Ko te whārite o te waka rererangi: me pehea ki te hanga? Types whārite plane

Ka taea te tautuhi i te wāhi rererangi i roto i ngā huarahi rerekē (kotahi ira me te pere, te vector me nga ngā rua, e toru ngā, me ētahi atu). Ko reira ki tenei i roto i te whakaaro, ka taea e whai i te whārite rererangi momo rerekē. i raro i hoki etahi tikanga kia rererangi whakarara, hāngai, rīpeka, me ētahi atu I runga i tenei a ka kōrero i roto i tenei tuhinga. Ka ako matou ki te hanga i te whārite whānui o te manureva, a e kore e anake.

Ko te ahua noa o te whārite

Tera pea R Ko te wāhi _3, e kua he tapawhā fakafekau'aki pūnaha xyz. tautuhi tatou he α pere, ka e kia tukua i te tīmatanga O. Na roto i te mutunga o nga α pere utu rererangi P i te mea hāngai ki reira.

Rawea, P i te noho pūwāhi Q = (x, y, z). Ko te pere pūtoro o te pūwāhi Q reta tohu p. Ko te roa o te pere ōrite α p = IαI me Ʋ = (cosα, cosβ, cosγ).

Tenei pere kōwae, whakahaua nei i roto i te aronga rite pere α. α, β me γ - he koki e kua hanga e i waenganui i te pere me nga tohutohu pai Ʋ toki wāhi x, y, z te whakaatu i. Ko te ngä o te pūwāhi i runga i pere QεP Ʋ ko te tamau i te mea e rite ana ki te p (p, Ʋ) = p (r≥0).

He whai kiko te whārite i runga, ka p = 0. Ko te rererangi n anake i roto i tenei take, e whiti ira e (α = 0), i te mea te takenga, me wae vector Ʋ, tukua i te e wāhi ka kia hāngai ki P, ahakoa tona aronga, e te tikanga e takoto te Ʋ pere ki runga ki te tohu. whārite o mua ko to tatou P rererangi, faaite i roto i te puka pere. Otiia i roto i te tirohanga o ona taunga ko:

he nui ake rite ki te 0. Kua kitea e matou te whārite rererangi i roto i te puka noa ranei P.

Ko te whārite whānui

Ki te te whārite i roto i te taunga tini i tetahi tau e kore he e rite ki te kore, te whiwhi tatou i te ōrite whārite ki tenei e tautuhi ana i te manureva rawa. ka whai te reira i te puka e whai ake nei:

Here, A, B, C - ko te maha o te wā kotahi rerekē i te kore. huaina ana tēnei whārite ko te whārite o te ahua whānui o te manureva.

Ko te whārite o te rererangi. take motuhake

te tikanga e taea te whakakē i te whārite ki te tikanga atu. A feruri i etahi o ratou.

Me kī e te whakarea te he 0. tohu tenei e te whakarara manureva ki te kau tuaka tomu'a. I roto i tenei take, te ahua o te whārite taui: Wu + CZ + D = 0.

Waihoki, ka rerekē te ahua o te whārite, me te ki te tikanga e whai ake nei:

• Tuatahi, ki te B = 0, te huringa whārite ki Tuaina + CZ + D = 0, e e tohu i te parallelism ki te tuaka Oy.
• Tuarua, ki te C = 0, kua puta te whārite ki Tuaina + Na + D = 0, e ko te ki te mea e pā ana ki whakarara ki te tuaka tomu'a Oz.
• Tuatoru, ki te D = 0, ka puta te whārite rite Tuaina + Na + CZ = 0, e pai te tikanga e pūtahi ana te manureva e (te take).
• A maha, ki te A = B = 0, te huringa whārite ki CZ + D = 0, e ka whakamatau ki te parallelism Oxy.
• A pae, ki te B = C = 0, riro te whārite Tuaina + D = 0, i te tikanga e he whakarara ki Oyz te manureva.
• Tuaono-, ki te A = C = 0, e te whārite i te puka Wu + D = 0, i.e., ka pūrongo ki te Oxz parallelism.

Puka o te whārite i roto i wāhanga

I roto i te take i reira tau A, B, C, rerekē D i te kore, i te ahua o te whārite (0) kia rite whai:

x / he + y / b + z / c = 1,

ai he = -D / A, b = -D / B, c = -D / C.

riro tatou rite te whārite hua o te manureva i roto i te mongamonga. Me kï reira e ka whakawhiti tenei waka rererangi te tuaka x-i te wāhi ki te taunga (he, 0,0), Oy - (0, b, 0), ko Oz - (0,0, s).

Homai te whārite x / he + y / b + z / c = 1, e kore he uaua ki te sioloto te whanaunga türanga rererangi ki te pūnaha tomu'a taunga.

