Dav hlau hauv qhov chaw. Qhov chaw ntawm cov dav hlau hauv qhov chaw

2024 Tus sau: Angel Austin | [email protected]. Kawg hloov kho: 2024-01-02 17:48

Ib dav hlau yog ib yam khoom geometric uas nws cov khoom siv thaum tsim qhov projections ntawm cov ntsiab lus thiab kab, nrog rau thaum xam qhov deb thiab dihedral kaum sab xis ntawm cov duab peb sab. Cia peb xav txog hauv kab lus no seb qhov sib npaug yuav siv tau los kawm qhov chaw ntawm cov dav hlau hauv qhov chaw.

dav hlau txhais

Txhua tus neeg nkag siab xav xav txog yam khoom twg yuav tham. Los ntawm ib tug geometric taw tes ntawm view, lub dav hlau yog ib tug sau ntawm cov ntsiab lus, tej vectors ntawm uas yuav tsum tau perpendicular rau ib co vectors. Piv txwv li, yog tias muaj m txawv cov ntsiab lus hauv qhov chaw, ces m(m-1) / 2 txawv vectors tuaj yeem tsim los ntawm lawv, txuas cov ntsiab lus ua khub. Yog hais tias tag nrho cov vectors yog perpendicular rau ib qho kev taw qhia, ces qhov no yog ib tug txaus mob uas tag nrho cov ntsiab lus m belongs rau tib lub dav hlau.

kev sib npaug sib luag

Nyob rau hauv spatial geometry, ib lub dav hlau tau piav qhia siv qhov sib npaug uas feem ntau muaj peb qhov tsis paub ua haujlwm sib raug rau x, y thiab z axes. Rautau qhov kev sib npaug dav dav hauv dav hlau tswj hauv qhov chaw, xav tias muaj vector n¯(A; B; C) thiab taw tes M(x₀; y₀; z_{0). Siv ob yam khoom no, lub dav hlau tuaj yeem txhais tau tshwj xeeb.}

Tseeb tiag, xav tias muaj qee qhov taw tes thib ob P(x; y; z) uas nws tsis paub txog kev tswj hwm. Raws li cov lus txhais tau hais los saum toj no, vector MP¯ yuav tsum yog perpendicular rau n¯, uas yog, cov khoom scalar rau lawv yog sib npaug rau xoom. Tom qab ntawd peb tuaj yeem sau cov lus hauv qab no:

(n¯MP¯)=0 or

A(x-x₀) + B(y-y₀) + C(z-z₀₎₌₀

Qhib cov brackets thiab qhia tus tshiab coefficient D, peb tau txais cov lus qhia:

Ax + By + Cz + D=0 where D=-1(Ax₀+ By ₀ + Cz₀₎

Qhov kev qhia no yog hu ua kev sib npaug dav dav rau lub dav hlau. Nws yog ib qho tseem ceeb kom nco ntsoov tias cov coefficients nyob rau hauv pem hauv ntej ntawm x, y thiab z ua lub coordinates ntawm vector n¯ (A; B; C) perpendicular rau lub dav hlau. Nws coincides nrog qhov qub thiab yog ib qho kev qhia rau lub dav hlau. Txhawm rau txiav txim siab qhov sib npaug dav, nws tsis muaj teeb meem qhov twg qhov vector yog qhia. Ntawd yog, cov dav hlau ua rau ntawm vectors n¯ thiab -n¯ yuav zoo ib yam.

Daim duab saum toj no qhia txog lub dav hlau, vector ib txwm rau nws, thiab ib kab perpendicular rau lub dav hlau.

ntu txiav tawm los ntawm lub dav hlau ntawm lub axes thiab cov kab zauv sib txuas

Cov kab zauv dav dav tso cai siv cov lej yooj yim los txiav txim, hauvnyob rau hauv dab tsi cov ntsiab lus lub dav hlau yuav hla lub coordinate axes. Nws yog ib qho tseem ceeb kom paub cov ntaub ntawv no txhawm rau kom muaj lub tswv yim txog txoj haujlwm hauv qhov chaw ntawm lub dav hlau, nrog rau thaum piav qhia nws hauv cov duab kos.

txhawm rau txiav txim siab cov ntsiab lus sib tshuam, ib qho kev sib npaug hauv ntu yog siv. Nws yog li ntawd hu ua vim hais tias nws qhia meej meej muaj qhov tseem ceeb ntawm qhov ntev ntawm cov ntu txiav tawm los ntawm lub dav hlau ntawm qhov chaw sib koom tes, thaum suav los ntawm qhov taw tes (0; 0; 0). Wb tau qhov sib npaug no.

