问:一个曲面某点切平面的法向量方向余弦公式的问题? 怎么求一个空间曲面上一点的切平面的法向量与z轴的夹角的余弦?
\u5207\u5e73\u9762\u7684\u65b9\u5411\u4f59\u5f26\u7531F=f(x,y)-z\u5f97\u5230\u6cd5\u7ebf\u5411\u91cfn=(fx,fy,-1)\u8fd9\u4e2a\u662f\u6cd5\u7ebf\u5411\u91cf\u7684\u65b9\u5411\u4f59\u5f26\u4e0d\u5c31\u5e94\u8be5\u662fcosa=(fx)/[(1+fx^2+fy^2)^(1/2)]cosb=(fy)/[(1+fx^2+fy^2)^(1/2)]cosv=(-1)/[(1+fx^2+fy^2)^(1/2)]\u561b?
\u5148\u6c42\u51fa\u90a3\u70b9\u7684\u6cd5\u5411\u91cf\u597d\u50cf\u662f\u6c42\u504fx\uff0c\u504fy\uff0c\u504fz\u5bfc \u5e26\u5165\u90a3\u70b9\u5c31\u662f\u6cd5\u5411\u91cf \u7136\u540e\u8ddf\uff080\uff0c0\uff0c1\uff09\u6c42\u4f59\u5f26 \u8fd9\u4e2a\u76f4\u63a5\u516c\u5f0f\u5c31\u53ef\u4ee5\u4e86 \u6b22\u8fce\u91c7\u7eb3
斑竹说的意思是举个例子(1,1,2)与(-1,-1,-2)方向是不会改变的,我不太会表述。楼主做线代再方程解的时候应该有体会的由F=f(x,y)-z得到法线向量n=(fx,fy,-1)这个是法线向量的方向余弦不就应该是cosa=(fx)/[(1+fx^2+fy^2)^(1/2)]cosb=(fy)/[(1+fx^2+fy^2)^(1/2)]cosv=(-1)/[(1+fx^2+fy^2)^(1/2)]嘛?
是这样的:原本n=(fx,fy,-1) 这样的话-1表示n与z抽是呈钝角的。但是我们规定,为了方便起见,最好将n与z抽夹角变成锐角,所以。。。。。成了现在的(-fx,-fy,1)。。。。不知道这样解释懂否?[]
绛旓細璁鏇查潰鏂圭▼涓 F锛圶锛孻锛孼锛夈傚叾瀵筙 Y Z鐨勫亸瀵煎垎鍒负 Fx锛圶锛孻锛孼锛夛紝Fy锛圶锛孻锛孼锛 锛孎z锛圶锛孻锛孼锛夈傚皢鐐癸紙a锛宐锛宑锛変唬鍏ュ緱 n锛濓蓟Fx锛孎y锛孎z锛 锛鍒囧钩闈㈡硶鍚戦噺锛夈傚啀灏嗗垏鐐癸紙a锛宐锛宑锛変唬鍏ュ緱銆傚垏骞抽潰鏂圭▼Fx锛婏紙X锛峚锛夛紜Fy锛婏紙Y锛峛锛夛紜Fz锛圸锛峜锛夛紳0銆傦紙姹傚垏骞抽潰鏂圭▼鐨...
绛旓細濡傚浘鎵绀猴細
绛旓細娉曠嚎鏂圭▼锛歺−x0Fx(x0,y0,z0)=y−y0Fy(x0,y0,z0)=z−z0Fz(x0,y0,z0).鍒囧钩闈鏂圭▼锛欶x(x0,y0,z0)(x−x0)+Fy(x0,y0,z0)(y−y0)+Fz(x0,y0,z0)(z−z0)=0.娉ㄨ锛 蹇冧腑濮嬬粓鎯崇潃涓涓壒渚嬶紝鐞冮潰锛歺2+y2+z2=R2.鐨悆鏀惧湪鍦颁笂锛屽湴闈㈠氨...
