Cov kab zauv ntawm lub dav hlau: yuav ua li cas ua? Hom dav hlau sib npaug

Lub dav hlau qhov chaw no tau hais tseg nyob rau hauv txoj kev sib txawv (ib teev thiab vector, lub vector thiab lub ob lub ntsiab lus, peb cov ntsiab lus, thiab lwm yam). Nws yog nrog no nyob rau hauv lub siab, lub dav hlau kab zauv yuav muaj ntau hom. Tsis tas li ntawd nyob rau hauv tej yam tej yam kev mob dav hlau tej zaum yuav thaum uas tig mus, perpendicular, kev sib tshuam, thiab lwm yam Nyob rau qhov no thiab yuav tham nyob rau hauv no tsab xov xwm. Peb yuav kawm tau mus ua lub general kab zauv ntawm lub dav hlau thiab tsis yog.

Lub daim ntawv ntawm cov kab zauv

Piv txwv R yog qhov chaw _3, uas muaj ib lub coordinate system XYZ. Peb txhais tau ib tug vector α, uas yuav raug tso tawm los ntawm lub starting taw tes O. Los ntawm lub kawg ntawm lub vector α kos dav hlau P uas yog perpendicular mus rau nws.

Txhais P nyob rau ntawm ib arbitrary point Q = (x, y, z). Lub voos kheej-kheej vector ntawm tus taw tes Q kos npe rau tsab ntawv p. Qhov ntev ntawm lub vector sib npaug α p = IαI thiab Ʋ = (cosα, cosβ, cosγ).

Qhov no unit vector, uas yog qhia nyob rau hauv cov kev taw qhia raws li vector α. α, β thiab γ - yog cov ces kaum uas yog tsim nyob rau nruab nrab ntawm cov vector thiab qhov zoo qhia Ʋ chaw axes x, y, z feem. Lub projection ntawm ib tug taw tes nyob rau hauv vector QεP Ʋ yog ib qhov uas yog sib npaug zos rau p (p, Ʋ) = p (r≥0).

Qhov saum toj no kab zauv nqis thaum p = 0. Lub tsuas n dav hlau nyob rau hauv cov ntaub ntawv no, yuav hla taw tes O (α = 0), uas yog lub hauv paus chiv keeb, thiab chav tsev vector Ʋ, tso tawm los ntawm tus taw tes O yuav perpendicular mus rau P, txawm hais tias nws kev taw qhia, uas txhais tau tias cov vector Ʋ txiav txim mus txog rau lub kos npe rau. Previous kab zauv yog peb dav hlau P, qhia nyob rau hauv vector daim ntawv. Tab sis nyob rau hauv view ntawm nws coordinates yog:

P yog ntau dua los yog sib npaug zos rau 0. Peb tau pom lub dav hlau kab zauv nyob rau hauv ib txwm daim ntawv.

Cov kev kab zauv

Yog hais tias lub equation nyob rau hauv lub coordinates muab los ntawm tej yam muaj pes tsawg tus uas yog tsis sib npaug zos rau zero, peb muab tau cov kab zauv sib npaug rau no uas qhia cov heev dav hlau. Nws yuav muaj cov nram qab daim ntawv:

Ntawm no, A, B, C - yog tus naj npawb ntawm ib txhij sib txawv ntawm xoom. Qhov no kab zauv yog hu ua tus kab zauv ntawm lub dav daim ntawv ntawm lub dav hlau.

Lub sib npaug ntawm lub dav hlau. Tshwj xeeb mob

Cov kab zauv yuav feem ntau yuav hloov nrog ntxiv tej yam kev mob. Xav txog tej ntawm lawv.

Xav hais tias cov coefficient A yog 0. Qhov no indicates tias lub dav hlau thaum uas tig mus rau lub predetermined axis twm. Nyob rau hauv cov ntaub ntawv no, rau hauv daim ntawv ntawm cov kab zauv hloov: Wu + CZ + D = 0.

Ib yam li ntawd, hauv daim ntawv ntawm kab zauv thiab yuav ua raws li qhov nram qab no tej yam kev mob:

• Firstly, Yog hais tias B = 0, kab zauv hloov mus rau Ax + CZ + D = 0, uas yuav qhia tau tias lub parallelism mus rau lub axis Oy.
• Secondly, yog tias C = 0, kab zauv yog hloov mus rau Ax + By + D = 0, uas yog hais txog thaum uas tig mus rau lub predetermined axis Oz.
• Peb, Yog hais tias D = 0, kab zauv yuav tshwm sim raws li Ax + By + CZ = 0, uas yuav txhais tau hais tias lub dav hlau tshuam O (lub hauv paus chiv keeb).
• Plaub, yog A = B = 0, kab zauv hloov mus rau CZ + D = 0, uas yuav ua pov thawj rau parallelism Oxy.
• Fifth, Yog hais tias B = C = 0, kab zauv yuav Ax + D = 0, uas txhais tau tias lub dav hlau yog thaum uas tig mus rau Oyz.
• Sixthly, yog A = C = 0, kab zauv yuav siv sij hawm rau hauv daim ntawv Wu + D = 0, i.e., yuav qhia mus rau lub parallelism Oxz.