Sau cov lus dav dav rau lub dav hlau raws li hauv qab no:

Ax + By + Cz=-D

Qhov sab laug thiab sab xis tuaj yeem faib los ntawm -D yam tsis ua txhaum kev sib luag. Peb muaj:

A/(-D)x + B/(-D)y + C/(-D)z=1 or

x/(-D/A) + y/(-D/B) + z/(-D/C)=1

Tsim tus lej ntawm txhua nqe lus nrog lub cim tshiab, peb tau txais:

p=-D/A; q=-D/B; r=-D/C then

x/p + y/q + z/r=1

Qhov no yog qhov sib npaug uas tau hais los saum toj no hauv ntu. Nws ua raws los ntawm nws tias tus nqi ntawm tus denominator ntawm txhua lub sij hawm qhia txog kev sib koom ua ke ntawm kev sib tshuam nrog cov axis sib thooj ntawm lub dav hlau. Piv txwv li, nws hla y-axis ntawm qhov taw tes (0; q; 0). Qhov no yooj yim to taub yog tias koj hloov qhov xoom x thiab z ua haujlwm rau hauv kab zauv.

Nco ntsoov tias yog tias tsis muaj qhov sib txawv ntawm qhov sib npaug hauv ntu, qhov no txhais tau tias lub dav hlau tsis cuam tshuam cov axis sib thooj. Piv txwv li, muab cov lus qhia:

x/p + y/q=1

Qhov no txhais tau hais tias lub dav hlau yuav txiav cov ntu p thiab q ntawm x thiab y axes raws li, tab sis nws yuav ua rau tib lub z axis.

Xaiv txog tus cwj pwm ntawm lub dav hlau thaumqhov tsis muaj qee qhov sib txawv hauv nws qhov sib npaug kuj muaj tseeb rau cov lus qhia dav dav, raws li pom hauv daim duab hauv qab no.

Vector parametric equation

Muaj qhov sib npaug thib peb uas tso cai piav qhia lub dav hlau hauv qhov chaw. Nws yog hu ua parametric vector vim hais tias nws yog muab los ntawm ob vectors dag nyob rau hauv lub dav hlau thiab ob tsis muaj peev xwm coj arbitrary ywj siab qhov tseem ceeb. Cia peb qhia seb qhov sib npaug no tuaj yeem tau li cas.

xav tias muaj ob peb tug paub vectors u ¯(a₁; b₁; c₁) and v¯(a₂; b₂; c₂). Yog tias lawv tsis sib npaug, ces lawv tuaj yeem siv los teeb tsa lub dav hlau tshwj xeeb los ntawm kev kho qhov pib ntawm ib qho ntawm cov vectors ntawm qhov chaw paub M(x₀; y₀; z_{0). Yog hais tias ib tug arbitrary vector MP¯ tuaj yeem sawv cev ua ke ntawm linear vectors u¯ thiab v¯, qhov no txhais tau hais tias qhov point P(x; y; z) belongs rau tib lub dav hlau li u¯, v¯. Yog li, peb tuaj yeem sau qhov sib npaug:}

MP¯=αu¯ + βv¯

Los yog sau qhov sib npaug ntawm kev tswj xyuas, peb tau txais:

(x; y; z)=(x₀; y₀; z₀) + α(a₁; b₁; c₁) + β(a ₂; b₂; c₂₎

Qhov kev sib npaug tau nthuav tawm yog parametric vector sib npaug rau lub dav hlau. ATvector qhov chaw ntawm lub dav hlau u¯ thiab v¯ yog hu ua generators.

Tom ntej no, thaum daws qhov teeb meem, nws yuav qhia tias qhov sib npaug no tuaj yeem txo qis rau hauv daim ntawv dav dav rau lub dav hlau.

Lub kaum sab xis ntawm dav hlau hauv qhov chaw

Intuitively, dav hlau hauv 3D qhov chaw tuaj yeem hla lossis tsis tau. Hauv thawj kis, nws yog qhov txaus siab los nrhiav lub kaum sab xis ntawm lawv. Kev suav ntawm lub kaum sab xis no yog qhov nyuaj dua li lub kaum sab xis ntawm kab, txij li peb tab tom tham txog cov khoom dihedral geometric. Txawm li cas los xij, twb tau hais qhia cov vector rau lub dav hlau los cawm.

