{"id":94,"date":"2013-05-29T21:49:55","date_gmt":"2013-05-29T18:49:55","guid":{"rendered":"http:\/\/mednis.info\/wp\/?p=94"},"modified":"2014-07-23T09:41:00","modified_gmt":"2014-07-23T06:41:00","slug":"how-to-trajektorijas-pagrieziena-radiusa-aprekinasana-zinot-3-punktu-koordinates","status":"publish","type":"post","link":"http:\/\/mednis.info\/wp\/?p=94","title":{"rendered":"[How to] Trajektorijas pagrieziena r\u0101diusa apr\u0113\u0137in\u0101\u0161ana zinot 3 punktu koordin\u0101tes"},"content":{"rendered":"

\"pagrieziena_radiuss\"<\/a><\/p>\n

Punktu A, B, C koordin\u0101tes ir zin\u0101mas no m\u0113r\u012bjumiem\u2013 t\u0101s ir attiec\u012bgi x1, y1, x2, y2, x3, y3. Ri\u0146\u0137a l\u012bnijas centrs ir punkts O, kura koordin\u0101tes x, y nav zin\u0101mas. Izmantojot Pitagora teor\u0113mu, varam uzrakst\u012bt 3 vien\u0101dojumu sist\u0113mu, kas saista zin\u0101m\u0101s koordin\u0101tes ar x, y un mekl\u0113jamo r\u0101diusu r:<\/p>\n

(x – x1<\/sub>)2<\/sup> + (y – y1)2<\/sup> = r2<\/sup>
\n(x – x2<\/sub>)2<\/sup> + (y – y2)2<\/sup> = r2<\/sup>
\n(x – x3<\/sub>)2<\/sup> + (y – y3)2<\/sup> = r2<\/sup> <\/p>\n

Vien\u0101dojumu sist\u0113mu var risin\u0101t, piem\u0113ram, no pirm\u0101 vien\u0101dojuma at\u0146emot otro \u2013 ieg\u016bsim sakar\u012bbu y(x), ko t\u0101l\u0101k var ievietot tre\u0161aj\u0101 vien\u0101dojum\u0101. Tom\u0113r izteiksmes \u0101tri k\u013c\u016bst p\u0101r\u0101k garas apr\u0113\u0137iniem \u201car roku\u201d. Izmantosim vienu no br\u012bvi pieejamaj\u0101m datoralgebras programm\u0101m – Maxima (versija 5.27.0, http:\/\/maxima.sourceforge.net).<\/p>\n

Vien\u0101dojumu sist\u0113ma k\u0101 uzdevums programmai Maxima izskat\u0101s \u0161\u0101di:<\/p>\n

eq_1: (x - x1)^2 + (y - y1)^2 = r^2$
\neq_2: (x - x2)^2 + (y - y2)^2 = r^2$
\neq_3: (x - x3)^2 + (y - y3)^2 = r^2$
\nsolve ([eq_1, eq_2, eq_3],[x,y,r]); <\/code><\/p>\n

