The aone kuti pali ndege: bwanji kuti? Mitundu ndege maikwezhoni

Mu danga, ndege imatha kufotokozedwa m'njira zosiyanasiyana (mfundo imodzi ndi vector, mfundo ziwiri ndi vector, mfundo zitatu, etc.). Zili ndi izi m'maganizo kuti kulinganirana kwa ndege kungakhale ndi mitundu yosiyanasiyana. Komanso, ngati zikhalidwe zina zatha, ndege zingakhale zofanana, zozizwitsa, zotsutsana, ndi zina zotero. Tidzakambirana za izi m'nkhaniyi. Tidzaphunzira momwe tingapangire kulinganirana kwa ndege osati osati kokha.

Machitidwe oyenera a equation

Tiyerekeze kuti pali danga R ₃ lomwe lili ndi dongosolo logwirizana ndi XYZ. Fotokozerani vector α, yomwe idzamasulidwe kuchokera pachiyambi O. Kupyolera kumapeto kwa vector α kukoka ndege Π, yomwe idzakhala yophiphiritsira.

Timatanthawuza ndi Π mfundo yotsutsa Q = (x, y, z). Tikalemba tsamba la Q ndi kalata p. Pankhaniyi, kutalika kwa vector α kuli p = IαI ndi Ʋ = (cosa, cosβ, cosγ).

Ndilo vector yomwe imayendetsedwa kumbali, ngati vector α. Α, β ndi γ ndizomwe zimapangidwira pakati pa vector Ʋ ndi njira zabwino zokhudzana ndi malo x, y, z, motsatira. Kuwonetsa kwa mfundo ina QENP pa vector Ʋ ndi nthawi zonse yomwe ili yofanana ndi p: (p, Ʋ) = p (p≥0).

Kugwirizana uku kumapangitsa kuti p = 0. Ndege yokhayo P yomwe ilipoyi idzaphatikiza mfundo O (α = 0), yomwe imayambira, ndipo unit vector Ʋ yotulutsidwa kuchokera kumalo O idzakhala yeniyeni kwa Π, ngakhale kuti ikutsogolera, kutanthauza kuti vector Ʋ ikufotokozedwa ndi Kulondola kwa chizindikiro. Mgwirizano wapitawo ndi equation ya ndege Yathu II, yomwe ikuwonetsedwa mu vector form. Koma mu makonzedwe ake akuwoneka ngati awa:

P ndi wamkulu kapena wofanana ndi 0. Tapeza chiwerengero cha ndege mumlengalenga.

Msonkhano waukulu

Ngati mgwirizano mu makonzedwe ukuchulukitsidwa ndi nambala iliyonse yosalingana ndi zero, timapeza mgwirizano wofanana ndi wopatsidwa, womwe umapanga ndege yomweyo. Zidzawoneka ngati izi:

Pano A, B, C ndi manambala omwe sali nthawi. Mtsinje uwu umatchedwa kuti equation ya ndege yaikulu.

Miyeso ya ndege. Milandu yapadera

Ma equation mu mawonekedwe ambiri akhoza kusinthidwa pamaso pa zinthu zina. Tiyeni tione ena mwa iwo.

Tangoganizani kuti coefficient A ndi 0. Izi zikutanthauza kuti ndege yoperekedwayo ikufanana ndi oxeni Ox. Pachifukwa ichi mawonekedwe a equation adzasintha: Boo + Cz + D = 0.

Mofananamo, mawonekedwe a equation adzasintha pansi pazifukwa izi:

• Choyamba, ngati B = 0, equation ikasintha ku Ax + Cz + D = 0, yomwe idzakhala umboni wa kufanana kwa Oy Oy.
• Chachiwiri, ngati C = 0, ndiye kuti equation imasinthidwa kukhala Ax + Boo + D = 0, yomwe idzakhala yofanana kufanana ndi Oz.
• Chachitatu, ngati D = 0, equation idzawoneka ngati Ax + Boo + Cz = 0, zomwe zikutanthauza kuti ndege ikudutsa O (chiyambi).
• Chachinayi, ngati A = B = 0, ndiye kuti equation ikasintha ku Cz + D = 0, yomwe idzafanana ndi Oxy.
• Chachisanu, ngati B = C = 0, equation imakhala Ax + D = 0, kutanthauza kuti ndege yopita ku Oyz ikufanana.
• Chachisanu ndi chimodzi, ngati A = C = 0, ndiye equation idzatenga mawonekedwe a Boo + D = 0, ndiko kuti, idzafotokozera kufanana kwa Oxz.