Ko te taunga o te pere noa

Ko te pere n noa ki te P rererangi he taunga e te hunga i te whakarea o te whārite whānui o te manureva, i.e. n (A, B, C).

I roto i te tikanga ki te whakatau i te taunga o te n tonu, he reira rawaka ki mohio te whārite whānui homai rererangi.

A, no te whakamahi i te whārite i roto i wāhanga, e kua te puka x / he + y / b + z / c = 1, rite ina whakamahi i te whārite whānui e taea te tuhituhi taunga o tetahi pere noa te waka rererangi i homai: (1 / he + 1 / b + 1 / c).

Me kï reira e ki te pere noa o te āwhina whakaoti ngā raruraru. Kei te arā, i te raruraru tino noa i roto i rererangi hāngai whakarara ranei tohu, te mahi o te kimi i te koki i waenganui i te rererangi i te koki i waenganui i te rererangi me ngā rārangi torotika ranei.

Patoa rite ki te whārite rererangi me ngā taunga o te pūwāhi pere noa

He n kahorekore pere, hāngai ki te waka rererangi i homai, ka karanga noa (noa) ki te waka rererangi tomu'a.

Tera pea e i roto i te wāhi fakafekau'aki (he tapawhā fakafekau'aki pūnaha) Oxyz whakaturia:

• Mₒ wāhi ki taunga (hₒ, uₒ, zₒ);
• kore pere n = He * + i B * j + C * k.

Me koe ki te hanga whārite o te manureva e haere i roto i Mₒ wāhi hāngai ki te n noa.

I roto i te wāhi whiriwhiri tatou i tetahi wāhi noho, me te rawea, M (x, y, z). Kia te pere pūtoro o ia M pūwāhi (x, y, z) ka hei r = x * i + y * j + z * k, me te vector pūtoro o te Mₒ pūwāhi (hₒ, uₒ, zₒ) - rₒ = hₒ * + i uₒ * j + zₒ * k. Ka riro i te M ira ki te waka rererangi i homai, ki te kia hāngai ki te n pere te MₒM pere. tuhituhi tatou i te huru o te orthogonality te whakamahi i te hua scalar:

[MₒM, n] = 0.

Mai MₒM = r-rₒ, ka titiro i te whārite pere o te rererangi rite tenei:

[R - rₒ, n] = 0.

Ka taea hoki i tetahi āhua tenei whārite. Hoki tenei whakaaro, nga āhuatanga o te hua scalar, a tahuri te taha maui o te whārite. [R - rₒ, n] = [r, n] - [rₒ, n]. Ki te denoted [rₒ, n] rite s, whiwhi tatou i te whārite e whai ake nei: [r, n] - he = 0 [r, n] = s, e faaite te tamau o te haurangi ki runga ki te pere noa o te pūtoro-tahumaero o nga ngā homai e no manureva ranei.

Na e taea te tiki e koe i te taururuku momo tuhi rererangi to tatou whārite pere [r - rₒ, n] = 0. Mai r-rₒ = (x-hₒ) * i + (y-uₒ) * j + (z-zₒ) * k, me n = + i B * j + C * k te *, to tatou:

Huri i te reira i roto i taua whai tatou i hanga te whārite te rererangi e haere ana i roto i te wāhi hāngai ki te n noa:

He * (x hₒ) + B * (y uₒ) S * (z-zₒ) = 0.

Patoa rite ki te whārite rererangi me ngā taunga o rua ngā o te rārangi torotika pere rererangi

tautuhi tatou e rua ngā noho M '(x', y ', z') ko M "(x", y ", z"), kia rite ki te pai kia rite ki te pere (te ', he ", he ‴).

Na e taea tatou te tuhituhi whārite tomu'a rererangi e haere i roto i te M pūwāhi ngā 'ko M ", me ia wāhi ki te M taunga (x, y, z) whakarara ki te pere i homai.

Ko te kupu pere M'M x = {x ', y-y'; zz '} ko M "M = {x" -x', y 'y'; z "-z '} kia kia paparite ki te pere he = (he ', he ", he ‴), e te tikanga e (M'M M" M, he) = 0.

Na ka titiro i to tatou whārite o te rererangi i roto i te wāhi rite tenei:

Momo o whārite plane, whiti e toru ngā

Kia mea a to tatou e toru ngā: (x ', y', z '), (x', y ', z'), (x ‴ whai ‴, z ‴), e kore e nei no ki te raina taua. He mea tika ki te tuhituhi whārite o te waka rererangi e haere ana i roto i te ngā toru i tohua. Tohe āhuahanga ariā e tenei ahua o te rererangi e tīariari, te reira noa kotahi me anake. Mai tenei waka rererangi e pūtahi ana te pūwāhi (x ', y', z '), e kia tona puka whārite:

Here, He rerekē i kore i te wa taua A, B, me C. Hoki tapahi haere rererangi hoatu e rua atu ngā (x ", y", z ") me (x ‴, y ‴, z ‴). I roto i tenei hononga kia kawea i roto i tenei ahua o tikanga:

Na ka taea e tatou te hanga i te pūnaha ōrite o whārite (rārangi) ki taurangi u, v, w:

I roto i to tatou take x, y z ranei tu wāhi noho e makona whārite (1). Whakaaro whārite (1) me te pūnaha o ngā whārite (2) me te (3) i te pūnaha o ngā whārite tohua i roto i te ahua i runga, nga makona vector N (A, B, C) e te mea nontrivial. Ko reira hoki te tokoingoa o te pūnaha, ko te kore.