R\u0101diusam r programma sniedz risin\u0101jumu:<\/p>\n

r=(sqrt(y2^2*y3^4-2*y1*y2*y3^4+y1^2*y3^4+x2^2*y3^4-2*x1*x2*y3^4+x1^2*
\ny3^4-2*y2^3*y3^3+2*y1*y2^2*y3^3+2*y1^2*y2*y3^3-2*x2^2*y2*y3^3+4*x1*x2*y2*y3^3-2*x1^2*y2*y3^3-2*y1^3*y3^3-2*x2^2*y1*y3^3+4*x1*x2*y1*y3^3-2*x1^2*y1*y3^3+
\ny2^4*y3^2+2*y1*y2^3*y3^2-6*y1^2*y2^2*y3^2+2*x3^2*y2^2*y3^2-2*x2*x3*y2^2*y3^2-2*x1*x3*y2^2*y3^2+2*x2^2*y2^2*y3^2-2*x1*x2*y2^2*y3^2+2*x1^2*y2^2*y3^2+2*y1^3*y2*
\ny3^2-4*x3^2*y1*y2*y3^2+4*x2*x3*y1*y2*y3^2+4*x1*x3*y1*y2*y3^2+2*x2^2*y1*y2*y3^2-8*x1*x2*y1*y2*y3^2+2*x1^2*y1*y2*y3^2+y1^4*y3^2+2*x3^2*y1^2*y3^2-2*x2*x3*
\ny1^2*y3^2-2*x1*x3*y1^2*y3^2+2*x2^2*y1^2*y3^2-2*x1*x2*y1^2*y3^2+2*x1^2*y1^2*y3^2+2*x2^2*x3^2*y3^2-4*x1*x2*x3^2*y3^2+2*x1^2*x3^2*y3^2-2*x2^3*x3*y3^2+2*x1*x2^2*x3
\n*y3^2+2*x1^2*x2*x3*y3^2-2*x1^3*x3*y3^2+x2^4*y3^2-2*x1*x2^3*y3^2+2*x1^2*x2^2*y3^2-2*x1^3*x2*y3^2+x1^4*y3^2-2*y1*y2^4*y3+2*y1^2*y2^3*y3-2*x3^2*y2^3*y3+4*x1*
\nx3*y2^3*y3-2*x1^2*y2^3*y3+2*y1^3*y2^2*y3+2*x3^2*y1*y2^2*y3+4*x2*x3*y1*y2^2*y3-8*x1*x3*y1*y2^2*y3-4*x2^2*y1*y2^2*y3+4*x1*x2*y1*y2^2*y3+2*x1^2*y1*y2^2*y3
\n-2*y1^4*y2*y3+2*x3^2*y1^2*y2*y3-8*x2*x3*y1^2*y2*y3+4*x1*x3*y1^2*y2*y3+2*x2^2*y1^2*y2*y3+4*x1*x2*y1^2*y2*y3-4*x1^2*y1^2*y2*y3-2*x2^2*x3^2*y2*y3+4*x1*
\nx2*x3^2*y2*y3-2*x1^2*x3^2*y2*y3+4*x1*x2^2*x3*y2*y3-8*x1^2*x2*x3*y2*y3+4*x1^3*x3*y2*y3-2*x1^2*x2^2*y2*y3+4*x1^3*x2*y2*y3-2*x1^4*y2*y3-2*x3^2*y1^3*y3+4
\n*x2*x3*y1^3*y3-2*x2^2*y1^3*y3-2*x2^2*x3^2*y1*y3+4*x1*x2*x3^2*y1*y3-2*x1^2*x3^2*y1*y3+4*x2^3*x3*y1*y3-8*x1*x2^2*x3*y1*y3+4*x1^2*x2*x3*y1*y3-2*x2^4*y1*
\ny3+4*x1*x2^3*y1*y3-2*x1^2*x2^2*y1*y3+y1^2*y2^4+x3^2*y2^4-2*x1*x3*y2^4+x1^2*y2^4-2*y1^3*y2^3-2*x3^2*y1*y2^3+4*x1*x3*y1*y2^3-2*x1^2*y1*y2^3+y1^4*y2^2+2*
\nx3^2*y1^2*y2^2-2*x2*x3*y1^2*y2^2-2*x1*x3*y1^2*y2^2+2*x2^2*y1^2*y2^2-2*x1*x2*y1^2*y2^2+2*x1^2*y1^2*y2^2+x3^4*y2^2-2*x2*x3^3*y2^2-2*x1*x3^3*y2^2+2*x2^2*x3^2*y2^2
\n+2*x1*x2*x3^2*y2^2+2*x1^2*x3^2*y2^2-4*x1*x2^2*x3*y2^2+2*x1^2*x2*x3*y2^2-2*x1^3*x3*y2^2+2*x1^2*x2^2*y2^2-2*x1^3*x2*y2^2+x1^4*y2^2-2*x3^2*y1^3*y2+4*x2*x3*y1^3*
\ny2-2*x2^2*y1^3*y2-2*x3^4*y1*y2+4*x2*x3^3*y1*y2+4*x1*x3^3*y1*y2-2*x2^2*x3^2*y1*y2-8*x1*x2*x3^2*y1*y2-2*x1^2*x3^2*y1*y2+4*x1*x2^2*x3*y1*y2+4*x1^2*x2*x3
\n*y1*y2-2*x1^2*x2^2*y1*y2+x3^2*y1^4-2*x2*x3*y1^4+x2^2*y1^4+x3^4*y1^2-2*x2*x3^3*y1^2-2*x1*x3^3*y1^2+2*x2^2*x3^2*y1^2+2*x1*x2*x3^2*y1^2+2*x1^2*x3^2*y1^2-2*x2^3*
\nx3*y1^2+2*x1*x2^2*x3*y1^2-4*x1^2*x2*x3*y1^2+x2^4*y1^2-2*x1*x2^3*y1^2+2*x1^2*x2^2*y1^2+x2^2*x3^4-2*x1*x2*x3^4+x1^2*x3^4-2*x2^3*x3^3+2*x1*x2^2*x3^3+2*x1^2*x2*
\nx3^3-2*x1^3*x3^3+x2^4*x3^2+2*x1*x2^3*x3^2-6*x1^2*x2^2*x3^2+2*x1^3*x2*x3^2+x1^4*x3^2-2*x1*x2^4*x3+2*x1^2*x2^3*x3+2*x1^3*x2^2*x3-2*x1^4*x2*x3+x1^2*x2^4-2*x1^3
\n*x2^3+x1^4*x2^2))\/((2*x2-2*x1)*y3+(2*x1-2*x3)*y2+(2*x3-2*x2)*y1)<\/code><\/p>\n

