Ka helu kaulike o ka pelane: pehea e hana? Kind pelane nā helu kaulike

Ke pelane makahiki hiki ke ho'ākāka 'ia i loko o naʻano likeʻole (kekahi kiko a me ka maʻi lele, i ka maʻi lele, a me ka mau kumu, ekolu kumu, etc.). He me kēia i loko o ka naau, i ka pelane helu kaulike hiki i nāʻano. Hoi kekahi poe kanaka ma lalo o kekahi mau kumu pelane paha e laua, ÷ haku, intersecting, etc. Ma keia a, e kamaʻilio pū i loko o kēia 'atikala. E aʻo mākou e hana i ka nui helu kaulike o ka pelane, aʻaʻole wale.

Ka maʻamau ano o ka helu kaulike

Ina R o ka makahiki _3, a loaʻa he mau huinahń loa i kahi kūlike o kahua XYZ. E hoakaka i ka maʻi lele α, a e e hookuu mai i ka ho omaka O. Ma ka pau ana o ka maʻi lele α huki pelane P i mea p ÷ haku ia mea.

Kû ke P ma ka ākeʻakeʻa kumu wahi Q = (m, y me, Z). Ke kahahńnai maʻi lele ana o ka wahi Q hoailona letter p. lākou, i ka lōʻihi o ka maʻi lele α P = IαI a me Ʋ = (cosα, cosβ, cosγ).

Keia pa alima maʻi lele, a ua kuhikuhi ia ma ke ala e hiki aku i maʻi lele α. α, β a me ka γ - poe pana pua i i hanaʻia ma waena o ka maʻi lele, a me ka maikaʻi kuhikuhi Ʋ makahiki lipi m, y me, Z niioaaonoaaiii. Ka wānana o ka wahi ma luna maʻi lele QεP Ʋ mea he ikaika i mea like i ka pa (p, Ʋ) = P (r≥0).

Ka luna helu kaulike i ka manaʻoi ka wā P = 0. Ka mea wale nō N pelane ma keia hihia, e kapa wahi O (α = 0), a o ka molekumu o, a me ka pa alima maʻi lele Ʋ, nolaila ia i kuu mai i ka wahi O e e p ÷ haku i ka P, eia nae kona kuhikuhi ana, a 'o ia hoʻi i ka maʻi lele Ʋ hoʻoholo mai ai i ka hoailona. Previous helu kaulike o mākou pelane P, kaheaia ma ka maʻi lele palapala. Akā, ma ka nānaina o kona wahi mea:

P, ua oi aku ma mua o paha e like me 0. mākou i loaʻa ai ka pelane helu kaulike i loko o maʻamau palapala.

O kekahi mau helu kaulike

Inā ka helu kaulike i loko o nā wahi hoonui ma kekahi helu i mea ole like ke 'Aʻohe, ua loaa i ka helu kaulike like paha a hiki i kēia mea Ho'ākāka' ka loa pelane. Ka mea, e loaʻa i ka penei ke ano:

Here, A, B, C - o ka helu o ka poʻe okoa mai Aʻohe. Keia helu kaulike ua kapaʻia ka helu kaulike o na mau palapala o ka pelane.

ʻO nā helu kaulike o na Kekoa. Special hihia

Ka helu kaulike hiki nui ke Hoʻololi 'ia me nā ana. E noonoo i kekahi o ia mau mea.

Kuhi i ke kaʻi lau waiwai A o 0. kēia Hōʻike i ka pelane laua a hiki i ka hoʻoholo ma mua iho (axis) bipi. Ma keia hihia, ke ano o ka helu kaulike hoʻololi: Wu + HA + D = 0.

Like me, ka palapala o ka helu kaulike a me E oko a me ke ana ma lalo nei:

• Mua, ina B = 0, ke helu kaulike hoʻololi i ke koʻi + HA + D = 0, a e hoike i ka parallelism a hiki i ka iho (axis) Oy.
• Ka lua, ina C = 0, ia iho me ka helu kaulike i loko o koʻi lipi + By + D = 0, i ka mea e olelo aku ai ia laua a hiki i ka hoʻoholo ma mua iho (axis) oz.
• Apana, ina D = 0, ka helu kaulike e hōʻike me ke koʻi + By + HA = 0, i makemake aha i ka pelane intersects O (i ke kumu).
• Ha, ina A = B = 0, ke helu kaulike hoʻololi i HA + D = 0, a e ho'āʻo i ka parallelism Oxy.
• Elima, ina B = C = 0, ka helu kaulike i koʻi lipi + D = 0, a 'o ia hoʻi i ka pelane mea ia laua ke Oyz.
• Sixthly, ina A = C = 0, ka helu kaulike i ka palapala Wu + D = 0, a laila, e hōʻike i ka parallelism Oxz.