Daim ntawv ntawm cov kab zauv nyob rau hauv feem

Nyob rau hauv lub rooj plaub uas tus xov tooj A, B, C, D txawv ntawm xoom, rau hauv daim ntawv ntawm kab zauv (0) tej zaum yuav raws li nram no:

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

nyob rau ib = -D / A, b = -D / B, c = -D / C.

Peb yuav tau txais raws li ib tug tshwm sim kab zauv ntawm lub dav hlau nyob rau hauv daim. Nws yuav tsum tau muab sau tseg hais tias qhov no dav hlau yuav tshuam tus x-axis ntawm tus taw tes nrog coordinates (a, 0,0), Oy - (0, b, 0), thiab Oz - (0,0, s).

Muab cov kab zauv x / a + y / b + z / c = 1, nws tsis yog nyuaj rau pom qhov kev tso kawm dav hlau txheeb ze mus rau ib tug predetermined tswj system.

Lub coordinates ntawm lub cev vector

Lub vector n mus rau lub dav hlau P muaj coordinates uas yog cov coefficients ntawm lub dav kab zauv ntawm lub dav hlau, i.e. n (A, B, C).

Nyob rau hauv thiaj li yuav txiav txim seb lub coordinates ntawm lub cev n, nws yog txaus kom paub tias cov general kab zauv muab dav hlau.

Thaum uas siv cov kab zauv nyob rau hauv feem, uas muaj rau hauv daim x / a + y / b + z / c = 1, raws li thaum uas siv cov kev kab zauv yuav sau coordinates ntawm tej qub vector ib tug muab dav hlau: (1 / ib tug + 1 / b + 1 / c).

Nws yuav tsum tau muab sau tseg tias ib txwm vector ntawm pab daws tau ntau yam teeb meem. Feem ntau cov teeb meem yog muaj nyob rau hauv cov ntawv pov thawj perpendicular los yog thaum uas tig mus dav hlau, cov hauj lwm ntawm kev nrhiav cov ces kaum ntawm lub dav hlau los yog cov ces kaum ntawm lub dav hlau thiab cov kab ncaj nraim.

Ntaus raws li lub dav hlau kab zauv thiab coordinates ntawm tus taw tes ib txwm vector

Ib tug nonzero vector n, perpendicular mus rau ib tug muab dav hlau, hu ua dab tsi (tej) mus rau ib tug predetermined dav hlau.

Piv txwv tias nyob rau hauv lub coordinate qhov chaw (ib lub coordinate system) Oxyz teev:

• Mₒ point nrog coordinates (hₒ, uₒ, zₒ);
• zero vector n = A * kuv + B * j + C * k.

Koj yuav tsum ua kab zauv ntawm lub dav hlau uas kis tau los ntawm Mₒ point perpendicular mus rau lub qub n.

Nyob rau hauv lub qhov chaw peb xaiv tej arbitrary point thiab txhais M (x, y, z). Cia lub voos kheej-kheej vector ntawm txhua point M (x, y, z) yuav r = x * kuv + y * j + z * k, thiab cov voos kheej-kheej vector ntawm ib tug taw tes Mₒ (hₒ, uₒ, zₒ) - rₒ = hₒ * kuv + uₒ * j + zₒ * k. Tus taw tes M yuav muaj mus rau ib tug muab dav hlau, yog hais tias tus vector MₒM yuav perpendicular mus rau lub vector n. Peb sau cov mob ntawm orthogonality siv cov scalar khoom:

[MₒM, n] = 0.

Txij li thaum MₒM = r-rₒ, lub vector kab zauv ntawm lub dav hlau yuav zoo li no:

[R - rₒ, n] = 0.

Qhov no kab zauv kuj muaj lwm zoo. Rau lub hom phiaj no, lub zog ntawm cov scalar khoom, thiab hloov dua siab tshiab rau sab laug sab ntawm lub kab zauv. [R - rₒ, n] = [r, n] - [rₒ, n]. Yog hais tias [rₒ, n] denoted li s, peb muab tau cov nram qab no kab zauv: [r, n] - a = 0 los yog [r, n] = s, uas qhia rau hauv lub constancy ntawm lub qhov nyob rau hauv lub cev vector ntawm lub voos kheej-kheej-vectors ntawm qhov muab cov ntsiab lus uas yuav dav hlau.