Redzams, ka mekl\u0113to r\u0101diusu r var apr\u0113\u0137in\u0101t no 6 zin\u0101majiem skait\u013ciem – koordin\u0101t\u0113m x1, y1, x2, y2, x3, y3 \u2013 izmantojot tikai vienk\u0101r\u0161as aritm\u0113tisk\u0101s darb\u012bbas.
\nAtrisin\u0101jum\u0101 ir dal\u012b\u0161ana ar izteiksmi<\/p>\n

(2*x2-2*x1)*y3+(2*x1-2*x3)*y2+(2*x3-2*x2)*y1<\/p>\n

\u0160\u012bs izteiksmes v\u0113rt\u012bba k\u013c\u016bst vien\u0101da ar 0, ja visi tr\u012bs punkti A, B un C atrodas uz vienas taisnes \u2013 to var izt\u0113loties k\u0101 pagriezienu, kura r\u0101diuss ir bezgal\u012bgi liels.
\nVaram p\u0101rbaud\u012bt izteiksmes pareiz\u012bbu ar vienk\u0101r\u0161u piem\u0113ru programm\u0101 Maxima: <\/p>\n

x1: 1;
\ny1: 0;
\nx2: 0;
\ny2: 1;
\nx3: -1;
\ny3: 0;<\/code><\/p>\n

r: (sqrt(y2^2*y3^4-2*y1*y2*y3^4+y1^2*y3^4+x2^2*y3^4-2*x1*x2*y3^4+x1^2*
\ny3^4-2*y2^3*y3^3+2*y1*y2^2*y3^3+2*y1^2*y2*y3^3-2*x2^2*y2*y3^3+4*x1*x2*y2*y3^3-2*x1^2*y2*y3^3-2*y1^3*y3^3-2*x2^2*y1*y3^3+4*x1*x2*y1*y3^3-2*x1^2*y1*y3^3+
\ny2^4*y3^2+2*y1*y2^3*y3^2-6*y1^2*y2^2*y3^2+2*x3^2*y2^2*y3^2-2*x2*x3*y2^2*y3^2-2*x1*x3*y2^2*y3^2+2*x2^2*y2^2*y3^2-2*x1*x2*y2^2*y3^2+2*x1^2*y2^2*y3^2+2*y1^3*y2*
\ny3^2-4*x3^2*y1*y2*y3^2+4*x2*x3*y1*y2*y3^2+4*x1*x3*y1*y2*y3^2+2*x2^2*y1*y2*y3^2-8*x1*x2*y1*y2*y3^2+2*x1^2*y1*y2*y3^2+y1^4*y3^2+2*x3^2*y1^2*y3^2-2*x2*x3*
\ny1^2*y3^2-2*x1*x3*y1^2*y3^2+2*x2^2*y1^2*y3^2-2*x1*x2*y1^2*y3^2+2*x1^2*y1^2*y3^2+2*x2^2*x3^2*y3^2-4*x1*x2*x3^2*y3^2+2*x1^2*x3^2*y3^2-2*x2^3*x3*y3^2+2*x1*x2^2*x3
\n*y3^2+2*x1^2*x2*x3*y3^2-2*x1^3*x3*y3^2+x2^4*y3^2-2*x1*x2^3*y3^2+2*x1^2*x2^2*y3^2-2*x1^3*x2*y3^2+x1^4*y3^2-2*y1*y2^4*y3+2*y1^2*y2^3*y3-2*x3^2*y2^3*y3+4*x1*
\nx3*y2^3*y3-2*x1^2*y2^3*y3+2*y1^3*y2^2*y3+2*x3^2*y1*y2^2*y3+4*x2*x3*y1*y2^2*y3-8*x1*x3*y1*y2^2*y3-4*x2^2*y1*y2^2*y3+4*x1*x2*y1*y2^2*y3+2*x1^2*y1*y2^2*y3