Ano o ka helu kaulike i loko o Hoʻohana

Ma ka hihia ma ka huina A, B, C, D okoa mai Aʻohe, i ke ano o ka helu kaulike (0) i e like penei:

m / He + y me / b + Z / c. = 1,

kahi i = -D / A, e = -D / B, pela aku = -D / C.

E loaa like me ka hopena helu kaulike o ka pelane ma ka apana. It E e kaulana i keia pelane, e intersect i ka axis x ma ka wahi me nā wahi (ka, 0,0), Oy - (0, b, 0), a me oz - (0,0, mau).

Ua haawiia ka helu kaulike m / he + y me / b + Z / c. = 1, ia mea,ʻaʻole paʻakikī e visualize i ka hoʻokomo pelane pili i ka hoʻoholo ma mua i kahi kūlike o kahua.

Nā hŘnń ia o ka mea maʻamau maʻi lele

Ka mea maʻamau maʻi lele N a hiki i ka pelane P he wahi i mea na coefficients o ka mau helu kaulike o ka pelane, a laila n. (A, B, C).

I mea e ho oholo i ka wahi o ka mea maʻamau n, ia mea lawa paha kana e ike i ka mau helu kaulike haawiia pelane.

I ka hoʻohana 'ana i ka helu kaulike i loko o ka pauku i loaʻa ka palapala m / he + y me / b + Z / c. = 1, like me ka mea mau helu kaulike, ua hiki ke kākau i nā wahi a kekahi o ka maʻamau maʻi lele o ka haawiia pelane: (1 / he + 1 / b + 1 / c).

It E e hoʻomaopopo i ka mea maʻamau maʻi lele o kōkuaʻana, e hoʻoponopono i nā pilikia. Ka loa, he pono ole pilikia e oia i maopopo ÷ haku a laua Kekoa, i ka hana o ka loaa i ka pana pua ma waena o nā Kekoa a me ka pana pua ma waena o nā Kekoa a me ka pololei laina.

E kikokiko e like me ka pelane helu kaulike a me ka wahi o ka wahi maʻamau maʻi lele

A nonzero maʻi lele n, p ÷ haku i ka haawiia mai pelane, kapaia maʻamau (maʻamau) a hiki i ka hoʻoholo ma mua pelane.

Ina i loko o ka wahi makahiki (he mau huinahń loa i kahi kūlike o ka'ōnaehana pūnaewele) Oxyz hoʻonoho:

• Mₒ wahi me nā wahi (hₒ, uₒ, zₒ);
• 'Aʻohe maʻi lele N = A * au + B * hu + C * ke k.

Oe Pono e e helu kaulike o ka pelane, e hele ma Mₒ lae p ÷ haku i ka maʻamau n.

Ma ka makahiki, ua koho kekahi ākeʻakeʻa kumu wahi a Kû ke M (m, y me, Z). E ka kahahńnai maʻi lele o kela a me keia wahi M (m, y me, Z) e e R = m * au + y me * hu + Z * ke k, a me ka ke kahahńnai maʻi lele ana o ka wahi Mₒ (hₒ, uₒ, zₒ) - rₒ = hₒ * au + uₒ * hu + zₒ * ho. Ka wahi M, e pili ana i ka haawiia mai pelane, ina i ka maʻi lele MₒM e p ÷ haku i ka maʻi lele n. E kākau i ka ano o ka hoʻohana 'ana i ka orthogonality scalar huahana.

[MₒM, n] = 0.

No ka MₒM = R-rₒ, i ka maʻi lele helu kaulike o ka pelane, e nana me keia:

[R - rₒ, n] = 0.

Kēia helu kaulike hiki no hoi i kekahi shape. No keia mea, i ka waiwai o ka scalar huina hoonui, a ua hoohuliia mai i kaʻaoʻao hema o ka helu kaulike. [R - rₒ, n] = [R, n] - [rₒ, n]. Inā [rₒ, n] denoted me ke, ua loaa i ka kēia helu kaulike. [R, n] - He = 0 a [R, n] = mau, a expresses ke kūpaʻa o na nā wānana waiwai ma luna o ka mea maʻamau maʻi lele ana o ke kahahńnai o-vectors o ka haawi mai ai kumu e pili ana pelane.

