㄧ瑳鍗″効鍧愭爣绯讳腑锛蹇冭剰绾跨殑鍙傛暟鏂圭▼涓猴細x(t)=a(2cost-cos2t)y(t)=a(...
绛旓細蹇冨舰绾匡紝鏄湁涓涓皷鐐圭殑澶栨憜绾裤備篃灏辨槸璇达紝涓涓渾娌跨潃鍙︿竴涓崐寰勭浉鍚岀殑鍦嗘粴鍔ㄦ椂锛屽渾涓婁竴鐐圭殑杞ㄨ抗灏辨槸蹇冭剰绾裤傚湪绗涘崱鍎垮潗鏍囩郴涓紝蹇冭剰绾跨殑鍙傛暟鏂圭▼涓猴細鍏紡1 鍏朵腑r鏄渾鐨勫崐寰勩傛洸绾跨殑灏栫偣浣嶄簬锛坮锛0锛夈傚湪鏋佸潗鏍囩郴涓殑鏂圭▼涓猴細鍏紡2 蹇冭剰绾 鍥涗釜鏈濈潃涓嶅悓鏂瑰悜鐨勫績鑴忕嚎 鍏充簬蹇冨舰绾夸紶闂伙細浼犻椈锛屾硶鍥...
绛旓細鍦ㄧ瑳鍗″効鍧愭爣绯讳腑锛蹇冭剰绾跨殑鍙傛暟鏂圭▼涓猴細x(t)=a(2cost-cos2t)y(t)=a(2sint-sin2t)涓鑸柟绋嬩负锛歺²+y²+ax=a*sqrt(x²+y²) 鍜 x²+y²-ax=a*sqrt(x²+y²锛夊湪鏋佸潗鏍囩郴涓殑鏂圭▼涓猴細蟻(胃)=2r(1+/-cos胃)P(胃)=2r(1+/-sin...
绛旓細x=rcos胃=a(1+cos胃)cos胃 y=rsin胃=a(1+cos胃)sin胃 (x,y)涓哄潗鏍囷紝胃涓鍙傛暟銆蹇冨舰绾锛屾槸涓涓渾涓婄殑鍥哄畾涓鐐瑰湪瀹冪粫鐫涓庡叾鐩稿垏涓斿崐寰勭浉鍚岀殑鍙﹀涓涓渾鍛ㄦ粴鍔ㄦ椂鎵褰㈡垚鐨勮建杩癸紝鍥犲叾褰㈢姸鍍忓績褰㈣屽緱鍚嶃傛瀬鍧愭爣鏂圭▼ 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=...
绛旓細鍦ㄦ暟瀛︾殑绗涘崱鍎垮潗鏍囦綋绯讳腑锛蹇冭剰绾跨殑杞ㄨ抗灞曠幇浜嗕竴绉嶇嫭鐗圭殑鍑犱綍缇庛傚畠鐨勫弬鏁版柟绋褰㈠紡濡備笅锛氬湪x杞村拰y杞寸殑浜ょ粐涓紝蹇冭剰绾跨殑x鍧愭爣琛ㄨ揪涓 x(t)=a(2cost-cos2t)锛岃繖閲岋紝a鏄渾鐨勫崐寰勶紝t鏄弬鏁帮紝鑰宑ost鍜宑os2t鐨勭粍鍚堝舰鎴愪簡鏇茬嚎鐨勫姩鎬佸彉鍖栥傜浉搴斿湴锛寉鍧愭爣鍒欐槸 y(t)=a(2sint-sin2t)锛宻int鍜宻in2t鐨...
绛旓細璇
绛旓細sin胃 锛坸锛寉锛変负鍧愭爣锛屛镐负鍙傛暟銆傚渾鐨勫弬鏁版柟绋 x=a+r cos胃 y=b+r sin胃锛埼糕垐 [0锛2蟺) 锛 (a,b) 涓哄渾蹇冨潗鏍囷紝r 涓哄渾鍗婂緞锛屛 涓哄弬鏁帮紝(x锛寉) 涓虹粡杩囩偣鐨勫潗鏍囥傛き鍦嗙殑鍙傛暟鏂圭▼ x=a cos胃銆 y=b sin胃锛埼糕垐[0锛2蟺锛夛級 a涓洪暱鍗婅酱闀 b涓虹煭鍗婅酱闀 胃涓哄弬鏁般
绛旓細鎸夌収濡備笅鏋佸潗鏍囨柟绋嬶紝鐒跺悗甯﹀叆涓嶅悓鍙傛暟鍗冲彲寰楀埌涓涓績鑴忕嚎鐢诲嚭鐨勫績褰傁=a(1+cos胃)锛堝績鍨嬫湞鍙筹級蟻=a(1-cos胃)锛堝績鍨嬫湞宸︼級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锛鍙傛暟鏂圭▼ -pi<=t<=pi 鎴 0<=t<=2*...
绛旓細鍦∕ATLAB涓紝杈撳叆涓嬪垪鎸囦护锛歩=-pi:0.1:pi;x=2.*(sin(i)-sin(2*i)./2);y=2.*(cos(i)-cos(i).^2);plot(x,y)鍗冲彲寰楀埌蹇冭剰绾銆