Tam sim no koj muaj peev xwm tau txais cov kev pab tswj xyuas hom ntaubntawv povthawj siv dav hlau peb vector kab zauv [r - rₒ, n] = 0. Vim r-rₒ = (x-hₒ) * kuv + (y-uₒ) * j + (z-zₒ) * k, thiab n = A * kuv + B * j + C * k, peb tau:

Nws hloov tawm hais tias peb muaj lub equation yog tsim dav hlau dua los ntawm tus taw tes perpendicular mus rau lub qub n:

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

Ntaus raws li lub dav hlau kab zauv thiab coordinates ntawm ob lub ntsiab lus ntawm lub vector dav hlau collinear

Peb txhais tau ob arbitrary ntsiab lus M '(x', y ', z') thiab M "(x", y ", z"), raws li zoo raws li lub vector (a ', ib ", ib tug ‴).

Tam sim no peb yuav sau kab zauv predetermined dav hlau uas kis los ntawm tus uas twb muaj lawm point M 'thiab M ", thiab txhua tus taw tes nrog cov coordinates M (x, y, z) thaum uas tig mus rau ib tug muab vector.

Yog li M'M vectors x = {x ', y-y'; ZZ '} thiab M "M = {x" -hluav', y 'y'; z "-z '} yuav tsum coplanar nrog lub vector ib = (ib tug ', ib ", ib tug ‴), uas txhais tau tias (M'M M" M, a) = 0.

Yog li ntawd peb kab zauv ntawm ib tug dav hlau nyob rau hauv qhov chaw yuav zoo li no:

Hom dav hlau kab zauv, hla peb cov ntsiab lus

Wb hais tias peb muaj peb cov ntsiab lus: (x ', y', z '), (x', y ', z'), (x ‴ Muaj ‴, z ‴), uas tsis yog mus rau tib kab. Nws yog tsim nyog los sau kab zauv ntawm lub dav hlau dua los ntawm cov peb cov ntsiab lus teev. geometry kev tshawb xav cav hais tias qhov no zoo ntawm lub dav hlau nyob ua ib ke, nws yog ib tug thiab xwb. Vim qhov no dav hlau tshuam lub point (x ', y', z '), nws equation daim ntawv yuav tsum:

Ntawm no, A, B, thiab C yog cov sib txawv ntawm xoom nyob rau tib lub sij hawm. Tsis tas li ntawd muab dav hlau tshuam ob ntsiab lus ntau ntxiv (x ", y", z ") thiab (x ‴, y ‴, z ‴). Nyob rau hauv no kev twb kev txuas yuav tsum tau nqa tawm no zoo tej yam kev mob:

Tam sim no peb muaj peev xwm tsim ib tug uniform system sib npaug (linear) nrog unknowns u, v, w:

Nyob rau hauv peb cov ntaub ntawv x, y los yog z sawv arbitrary point uas satisfies kab zauv (1). Xav kab zauv (1) thiab ib tug system ntawm kev sib npaug (2) thiab (3) lub system ntawm equations qhia nyob rau hauv cov duab saum toj, lub vector satisfies N (A, B, C) uas yog nontrivial. Nws yog vim hais tias cov determinant ntawm lub system yog pes tsawg.

Kab zauv (1) uas peb twb tau txais, qhov no yog qhov sib npaug ntawm lub dav hlau. 3 taw tes nws yeej mus, thiab nws yog ib qho yooj yim mus xyuas. Ua li no, peb nthuav lub determinant ntawm lub ntsiab nyob rau hauv thawj kab. Ntawm uas twb muaj lawm cov khoom determinant li nram no hais tias peb lub dav hlau ib txhij tshuam rau peb Ameslikas predetermined point (x ', y', z '), (x ", y", z "), (x ‴, y ‴, z ‴). Yog li ntawd peb txiav txim siab los ua hauj lwm nyob rau hauv pem hauv ntej ntawm peb.

Dihedral lub ntawm lub dav hlau

Dihedral lub yog ib tug spatial geometric zoo tsim los ntawm ob tug ib nrab-dav hlau uas emanate los ntawm ib tug ncaj kab. Nyob rau hauv lwm yam lus, ib feem ntawm lub chaw uas yog tas rau cov ib nrab-dav hlau.