Mtundu wa equation mu zigawo

Pankhaniyi pamene manambala A, B, C, D ali osiyana ndi zero, mawonekedwe a equation (0) akhoza kukhala motere:

X / a + y / b + z / c = 1,

Pomwe = =D / A, b = -D / B, c = -D / C.

Chotsatira chake, timapeza mgwirizano wa ndege mu zigawozo. Tiyenera kukumbukira kuti ndegeyi idzayendetsa mbali ya Ox kumbali ndi zolemba (a, 0,0), Oy - (0, b, 0), ndi Oz - (0,0, c).

Poganizira equation x / a + y / b + z / c = 1, sizili zovuta kufotokoza momwe dongosolo la ndege likuyendera pa dongosolo lokonzekera.

Maofesi a vector wabwino

Vector yachibadwa ndi ndege Π ili ndi zochitika zomwe zimagwirizanitsa ndi ndege, yomwe ndi n, A, B, C).

Kuti muzindikire zolumikizana zachizolowezi n, ndikwanira kudziwa momwe kulingalira kwa ndege yoperekedwa.

Pogwiritsa ntchito equation m'magulu, omwe ali ndi mawonekedwe x / a + y / b + z / c = 1, monga ndi equation, tikhoza kulemba makonzedwe a mtundu uliwonse wa ndege yopatsidwa: (1 / a + 1 / b + 1 / C).

Ndikoyenera kudziwa kuti vector yabwino imathandiza kuthetsa ntchito zosiyanasiyana. Mavuto omwe amapezeka kwambiri akuphatikizapo vuto lowonetsetsa kuti pali ndege, vuto la kupeza mapepala pakati pa ndege kapena mapulaneti pakati pa ndege ndi mizere.

Maonekedwe a ndege molingana ndi makonzedwe a mfundoyo ndi vector yachibadwa

Mphepete mwachitsulo chosagwirizana ndi ndege yomwe yapatsidwa imatchedwa yachibadwa (yachibadwa) kwa ndege yopatsidwa.

Tangoganizirani kuti mu malo okonzedwa bwino (octetular coordinate system) Oxyz amapatsidwa:

• Lembani Mₒ ndi makonzedwe (xₒ, yₒ, z );
• Zero vector ndi n = A * i + B * j + C * k.

Ndikofunika kulembetsa equation ya ndege, yomwe idzadutsa muzomwe M ___yomwe ikugwiritsidwa ntchito n.

Mu malo omwe timasankha mfundo iliyonse yosamveka ndikutanthauzira ndi M (xy, z). Lolani makwerero a mtundu uliwonse M (x, y, z) akhale r = x * i + y * j + z * k, ndi mawonekedwe a chithunzi Mₒ (xₒ, yₒ, zₒ) - rₒ = xₒ * i + yₒ * J + zₒ * k. Mfundo M idzakhala ya ndege yomwe inapatsidwa ngati vector MKM ili yosiyana ndi vector n. Tiyeni tilembere chikhalidwe cha chikhalidwe kudzera mwa mankhwala osokoneza bongo:

[MM, n] = 0.

Kuchokera MMM = r-rₒ, kugwirizana kwa ndege kumakhala ngati:

[R - rₒ, n] = 0.

Kugwirizana uku kungakhale ndi mawonekedwe ena. Kuti tichite izi, timagwiritsa ntchito katundu wa scalar mankhwala, ndipo mbali ya kumanzere ya equation amasinthidwa. [R - rₒ, n] = [r, n] - [rₒ, n]. Ngati [rₒ, n] idatchulidwa monga c, ndiye kuti zotsatirazi zikupezeka: [r, n] - c = 0 kapena [r, n] = c, zomwe zimasonyeza nthawi zonse zomwe zimagwiritsidwa ntchito pazithunzi zapakati pa ndege.