\n-2*y1^4*y2*y3+2*x3^2*y1^2*y2*y3-8*x2*x3*y1^2*y2*y3+4*x1*x3*y1^2*y2*y3+2*x2^2*y1^2*y2*y3+4*x1*x2*y1^2*y2*y3-4*x1^2*y1^2*y2*y3-2*x2^2*x3^2*y2*y3+4*x1*
\nx2*x3^2*y2*y3-2*x1^2*x3^2*y2*y3+4*x1*x2^2*x3*y2*y3-8*x1^2*x2*x3*y2*y3+4*x1^3*x3*y2*y3-2*x1^2*x2^2*y2*y3+4*x1^3*x2*y2*y3-2*x1^4*y2*y3-2*x3^2*y1^3*y3+4
\n*x2*x3*y1^3*y3-2*x2^2*y1^3*y3-2*x2^2*x3^2*y1*y3+4*x1*x2*x3^2*y1*y3-2*x1^2*x3^2*y1*y3+4*x2^3*x3*y1*y3-8*x1*x2^2*x3*y1*y3+4*x1^2*x2*x3*y1*y3-2*x2^4*y1*
\ny3+4*x1*x2^3*y1*y3-2*x1^2*x2^2*y1*y3+y1^2*y2^4+x3^2*y2^4-2*x1*x3*y2^4+x1^2*y2^4-2*y1^3*y2^3-2*x3^2*y1*y2^3+4*x1*x3*y1*y2^3-2*x1^2*y1*y2^3+y1^4*y2^2+2*
\nx3^2*y1^2*y2^2-2*x2*x3*y1^2*y2^2-2*x1*x3*y1^2*y2^2+2*x2^2*y1^2*y2^2-2*x1*x2*y1^2*y2^2+2*x1^2*y1^2*y2^2+x3^4*y2^2-2*x2*x3^3*y2^2-2*x1*x3^3*y2^2+2*x2^2*x3^2*y2^2
\n+2*x1*x2*x3^2*y2^2+2*x1^2*x3^2*y2^2-4*x1*x2^2*x3*y2^2+2*x1^2*x2*x3*y2^2-2*x1^3*x3*y2^2+2*x1^2*x2^2*y2^2-2*x1^3*x2*y2^2+x1^4*y2^2-2*x3^2*y1^3*y2+4*x2*x3*y1^3*
\ny2-2*x2^2*y1^3*y2-2*x3^4*y1*y2+4*x2*x3^3*y1*y2+4*x1*x3^3*y1*y2-2*x2^2*x3^2*y1*y2-8*x1*x2*x3^2*y1*y2-2*x1^2*x3^2*y1*y2+4*x1*x2^2*x3*y1*y2+4*x1^2*x2*x3
\n*y1*y2-2*x1^2*x2^2*y1*y2+x3^2*y1^4-2*x2*x3*y1^4+x2^2*y1^4+x3^4*y1^2-2*x2*x3^3*y1^2-2*x1*x3^3*y1^2+2*x2^2*x3^2*y1^2+2*x1*x2*x3^2*y1^2+2*x1^2*x3^2*y1^2-2*x2^3*
\nx3*y1^2+2*x1*x2^2*x3*y1^2-4*x1^2*x2*x3*y1^2+x2^4*y1^2-2*x1*x2^3*y1^2+2*x1^2*x2^2*y1^2+x2^2*x3^4-2*x1*x2*x3^4+x1^2*x3^4-2*x2^3*x3^3+2*x1*x2^2*x3^3+2*x1^2*x2*
\nx3^3-2*x1^3*x3^3+x2^4*x3^2+2*x1*x2^3*x3^2-6*x1^2*x2^2*x3^2+2*x1^3*x2*x3^2+x1^4*x3^2-2*x1*x2^4*x3+2*x1^2*x2^3*x3+2*x1^3*x2^2*x3-2*x1^4*x2*x3+x1^2*x2^4-2*x1^3
\n*x2^3+x1^4*x2^2))\/((2*x2-2*x1)*y3+(2*x1-2*x3)*y2+(2*x3-2*x2)*y1);<\/code><\/p>\n