Piv txwv peb muaj ob lub dav hlau nrog rau cov nram qab no sib npaug:

Peb paub hais tias cov vector N = (A, B, C) thiab N¹ = (A¹, H¹, S¹) raws li predetermined dav hlau yog perpendicular. Nyob rau hauv no hais txog, lub kaum sab xis φ ntawm vectors N thiab N¹ sib npaug zos lub (dihedral), uas yog nyob nruab nrab ntawm cov dav hlau. Lub scalar khoom yog muab los ntawm:

NN¹ = | N || N¹ | cos φ,

precisely vim hais tias

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

Nws yog txaus los xav txog tias 0≤φ≤π.

Ua tau ob dav hlau uas tshuam, daim ntawv ob lub kaum sab xis (dihedral): φ ₁ thiab φ _2. Lawv zaum yog sib npaug zos rau π (φ ₁ + φ ₂ = π). Raws li rau lawv cosines, lawv tsis tseem ceeb yog muaj sib npaug, tab sis lawv yog cov sib txawv tej yam tshwm sim, uas yog, cos φ ₁ = -cos φ _2. Yog hais tias nyob rau hauv kab zauv (0) yog hloov los ntawm A, B thiab C ntawm -Ib, -B thiab C feem, kab zauv, peb tau, yuav txiav txim seb lub tib lub dav hlau, tsuas muaj kaum sab xis φ nyob rau hauv kab zauv cos φ = NN ¹ / | N || N ¹ | Nws yuav tsum tau hloov los ntawm π-φ.

Cov kab zauv ntawm lub perpendicular dav hlau

Hu perpendicular dav hlau, ntawm uas lub kaum sab xis yog 90 degrees. Siv cov ntaub ntawv uas hais saum toj no, peb yuav nrhiav tau cov kab zauv ntawm ib tug dav hlau perpendicular mus rau lwm yam. Puas xav tias peb muaj ob dav hlau: Ax + By + CZ + D = 0, thiab + A¹h V¹u S¹z + + D = 0. Peb yuav hais tias lawv yog orthogonal yog cos = 0. Qhov no txhais tau tias NN¹ = AA¹ + VV¹ SS¹ + = 0.

Cov kab zauv ntawm ib tug thaum uas tig mus dav hlau

Nws xa mus rau ob thaum uas tig mus dav hlau uas muaj tsis muaj ntsiab lus nyob rau hauv ntau.

Tus mob ntawm thaum uas tig mus dav hlau (lawv sib npaug yog tib yam li nyob rau hauv lub yav dhau los cov pawg lus) yog hais tias lub vectors N thiab N¹, uas yog lawv mus rau lawv, collinear. Qhov no txhais tau tias cov nram qab no tej yam kev mob uas tau ntsib proportionality:

A / A¹ = B / C = H¹ / S¹.

Yog hais tias lub proportional cov nqe lus loj - A / A¹ = B / C = H¹ / S¹ = DD¹,

no qhia tau hais tias cov ntaub ntawv dav hlau ntawm tib. Qhov no txhais tau tias kab zauv Ax + By + CZ + D = 0 thiab + A¹h V¹u S¹z + + D¹ = 0 piav ib lub dav hlau.

Qhov kev ncua deb ntawm taw tes rau lub dav hlau

Piv txwv peb muaj ib tug dav hlau P, uas yog muab los ntawm (0). Nws yog tsim nyog los nrhiav qhov kev ncua deb ntawm tus taw tes nrog coordinates (hₒ, uₒ, zₒ) = Qₒ. , Koj yuav tau nqa cov kab zauv nyob rau hauv lub dav hlau II qub tsos kom nws:

(Ρ, v) = p (r≥0).

Nyob rau hauv cov ntaub ntawv no, ρ (x, y, z) yog lub voos kheej-kheej vector ntawm peb taw tes Q, nyob rau n p - n yog qhov ntev ntawm lub perpendicular, uas twb tso tawm los ntawm lub voj point, v - yog lub tsev vector, uas yog cov txheej txheem ntawm nyob rau hauv cov kev taw qhia a.

Qhov txawv ρ-ρº voos kheej-kheej vector ntawm ib tug taw tes Q = (x, y, z), teej tug mus rau n thiab lub voos kheej-kheej vector ntawm ib tug muab point Q ₀ = (hₒ, uₒ, zₒ) yog xws li ib tug vector, lub meej nqi ntawm cov projection ntawm uas nyob rau v sib npaug rau hauv lub deb d, uas yog tsim nyog los nrhiav tau los ntawm Q = ₀ (hₒ, uₒ, zₒ) rau P:

D = | (ρ-ρ ^0, v) |, tab sis

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

Yog li ntawd nws puv tawm,

d = | (ρ ^0, v) p |.