Tsopano titha kupeza mawonekedwe a makonzedwe a zolembera za ndege yathu [r - rₒ, n] = 0. Popeza r-rₒ = (x-xₒ) * i + (y-yₒ) * j + (z-zₒ) * k, ndi N = A * i + B * j + C * k, tili nawo:

Zili choncho kuti timakhala ndi kayendedwe ka ndege yomwe imadutsa pazomwe zimagwiritsidwa ntchito podutsa nthawi zonse.

A * (x - xₒ) + B * (y-yₒ) C * (z-zₒ) = 0.

Maonekedwe a ndege molingana ndi makonzedwe a zigawo ziwiri ndi vector, ndege yachilendo

Timafotokozera mfundo ziwiri zosamveka M '(x', y ', z') ndi M "(x", y ", z"), komanso vector (a ', a', a).

Tsopano tingathe kulembetsa ndondomeko ya ndege yomwe inaperekedwa, yomwe idzadutsa pazigawo zomwe zilipo M 'ndi M ", komanso mfundo iliyonse M ndi zigawo zogwirizana (x, y, z) zomwe zikugwirizana ndi vector.

Kuwonjezera pamenepo, zizindikiro za M'M = {x-x '; y-y'; zz '} ndi M "{=" -x "; y" -y'; z "-z '} ziyenera kukhala zojambula ndi vector A = (a,, a, a), ndipo izi zikutanthauza kuti (M'M, M, a) = 0.

Kotero, kulinganirana kwathu kwa ndege mu denga kudzawoneka ngati izi:

Maonekedwe a ndege ikuphatikizira mfundo zitatu

Tiyerekeze kuti tili ndi mfundo zitatu: (x ', y', z '), (x ", y", z "), (x ➝, y , z ) zomwe sizili mzere womwewo. Ndikofunikira kulemba kuwerengera kwa ndege kudutsa mu mfundo zitatu. Chiphunzitso cha geometry chimanena kuti ndege yotereyo imakhalapo, koma ndi yapadera komanso yosagwiritsidwa ntchito. Popeza ndegeyi ikuphatikizapo (x ', y', z '), mawonekedwe ake adzakhala motere:

Apa A, B, C ndi onse oseri. Ndege yomwe inapatsidwa imadutsa mfundo ziwiri: (x ", y", z ") ndi (x ➝, y , z ➝). Pachifukwa ichi, zikhalidwe zimenezi ziyenera kukwaniritsidwa:

Tsopano ife tikhoza kupanga dongosolo logwirizana logwirizana (loyambira) ndi osadziwika u, v, w:

Kwa ife, x, y kapena z ndi mfundo yosamveka yomwe imakhutitsa mgwirizano (1). Kuganizira equation (1) ndi dongosolo kuchokera ku equation (2) ndi (3), dongosolo la equation lomwe limasonyezedwa mu chithunzi pamwambapa limakhutiritsa vector N (A, B, C), yomwe ili yopanda malire. Ndicho chifukwa chake dongosolo ili ndi zero.

Kuyesa (1), komwe tinapeze, ndiko kulinganirana kwa ndege. Pambuyo pa mfundo zitatu, zikupita ndendende, ndipo n'zosavuta kufufuza. Kuti tichite izi, tifunika kufotokoza zomwe timachita pa mzere woyamba. Kuchokera pa zomwe zilipo kale, ndege yathu nthawi imodzi imatambasula zigawo zitatu zomwe poyamba zinaperekedwa (x ', y', z '), (x ", y", z "), (x ➝, y ➝, z ➝). Izi ndizo, tathetsa ntchito yomwe yatiika patsogolo pathu.

Mphepete mwa mbali ziwiri pakati pa ndege

Kona kozungulira kumbaliyi ikuimira chiwerengero cha mlengalenga chamakono chomwe chimapangidwa ndi ndege ziwiri. Mwa kuyankhula kwina, izi ndi gawo la malo omwe amamangidwira ndegeyi.