1
\n 0
\n 0
\n 1
\n -1
\n 0
\n1.0 <---- apr\u0113\u0137in\u0101tais r\u0101diuss ir pareizs.<\/code><\/p>\n

M\u0113\u0123in\u0101jumi vienk\u0101r\u0161ot atrisin\u0101jumu, savelkot l\u012bdz\u012bgos locek\u013cus vai iznesot pirms iekav\u0101m kop\u0113jos reizin\u0101t\u0101jus, nedod b\u016btiskus rezult\u0101tus. Apjom\u012bg\u0101 izteiksme var\u0113tu b\u016bt p\u0101rsteigums \u2013 k\u0101p\u0113c tik vienk\u0101r\u0161am uzdevumam ir tik sare\u017e\u0123\u012bts atrisin\u0101jums. Dom\u0101ju, ka izskaidrojums var\u0113tu b\u016bt taj\u0101, ka katrs no 6 skait\u013ciem (koordin\u0101t\u0113m) ietekm\u0113 rezult\u0101tu (r\u0101diusu) sare\u017e\u0123\u012bt\u0101 mijiedarb\u012bb\u0101 ar katru no p\u0101r\u0113jiem 5 skait\u013ciem, t\u0101d\u0113\u013c ar\u012b izteiksm\u0113 tik daudz locek\u013cu, kur katra koordin\u0101te reizin\u0101ta pati ar sevi un cit\u0101m koordin\u0101t\u0113m da\u017e\u0101d\u0101s pak\u0101p\u0113s.<\/p>\n

Mg.phys. Andris Mednis<\/p>\n","protected":false},"excerpt":{"rendered":"

Punktu A, B, C koordin\u0101tes ir zin\u0101mas no m\u0113r\u012bjumiem\u2013 t\u0101s ir attiec\u012bgi x1, y1, x2, y2, x3, y3. Ri\u0146\u0137a l\u012bnijas centrs ir punkts O, kura koordin\u0101tes x, y nav zin\u0101mas. Izmantojot Pitagora teor\u0113mu, varam uzrakst\u012bt 3 vien\u0101dojumu sist\u0113mu, kas saista zin\u0101m\u0101s koordin\u0101tes ar x, y un mekl\u0113jamo r\u0101diusu r: (x – x1)2 + (y – … Continue reading [How to] Trajektorijas pagrieziena r\u0101diusa apr\u0113\u0137in\u0101\u0161ana zinot 3 punktu koordin\u0101tes<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[15,6],"tags":[16,17,18],"_links":{"self":[{"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=\/wp\/v2\/posts\/94"}],"collection":[{"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=94"}],"version-history":[{"count":2,"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=\/wp\/v2\/posts\/94\/revisions"}],"predecessor-version":[{"id":98,"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=\/wp\/v2\/posts\/94\/revisions\/98"}],"wp:attachment":[{"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=94"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=94"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mednis.info\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}