绛旓細鍏堟眰鍑洪偅鐐鐨勬硶鍚戦噺濂藉儚鏄眰鍋弜锛屽亸y锛屽亸z瀵 甯﹀叆閭g偣灏辨槸娉曞悜閲 鐒跺悗璺燂紙0锛0锛1锛夋眰浣欏鸡 杩欎釜鐩存帴鍏紡灏卞彲浠ヤ簡 娆㈣繋閲囩撼
绛旓細*(1,2,1)锛1锛4a锛3a^2锛0锛屽緱a锛-1鎴-1/3锛屾墍浠ョ偣P鐨勫潗鏍囨槸(-1,1,-1)鎴(-1/3,1/9,-1/27)鎵浠ワ紝鏇茬嚎涓婄偣(-1,1,-1)鎴(-1/3,1/9,-1/27)澶勭殑鍒囩嚎骞宠浜庡钩闈+2y+z锛4 2銆佽鏇查潰涓婄偣P(a,b,c)澶勭殑鍒囧钩闈㈠钩琛屼簬骞抽潰x+4y+6z锛0锛屽垯鍒囧钩闈㈢殑娉曞悜閲骞宠浜(1,...
绛旓細姹鏇查潰鍦鏌愮偣鐨勫垏骞抽潰鍜娉曠嚎鏂圭▼鏂规硶濡備笅锛1銆佹洸闈㈡柟绋嬫槸y^2+z^2=2x銆傝鏇茬嚎鏂圭▼涓篎绛変簬0锛寉绛変簬0锛岄ザX杞存棆杞竴鍛紝鎵鐢熸垚鐨勬棆杞洸闈㈡柟绋嬪氨鏄疐绛変簬0锛岄ザz杞存棆杞竴鍛紝鎵鐢熸垚鐨勬棆杞洸闈㈡柟绋嬪氨鏄疐姝h礋sqrt绛変簬0銆2銆佺粫鍝釜杞存棆杞紝鏂圭▼涓摢涓彉閲忓氨涓嶅彉锛岃屽彟涓涓彉閲忔崲涓哄墿涓嬬殑涓や釜鍙橀噺鐨...
绛旓細杩欎釜娉曞悜閲忔柟鍚戞槸涓瀹氱殑锛屼絾鏄洜涓洪暱搴﹀彲鍙樺寲锛屽嵆锛岃娉曞悜閲忎负n锛屽垯浠绘剰闈為浂甯告暟a锛宎n閮借〃绀鸿骞抽潰娉曞悜閲
绛旓細璁剧┖闂鏇查潰鏂圭▼涓篎(x,y,z)=0 閭d箞瀹冨湪鐐(x0,y0,z0)澶勭殑鍒囧钩闈㈢殑娉曞悜閲鍙互琛ㄧず涓 n0=(F'x(x0,y0,z0),F'y(x0,y0,z0),F'z(x0,y0,z0))鎵浠ュ垏骞抽潰鏂圭▼涓 F'x(x0,y0,z0) (x-x0)+F'y(x0,y0,z0) (y-y0)+F'z(x0,y0,z0) (z-z0)=0 ...
绛旓細鏂规硶濡備笅锛1銆鏇查潰鐢辨柟绋婩锛坸锛寉锛寊锛=0鍐冲畾锛岀浉搴旂殑鏌愪竴鐐M鐨勬硶鍚戦噺浣犲彧闇瑕佸搴旂殑姹傚亸瀵兼暟灏卞彲浠ヤ簡銆2銆佺敱浜庢硶鍚戦噺鎵鍦ㄧ殑鏄竴鏉$洿绾匡紝鎵浠ユ柟鍚戞潵璁叉湁涓や釜锛屽鏋滄病鏈夌壒鍒姹備竴鑸槸鍙互闅忎究閫夋嫨鐨勶紝濡傛灉鏄潗鏍囩殑鏇查潰绉垎浠涔堢殑锛岄渶瑕佹敞鎰忎竴涓嬪拰xyz姝f柟鍚戜箣闂寸殑澶硅锛屽洜涓鸿繖鍏崇郴鍒伴潰绉姇褰辩殑姝h礋銆...
绛旓細濡傛灉鏄笉灏侀棴鐨鏇查潰锛屼竴鑸兘璇存洸闈笂渚ф垨鑰呬笅渚с傚皢鏇查潰鍐欐垚鍙傛暟鐨勫舰寮忥細z=f(x,y)锛屽啀姹傚畠鐨勫亸瀵兼暟锛∂f/∂x鍜∂f/∂y锛岃繖涓や釜鍚戦噺鏋勬垚浜鍒囧钩闈㈢殑涓缁勫熀锛屾墍浠娉曞悜閲=∂f/∂x脳∂f/∂y/||∂f/∂x脳∂f/∂y|...