Tiyerekeze kuti tili ndi ndege ziwiri zomwe zili ndi izi:

Tikudziwa kuti zizindikiro za N = (A, B, C) ndi N¹ = (А¹, В¹, С¹) zimagwirizana ndi ndege zoperekedwa. Pogwirizana ndi izi, mbali φ pakati pa makina N ndi N¹ ndi ofanana ndi mbali (mbali ziwiri) yomwe ili pakati pa ndegezi. Zamakono zomwe zili ndi mawonekedwe:

NN¹ = | N || N¹ | cos φ,

Ndendende chifukwa

(² (²²) ² + (¹) ² (² (²) ² (¹) (¹) (² (²²) ² (¹) ² (¹ (²) ² (²) ²).

Zokwanira kukumbukira kuti 0≤φ≤π.

Ndipotu, ndege ziwiri zomwe zimayendayenda zimapanga maanga awiri ( ₁₎ ndi φ ₂ . Chiwerengero chawo chifanana ndi π (φ ₁ + φ ₂ = π). Zosowa zawo, zomwe zimayendera bwino ndizofanana, koma zimasiyana ndi chizindikiro, ndicho cos φ ₁ = -cos φ ₂ . Ngati titengapo A, B ndi C mwa nambala -A, -B ndi -C, motero, mu equation (0), ndiye kuti equation yomwe timapeza iwonetsa ndege yomweyi, yokha, φ mu equation cos φ = NN ¹ / | N || N ¹ | Adzalandidwa ndi π-φ.

Kufanana kwa ndege yopitirira

Zopeka ndizo ndege zomwe pakati pake pamakhala madigiri 90. Pogwiritsira ntchito mfundo zomwe tazitchula pamwambapa, tikhoza kupeza mayendedwe a ndege perpendicular to the other. Tiyerekeze kuti tili ndi ndege ziwiri: Ax + Boo + Cz + D = 0 ndi A¹x + Bуy + Czz + D = 0. Titha kunena kuti iwo adzakhala perpendicular ngati cosφ = 0. Izi zikutanthauza kuti NN¹ = AA + + BB¹ + CC¹ = 0.

Kufanana kwa ndege yofanana

Kufanana ndi ndege ziwiri zomwe ziribe mfundo zofanana.

Mkhalidwe wa kufanana kwa ndege (zofanana zawo ziri chimodzimodzi ndi ndime yapitayi) ndikuti zizindikiro za N ndi N1, zomwe ziri zogwirizana ndizo, ndizomwe zili m'mphepete mwa nyanja. Ndipo izi zikutanthauza kuti zikhalidwe zotsatirazi zikukhutira:

A / A¹ = B / B¹ = C / C¹.

Ngati chiwerengerochi chikuwonjezeka - A / A¹ = B / B¹ = C / C¹ = DD¹,

Izi zikusonyeza kuti ndegezi zimagwirizana. Izi zikutanthauza kuti equations Ax + Boo + Cz + D = 0 ndi A¹x + Bуy + Czz + D¹ = 0 amafotokoza ndege imodzi.

Kutalikirana kwa ndege kuchokera kumalo

Tiyerekeze kuti tili ndi ndege Π, yomwe imaperekedwa ndi equation (0). Ndikofunika kuti musapeze mtunda kuchokera pamalopo ndi zigawo (xₒ, yₒ, zₒ) = Q . Kuti tichite izi, tifunika kuchepetsa kugwirizana kwa ndege Π ku mawonekedwe oyenera:

(Ρ, v) = p (p≥0).

Pachifukwa ichi, ρ (x, y, z) ndi vesi yeniyeni ya mfundo Y yathu yomwe ili pa II, p ndi kutalika kwa P kupatulapo P yomwe inamasulidwa kuchokera ku zero, v ndilo vector yomwe ili pambali ya.

Kusiyanitsa ρ - ρ kwazithunzi zapakati pazigawo zonse Q = (x, y, z) za Π, komanso mawonekedwe a pulogalamu ya Q ₀ = (xₒ, yₒ, zₒ) ndivector yomwe ikuwonetseratu V ndi ofanana ndi mtunda d, umene uyenera kupezeka kuchokera pa Q ₀ = (xₒ, yₒ, zₒ) ku Π:

D = | (ρ-ρ ⁰ , v) |, koma

(Ρ-ρ ⁰ , v) = (ρ, v) - (ρ ⁰ , v) = ρ - (ρ ⁰ , v).

Kotero izo zikutuluka,

D = | (ρ ⁰ , v) -p |.