Whārite (1) e kua ka tatou, ko te whārite o te rererangi tenei. 3 ira ia tino haere, a te mea ohie ki te tirohia. Ki te mahi i tenei, te whakawhānui i tatou te tokoingoa i te āhuatanga i roto i te rarangi tuatahi. O nga āhuatanga ngā tokoingoa whai e tatou manureva wā kotahi e pūtahi ana te mata e toru këtia tuatahi (x ', y', z '), (x ", y", z "), (x ‴, y ‴, z ‴). Na faaoti matou ki te tūmahi i roto i te mua o tatou.

koki Dihedral i waenganui i nga rererangi

koki Dihedral Ko te āhua āhuahanga mokowā i hanga e rua hawhe-rererangi e heke i te rārangi tika. I roto i te mau parau te tahi atu wahi o te wāhi iti nei te ki te hawhe-rererangi,.

Tera pea i tatou e rua waka rererangi ki te whārite e whai ake nei:

E matau ana tatou e te N pere = (A, B, C) me te N¹ = (A¹, H¹, S¹) rite ki rererangi këtia e hāngai. I roto i tenei whakaaro, i te koki φ waenganui i pere N me N¹ rite koki (dihedral), kei nei i waenganui i enei rererangi. hoatu te hua scalar e:

NN¹ = | N || N¹ | cos φ,

ïa no te mea

cosφ = NN¹ / | N || N¹ | = (AA¹ + VV¹ SS¹ +) / ((√ (A² + s² + V²)) * (√ (A¹) ² + (H¹) ² + (S¹) ²)).

Ko reira nui ki te whakaaro i taua 0≤φ≤π.

Mau e rua rererangi e whakawhiti, puka e rua koki (dihedral): φ ₁ me φ _2. He rite ki te π (φ ₁ + φ ₂ = π) ratou moni. Ka rite ki hoki o ratou cosine, he o ratou uara pūmau rite, engari he rerekē tohu ratou, e ko, cos φ ₁ = -cos φ _2. Ki te i roto i te whārite (0) whakakapia e te, B me C o -E, -B ko -C aua, te whārite, whiwhi matou, ka whakatau i te taua rererangi, anake te koki φ i roto i cos whārite φ = nn ¹ / | N || N ¹ | Ka whakakapia te reira e π-φ.

Ko te whārite o te waka rererangi hāngai

Ka karanga hāngai rererangi, i waenganui i te koki, ko te 90 nekehanga. Mā te whakamahi i te rauemi aroaro i runga, ka taea e kitea tatou te whārite o te waka rererangi hāngai ki te tahi atu. Akuanei pea to tatou e rua rererangi: Tuaina + Na + CZ + D = 0, a + A¹h V¹u S¹z + + D = 0. Ka taea e tatou te mea e he poutū ratou ki te cos = 0. tikanga o tēnei e NN¹ = AA¹ + VV¹ SS¹ + = 0.

Ko te whārite o te waka rererangi whakarara

tuku te reira ki te rua rererangi whakarara e roto i kahore ngā i roto i te noa.

Ko te huru o rererangi whakarara (ratou whārite ko te taua rite i roto i te paratarafa o mua) ko e te pere N ko N¹, e he hāngai ki a ratou, rārani. tikanga o tēnei e kua tutaki te tikanga e whai ake nei proportionality:

A / A¹ = B / C = H¹ / S¹.

Ki te faaaano i te ngā hautanga - A / A¹ = B / C = H¹ / S¹ = DD¹,

tohu tenei e te waka rererangi raraunga o te taua. Tenei te tikanga e whārite Tuaina + Na + CZ + D = 0 me + A¹h V¹u S¹z + + D¹ = 0 whakaahua kotahi waka rererangi.

Ko te tawhiti i te wāhi ki te waka rererangi

Pea to tatou he P plane, homai nei e (0). He mea tika ki te kitea te tawhiti i te wāhi ki te taunga (hₒ, uₒ, zₒ) = Qₒ. , Me koe ki te kawe i te whārite i roto i te manureva II ahua noa ki te hanga i taua mea:

(Ρ, v) = p (r≥0).