Tam sim no nws yog tseeb hais tias mus xam qhov kev ncua deb d los ntawm ₀ mus Q dav hlau P, nws yog tsim nyog los siv cov saib dav hlau kab zauv, cov ua haujlwm mus rau sab laug ntawm p, thiab qhov chaw kawg ntawm x, y, z neeg hloov zov (hₒ, uₒ, zₒ).

Yog li, peb yuav nrhiav tau lub meej nqi ntawm lub resulting qhia uas yuav tsum tau d.

Siv qhov tsis muaj lus, peb tau txais lub cuab:

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

Yog hais tias qhov kev cai tswjhwm point Q ₀ yog nyob rau lwm sab ntawm lub dav hlau P raws li lub hauv paus chiv keeb, ces ntawm lub vector ρ-ρ ⁰ thiab v yog ib tug obtuse angle, li no:

d = - (ρ-ρ ^0, v) = (ρ ^0, v) -p> 0.

Nyob rau hauv lub rooj plaub thaum lub point Q ₀ nyob rau hauv nrog cov keeb kwm nyob rau tib sab ntawm lub U, lub mob lub yog tsim, uas yog:

d = (ρ-ρ ^0, v) = p - (ρ ^0, v)> 0.

Cov no hais tias nyob rau hauv lub yav tas los cov ntaub ntawv (ρ ^0, v)> p, nyob rau hauv lub thib ob (ρ ^0, v)

Thiab nws tangent dav hlau kab zauv

Txog lub dav hlau mus rau qhov chaw nyob rau tus taw tes ntawm tangency Mº - ib tug dav hlau uas muaj tag nrho cov tau tangent rau qhov nkhaus twv los ntawm tus taw tes rau ntawm qhov chaw.

Nrog rau qhov no nto hauv daim ntawv ntawm cov kab zauv F (x, y, z) = 0 nyob rau hauv kab zauv ntawm lub tangent dav hlau tangent point Mº (hº, uº, zº) yuav tsum:

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

Yog hais tias qhov chaw yog teem ntsees z = f (x, y), ces lub tangent dav hlau yog piav los ntawm kab zauv:

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

Cov kev tshuam ntawm ob dav hlau

Nyob rau hauv peb-dimensional qhov chaw yog ib tug tswj system (lub) Oxyz, muab ob dav hlau P 'thiab P' uas sib tshooj thiab tsis coincide. Txij li thaum tej dav hlau, uas yog nyob rau hauv ib lub coordinate system txhais los ntawm cov kev kab zauv, peb xav hais tias n 'thiab n "yog txhais los ntawm qhov sib npaug A'x + V'u S'z + + D' = 0 thiab A" + B x '+ y nrog "z + D" = 0. Nyob rau hauv cov ntaub ntawv no peb muaj ib txwm n '(A', B, C ') ntawm lub dav hlau P' thiab lub cev n "(A", B ", C") ntawm lub dav hlau P '. Raws li peb lub dav hlau yog tsis thaum uas tig thiab tsis coincide, ces cov vectors tsis collinear. Siv cov lus ntawm kev kawm txog zauv, peb muaj tus mob no yuav tau sau ntawv raws li: n '≠ n "↔ (A', B, C ') ≠ (λ * Thiab", λ * Nyob rau hauv ", λ * C"), λεR. Qhia cov kab ncaj nraim uas yog lus dag ntawm cov kev tshuam P 'thiab P ", yuav tau denoted los ntawm tsab ntawv ib tug, nyob rau hauv cov ntaub ntawv no ib tug = P' ∩ P".

thiab - ib tug kab uas muaj ib tug plurality ntawm cov ntsiab lus (cov) dav hlau P 'thiab P ". Qhov no txhais tau tias lub coordinates ntawm tej taw tes teej tug mus rau txoj kab ib tug, yuav tsum ib txhij siav kab zauv A'x + V'u S'z + + D '= 0 thiab A "x + B' + C y" z + D "= 0. Qhov no txhais tau tias lub coordinates ntawm tus taw tes yuav ua tau ib tug kev tshuaj nram qab no sib npaug:

Cov no hais tias kev daws (zuag qhia tag nrho) ntawm no kev sib npaug yuav txiav txim seb lub coordinates ntawm txhua tus ntawm cov ntsiab lus ntawm txoj kab uas yuav ua raws li tus taw tes ntawm kev tshuam P 'thiab P ", thiab txiav txim seb ib tug kab nyob rau hauv ib tug tswj system Oxyz (lub) qhov chaw.