Ano, e hiki ke kiʻi i ka i kahi kūlike 'ano hoʻopaʻa pelane mākou maʻi lele helu kaulike [R - rₒ, n] = 0. No ka mea, R-rₒ = (m-hₒ) * au + (y-uₒ) * hu + (Z-zₒ) * K, a n = A * au + B * hu + C * ke k, ua loaʻa:

Ua huli mai ia mākou i ka helu kaulike ua i hana pelane, maalo ae la oia ma ka wahi p ÷ haku i ka maʻamau n:

A * (m hₒ) + B * (y me uₒ) S * (Z-zₒ) = 0.

E kikokiko e like me ka pelane helu kaulike a me ka wahi o nā kumu o ka maʻi lele pelane collinear

E hoakaka mau ākeʻakeʻa kumu heluʻai M '(m', y me ', Z') a me ka M "(m", y ", Z"), e like me ka maʻi lele (he ', he ", he ‴).

Ano, ua hiki palapala helu kaulike hoʻoholo ma mua pelane, a hele ma ka na wahi M 'a me M ", a me kēlā wahi a me na wahi M (m, y me, Z) laua i ka maʻi lele i haawiia mai.

Pela M'M vectors He m = {m ', y-y me'; zz '} a me M "M = {m" -x', y me 'y me'; Z "-z '} e e coplanar me ka maʻi lele he = (he ', he ", he ‴), a' o ia hoʻi i (M'M M" M, he) = 0.

No laila, e nānā mākou helu kaulike o ka pelane ma ka makahiki e like keia:

ʻAno o ka pelane helu kaulike, e holo ekolu puan

E ka olelo aku mākou i ekolu points: (m ', y me', Z '), (m', y me ', Z'), (m ‴ ka mea, ua ‴, Z ‴), i hui ole ia i ka ia laina. He mea pono e kākau helu kaulike o ka pelane, maalo ae la oia ma na mea nui ekolu i hoakaka ia. anahonua kumumanaʻo hoʻopaʻapaʻa i kēiaʻano o ka pelane, aole ia nei, ka mea, ke pono hoʻokahi a me ka wale. No keia pelane intersects ka wahi (m ', y me', Z '), kona helu kaulike palapala makemake e:

Here, A, B, a me Cʻokoʻa mai Aʻohe i ka ia manawa. I haawiia mai pelane intersects elua hou heluʻai (m ", y", Z ") a me (m ‴, y me ‴, Z ‴). Ma keia pili e halihaliia oia iwaho i kēiaʻano o ana:

Ano, aole makou ke hana i kekahiʻaʻahu kahua o nā helu kaulike (laina) me unknowns u, V, w,:

I ko makou hihia m, y me a Z ku ākeʻakeʻa kumu wahi a satisfies helu kaulike (1). E noonoo pono i helu kaulike (1) a me ke ano o nā helu kaulike (2) a me (3) ka'ōnaehana o nā helu kaulike hōʻike i loko o ka huahelu ma luna, i ka maʻi lele satisfies N (A, B, C) i mea nontrivial. He no ka mea, o ka determinant o ka'ōnaehana mea Aʻohe.

Helu kaulike (1) ia makou loaʻa, keia mea i ka helu kaulike o ka pelane. 3 wahi oia maoli hele, a he mea paakiki ole e nānā. E hana i kēia, ua hoʻonui i ka determinant ma na kumu mua i loko o ka lalani mua. O ka mea na ka waiwai determinant penei ua mākou pelane ka poʻe intersects ke kope hoʻoholo ma mua wahi ekolu (m ', y me', Z '), (m ", y", Z "), (m ‴, y me ‴, Z ‴). No laila, ua hoʻoholo i ka hana ma mua o mākou.

Dihedral huina ma waena o nā Kekoa

Dihedral huina mea he māhele geometric shape i hanaʻia e nā hapalua-Kekoa i emanate mai ka pololei o ka laina. Ma na olelo e ae, kekahi hapa o ka makahiki i ua kaupalena ai i ka hapa-Kekoa.

Ina mākou i mau pelane, a me ka hope o nā helu kaulike.