I roto i tenei take, ρ (x, y, z) ko te pere pūtoro o to tatou Q pūwāhi, kei runga i n p - n ko te roa o te hāngai, i tukua i te wāhi kore, v - ko te pere wae, whakaritea nei i roto i te ahunga i te.

Ko te rerekētanga ρ-ρº pere pūtoro o te Q pūwāhi = (x, y, z), no ki n me te pere pūtoro o te ira i homai Q ₀ = (hₒ, uₒ, zₒ) ko te pere taua, te uara pū o te ngä o nei i runga i v ōrite te tawhiti d, e te mea e tika ana ki te kitea i Q = ₀ (hₒ, uₒ, zₒ) ki P:

D = | (ρ-ρ ^0, v) |, engari

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

Na huri i te reira i roto i,

d = | (ρ ^0, v) p |.

Na ko reira mārama e ki te tātai i te tawhiti d i ₀ ki Q rererangi P, he mea e tika ana ki te whakamahi i te whārite view rererangi noa, te neke ki te maui o p, me te wahi whakamutunga o te x, y, z whakakapi (hₒ, uₒ, zₒ).

Ko te kupu, kitea tatou te uara pū o te faaiteraa hua e hiahiatia ana e te d.

Mā te whakamahi i te tawhā o te reo, te tiki matou te kitea:

d = | Ahₒ Vuₒ + + Czₒ | / √ (A² + V² + s²).

Ki te te wāhi Q i tohua ₀ Ko i te tahi atu taha o te P rererangi rite te takenga, ka waenganui i te pere ρ-ρ ⁰ me v ko te koki hāpūpū, te kupu:

d = - (ρ-ρ ^0, v) = (ρ ^0, v) -p> 0.

I roto i te take, ka te wāhi Q ₀ roto i te taha ki te takenga kei i te taha ano o te U, hanga te koki tāhapa ko, e ko:

d = (ρ-ρ ^0, v) = p - (ρ ^0, v)> 0.

Ko te hua ko e i roto i te take o mua (ρ ^0, v)> p, i roto i te rua o (ρ ^0, v)

Na tona whārite pātapa rererangi

Mo te waka rererangi ki te mata i te mata o te tangency Mº - he waka rererangi kei roto pātapa katoa taea ki te ānau unu i roto i taua wāhi i runga i te mata.

Ki tenei puka mata o te whārite F (x, y, z) = 0 i roto i te whārite o te Mº pūwāhi pātapa rererangi pātapa (hº, uº, zº) e kia:

F _x (hº, uº, zº) (hº x) + F _x (hº, uº, zº) (uº y) + F _x (hº, uº, zº) (z-zº) = 0.

Ki te whakaturia e te mata te āta z = f (x, y), ka whakaahuatia te rererangi pātapa e te whārite:

z-zº = f (hº, uº) (hº x) + f (hº, uº) (y uº).

Ko te pūtahitanga o rua rererangi

I roto i te wāhi e toru-ahu ko te pūnaha fakafekau'aki hoatu e rua rererangi P 'me te P' e īnaki me kore e hāngai (tapawhā) Oxyz,. Mai tetahi manureva, i te mea i roto i te tapawhā fakafekau'aki pūnaha tautuhia e te whārite whānui, amo tatou e n 'me n "e tautuhia e te whārite A'x + V'u S'z + + D' = 0 me te" + B x '+ y ki te "z + D" = 0. I roto i tenei take to tatou n noa '(A', B ', C') o te rererangi P 'me te n noa "(A", B ", C") o te P plane'. Ka rite ki to tatou manureva e kore e faitatau a kahore e hāngai, ka kore e rārani enei pere. Mā te whakamahi i te reo o pāngarau, to tatou e taea te tuhituhi tenei huru rite: n '≠ n "↔ (A', B ', C') ≠ (λ * A", λ * I roto i ", λ * C"), λεR. Kia te rārangi tika e takoto i te pūtahitanga P 'me te P "ka, kia denoted e te pukapuka he, i roto i tenei take i te = P' ∩ P".

a - he aho arā, o te plurality o ngā (noa) rererangi P 'me P ". tikanga o tēnei e te taunga o tetahi wāhi no ki te he raina, me te wā kotahi makona i te whārite A'x + V'u S'z + + D '= 0 me te "x + B' + C y" z + D "= 0. tikanga o tēnei e te taunga o te wāhi ka waiho he otinga ngā o te whārite e whai ake nei:

Ko te hua ko e te otinga (whānui) o pūnaha o ngā whārite tenei ka whakatau i te taunga o ia o ngā i runga i te raina e ka mahi rite te mata o te pūtahitanga P 'me P ", a ka whakatau i roto i te pūnaha fakafekau'aki Oxyz (tapawhā) wāhi he aho.