Tsopano tikuwona kuti kuti tiwerenge mtunda d kuchokera ku Q ₀ kupita ku ndege II, tiyenera kugwiritsa ntchito mawonekedwe a ndege, kenako tumizani ku dzanja lamanzere la p, ndikulowetsamo (xp, yp, zp) mmalo mwa x, y, z.

Kotero, ife tikupeza kufunikira kwathunthu kwa kufotokozera kumeneku, ndiko, kufunikira d.

Pogwiritsa ntchito chilankhulo cha magawo, timapeza momveka bwino:

D = | Axₒ + Vuₒ + Czₒ | / √ (A² + B² + C²).

Ngati mfundo Q ₀ ili kumbali ina ya ndege II, monga chiyambi, ndiye pakati pa vector ρ-ρ ⁰ ndi v paliponse paliponse :

D = - (ρ-ρ ⁰ , v) = (ρ ⁰ , v) -p> 0.

Pankhani yomwe mfundo ya Q ₀ pamodzi ndi magwero a makonzedwe ali pambali imodzi ya II, ndiye kuti malo opangidwawo ndi ovuta, ndiwo:

D = (ρ-ρ ⁰ , v) = ρ - (ρ ⁰ , v)> 0.

Zotsatira zake, zimakhala kuti pa choyamba (ρ ⁰ , v)> p, pachifukwa chachiwiri (ρ ⁰ , v)

Ndege yamtunduwu ndi yofanana

Ndege imafika pamwamba pamtunda wa M0 ndi ndege yomwe ili ndi zovuta zonse zomwe zimapezeka pamtundawu pamwambapa.

Ndi mawonekedwe a pamwamba equation F (x, y, z) = 0, mgwirizano wa ndege yaikulu pamtunda M0 (x, y, z0) udzawoneka ngati:

Fx ( _x °, yo, z0) (x - x0) + Fx (x0, y0, z0) (y-y0) + Fx (x0, y0, z0) (z-z0) = 0.

Ngati tifotokozera pamwamba pa fomu yeniyeni z = f (x, y), ndiye ndege yaikulu idzafotokozedwa ndi equation:

Z - z0 = f (x0, y0) (x - x0) + f (x0, y0) (y-y0).

Kusamvana kwa ndege ziwiri

Mu dera la magawo atatu ozungulira ozungulira (ophatikizira) Oxyz alipo, ndege ziwiri П 'ndi П' zimaperekedwa, zomwe zimadutsana komanso sizigwirizana. Popeza ndege iliyonse yokhala ndi makonzedwe ovomerezeka amagwiritsidwa ntchito ndi ofanana, timaganiza kuti Π 'ndi Π' zimaperekedwa ndi equations A'x + B'y + C'z + D '= 0 ndi A "x + B" y + Ndi "z + D" = 0. Pachifukwa ichi timakhala ndi n '(A', B ', C') ya ndege II 'ndi yachibadwa n "(A", B ", C") ya ndege II ". Popeza mapulaneti athu sali ofanana komanso osagwirizana, izi zimakhala sizinthu zozungulira. Pogwiritsa ntchito chinenero cha masamu, tingathe kulemba izi motere: n '≠ n "↔ (A', B ', C') ≠ (λ * A", λ * B ", λ * C"), λεR. Lembani mzere umene ukugona pamsewu wa П 'ndi П' uwonetsedwe ndi, p. = П '∩ P ".

A ndi mzere wopangidwa ndi zigawo zonse za (wamba) ndege II ndi II ". Izi zikutanthauza kuti makonzedwe a mfundo iliyonse ya mzere ayenera kuwonetsera panthawi imodzimodziyo Ax + B'y + C'z + D '= 0 ndi A "x + B" y + C "z + D" = 0. Choncho, makonzedwe a mfundoyi adzakhala yankho lenileni la kayendedwe ka zotsatirazi:

Zotsatira zake n'zakuti, zowonongeka (zofanana) zazomwezi zidzasintha ndondomeko ya mfundo zonse za mzere wolunjika, zomwe zidzakhala ngati mapepala a P 'ndi P', ndi kusankha mzere wolunjika mu oxyz (ophatikizira) mu malo.