Ua ike makou i ka maʻi lele N = (A, B, C) a me N¹ = (A¹, H¹, S¹) e like me ka hoʻoholo ma mua Kekoa i ÷ haku. Ma keia manao, o ka huina φ ma waena o vectors N a N¹ like huina (dihedral), i kahi ma waena o kēia mau Kekoa. Ua haawiia mai ia e ke scalar huahana.

NN¹ = | N || N¹ | cos φ,

kaʻoi loa no ka mea,

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

He nui iho la ia e noonoo i ka 0≤φ≤π.

Nae mau Kekoa i intersect, ano mau huina (dihedral): φ ₁ a me φ _2. Ko laua huina mea like e π (φ ₁ + φ ₂ = π). Like no kā lākou cosines, kā lākou mau loa nā loina e like, akā, ka mea, uaʻokoʻa hana mana,ʻo ia hoʻi, cos φ ₁ = -cos φ _2. Inā ma ka helu kaulike (0) Ua UAIAaIN e A, B a me C o -A, -B a me -C pakahi, i ka helu kaulike, ua loaa, e hooholo i ka ia pelane, ka mea wale huina φ ma ka helu kaulike cos φ = NN ¹ / | N || N ¹ | Ka mea, e auou caiaiai e π-φ.

Ka helu kaulike o ka p ÷ haku pelane

Kapaia ÷ haku pelane, ma waena o i ka huina o 90 degere. E ho ohana i ka mea hōʻike ma luna, ua hiki ke loaʻa i ka helu kaulike o ka pelane p ÷ haku i ka 'ē aʻe. Manao nei anei mākou i mau Kekoa: koʻi lipi + By + HA + D = 0, a + A¹h V¹u S¹z + + D = 0. Mākou ke i aku nei au ia mea, e orthogonal ina cos = 0. Kēia 'o ia hoʻi i NN¹ = AA¹ + VV¹ SS¹ + = 0.

Ka helu kaulike o ka laua pelane

Ua haawiia i elua laua Kekoa i pihaʻi meʻaʻohe kumu i loko, he pono ole.

Ke ano o ka laua Kekoa (lākou nā helu kaulike i ka mea like me ia ma ka mua paukū) mea i ka vectors N a me N¹, i mea p ÷ haku ia ia, collinear. Kēia 'o ia hoʻi i ka mea, ua halawai ka ana ma lalo nei proportionality:

A / A¹ = B / C = H¹ / S¹.

Inā ka hapaʻauhau hua'ōlelo i hoʻonui - A / A¹ = B / C = H¹ / S¹ = DD¹,

keia Hōʻike i ka 'ikepili pelane o ka ia. Kēia 'o ia hoʻi i helu kaulike koʻi + By + HA + D = 0, a + A¹h V¹u S¹z + + D¹ = 0 e wehewehe i kekahi pelane.

Ka mamao mai ka wahi e pelane

Ina mākou i ka pelane P, a ua haawiia mai e (0). He pono, e imi i ka mamao mai ka wahi me ka wahi (hₒ, uₒ, zₒ) = Qₒ. , Pono, e lawe mai i ka helu kaulike i loko o ka pelane II maʻamau nānā aku i ka hana ia.

(Ρ, V) = P (r≥0).

Ma keia hihia, ρ (m, y me, Z) o ka kahahńnai maʻi lele o ko mākou wahi Q, Aia ma luna o N pa - N o ka lōʻihi o ka p ÷ haku, i i hookuu ia mai ka 'Aʻohe wahi, V - o ka mokuna maʻi lele, a ua hoʻonohonoho i loko o ka ao ao o ka.

Ke koena unuhi ρ-ρº kahahńnai maʻi lele ana o ka wahi Q = (m, y me, Z), no ka N, a me ka ke kahahńnai maʻi lele o ka hāʻawi wahi Q ₀ = (hₒ, uₒ, zₒ) o ia maʻi lele he, ke kāpae cia o ka wānana o a ma luna o V lākou, i ka mamao n, i mea e pono e loaʻa mai Q = ₀ (hₒ, uₒ, zₒ) i ka P:

D = | (ρ-ρ ^0, V) |, akā,

(Ρ-ρ ^0, V) = (ρ, V ) - (ρ ^0, V) = P (ρ ^0, V).

No laila, e huli mai,

b = | (ρ ^0, V) P |.

Ano ka mea, ua ike ia e ho omaulia i ka mamao D mai ka ₀ a hiki i Q pelane P, ia mea pono e hoʻohana maʻamau nānaina pelane helu kaulike, i ka 'oni iki i ka hema o ka p, a me ka hope wahi o ka m, y me, Z ae pani (hₒ, uₒ, zₒ).

Pela, ua loaʻa i ka kāpae cia o ka kūponoʻana i ua koi d.

E ho ohana i ka mea kiko'î o ka 'ōlelo, ua loaa i ka maopopo:

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

Inā ka mea i hoakaka ia wahi Q ₀ mea ma ka kēlāʻaoʻao o ka pelane P like me ka molekumu o, laila, ma waena o ka maʻi lele ρ-ρ ⁰ a me ka V o ka obtuse huina, penei:

D = - (ρ-ρ ^0, V) = (ρ ^0, V) mutt> 0.

Ma ka hihia i ka wa a ka wahi Q ₀ loko o ka huipūʻana me ka ho omaka aia ma kaʻaoʻao hookahi o ka U, ua hana ka huʻi huina, i mea:

D = (ρ-ρ ^0, V) = P - (ρ ^0, V)> 0.

I ka hopena o ia i loko o ka mua hihia (ρ ^0, V)> p, i loko o ka lua o ka (ρ ^0, V)

A me kona tangent pelane helu kaulike

No ka pelane, a hiki i ka ili ma ka lae o ka tangency Mº - he pelane i loaʻa ka hiki a pau tangent i ka hālawa e holo ana ma ia wahi ma luna o kaʻili kai.

Me keiaʻilikai palapala o ka helu kaulike F (x, y me, Z) = 0 i loko o ka helu kaulike o ka tangent pelane tangent wahi Mº (hº, uº, zº) makemake e:

F _m (hº, uº, zº) (hº m) + F _m (hº, uº, zº) (uº y me) + F _m (hº, uº, zº) (Z-zº) = 0.

Inā ka 'ilikai ua i kūlike loa a Z = f (x, y me), a laila, ua ho'ākāka' ia ma ka helu kaulike ka tangent pelane:

Z-zº = f (hº, uº) (hº m) + f (hº, uº) (y me uº).

Ke kuʻina o nā Kekoa

I ekolu-dimensional la he mea kekahi i kahi kūlike o'ōnaehana (huinahń) Oxyz, hāʻawi mau Kekoa P 'a me ka P' 'ano a me ka mai ole coincide. Mai kekahi pelane, a mea i loko o kekahi mau huinahń loa i kahi kūlike o'ōnaehana i ho'ākāka 'ia e ka mau helu kaulike, ua kuhi ua N' a N "i ho'ākāka 'ia ma nā nā helu kaulike A'x + Vʻu S'z + + D' = 0 a me A" + B m '+ y me me "Z + D" = 0. Ma keia hihia mākou i maʻamau n. '(A', B ', C') o ka pelane P 'a me ka mea maʻamau n. "(A", B ", C") o ka pelane P'. E like me kā mākou pelane ka mea,ʻaʻole ia laua, a mai ole coincide, a laila, i kēia mau vectors i ole collinear. E ho ohana i ka 'ōlelo o ka makemakika, ke i keia kulana hiki ke palapalaia e like me: N' ≠ N "↔ (A ', B', C ') ≠ (λ * A", λ * In ", λ * C"), λεR. E hoopololei i ka laina e waiho nei ma ke kuʻina P 'a me ka P ", e e denoted ma ka palapala kekahi, ma keia hihia i = P' ∩ P".

a - he laina iloko olaila no ka ai iiiaanoaa o nā kumu i (pono) Kekoa P 'a me ka P ". Kēia 'o ia hoʻi i nā wahi o kekahi wahi pili ana i ka laina i kekahi, pono ka poʻe panai ai i na helu kaulike A'x + Vʻu S'z + + D' = 0 a me A "m + B '+ C y" Z + D "= 0. Kēia 'o ia hoʻi i nā wahi o ka wahi, e lilo i kekahi pāʻoihana o ke kēia nā helu kaulike.

I ka hopena o ia ka pāʻoihana (nohona ma) o keia nenoaiu o nā helu kaulike e hoʻoholo i nā wahi o kēlā me kēia o nā kumu ma ka laina a e hana e like me ka wahi o ka kuʻina P 'a me ka P ", a hooholo i ka laina ma ka i kahi kūlike o'ōnaehana Oxyz (huinahń loa) ka lewa.