Nws yog geometrically tsim kom lub dihedral lub kaum sab xis ntawm ob lub dav hlau sib tshuam yog sib npaug rau lub kaum sab xis ntawm lawv cov vectors qhia. Wb denote cov vectors as n₁¯(a₁; b₁; c₁) thiab n₂¯(a₂; b₂; c_{2). Lub cosine ntawm lub kaum sab xis ntawm lawv yog txiav txim los ntawm cov khoom scalar. Ntawd yog, lub kaum sab xis nws tus kheej hauv qhov chaw nruab nrab ntawm lub dav hlau tuaj yeem xam los ntawm tus qauv:}

φ=arccos(|(n₁¯n₂¯)|/(|n₁ ¯||n_2¯|)

Ntawm no cov qauv hauv tus lej yog siv los pov tseg tus nqi ntawm obtuse lub kaum sab xis (nruab nrab ntawm cov dav hlau sib tshuam nws ib txwm tsawg dua lossis sib npaug rau 90o).

Nyob hauv daim ntawv sib koom ua ke, cov lus qhia no tuaj yeem rov sau dua li hauv qab no:

φ=arccos(|a₁a₂ + b₁b ₂ +c₁c₂|/(√(a₁² + b₁² + c₁²)√(a₂² + b₂² + c ₂²⁾⁾

dav hlau perpendicular thiab parallel

Yog tias cov dav hlau sib tshuam thiab lub kaum sab xis uas tsim los ntawm lawv yog 90o, ces lawv yuav yog qhov sib npaug. Ib qho piv txwv ntawm cov dav hlau zoo li no yog lub voj voos prism lossis lub voos xwmfab. Cov duab no yog tsim los ntawm rau lub dav hlau. Ntawm txhua qhov vertex ntawm cov npe muaj npe muaj peb lub dav hlau perpendicular rau ib leeg.

Yuav kom paub seb cov dav hlau xav tau yog qhov sib npaug, nws txaus los xam cov khoom lag luam ntawm lawv cov vectors ib txwm. Ib qho xwm txheej txaus rau perpendicularity nyob rau hauv qhov chaw ntawm cov dav hlau yog xoom tus nqi ntawm cov khoom no.

Parallel yog hu ua cov dav hlau tsis sib tshuam. Qee lub sij hawm nws kuj tau hais tias cov dav hlau sib luag sib tshuam ntawm infinity. Cov xwm txheej ntawm parallelism nyob rau hauv qhov chaw ntawm cov dav hlau coincides nrog cov xwm txheej rau cov kev taw qhia vectors n₁¯ thiab n_{2¯. Koj tuaj yeem tshawb xyuas nws hauv ob txoj kev:}

1. Xaiv cov cosine ntawm lub dihedral kaum (cos(φ)) siv cov khoom scalar. Yog tias cov dav hlau sib npaug, ces tus nqi yuav yog 1.
2. Sim sawv cev rau ib tus vector los ntawm lwm tus los ntawm kev muab faib los ntawm qee tus lej, i.e. n₁¯=kn_{2¯. Yog tias qhov no tuaj yeem ua tiav, ces cov dav hlau sib txuas yogsib npaug.}

Daim duab qhia ob lub dav hlau sib luag.

Tam sim no cia peb muab piv txwv ntawm kev daws ob qhov teeb meem nthuav dav siv cov kev paub lej tau txais.

Yuav ua li cas kom tau txais daim ntawv dav dav los ntawm kev sib npaug vector?

Qhov no yog parametric vector qhia rau lub dav hlau. Txhawm rau ua kom yooj yim rau kev nkag siab txog kev khiav dej num thiab kev ua lej siv, xav txog ib qho piv txwv tshwj xeeb:

(x; y; z)=(1; 2; 0) + α(2; -1; 1) + β(0; 1; 3)

nthuav tawm qhov kev qhia no thiab nthuav qhia qhov tsis paub tsis meej:

x=1 + 2α;

y=2 - α + β;

z=α + 3β

Tam sim no:

α=(x - 1)/2;

β=y - 2 + (x - 1)/2;

z=(x - 1)/2 + 3 (y - 2 + (x - 1)/2)

Qhib cov kab ke hauv kab lus kawg, peb tau txais:

z=2x-2 + 3y - 6 or

2x + 3y - z - 8=0

Peb tau txais daim ntawv dav dav ntawm qhov sib npaug rau lub dav hlau tau teev tseg hauv cov lus qhia teeb meem hauv daim ntawv vector

Yuav ua li cas lub dav hlau hla peb lub ntsiab lus?

Nws tuaj yeem kos ib lub dav hlau los ntawm peb lub ntsiab lus yog tias cov ntsiab lus no tsis yog ib txoj kab ncaj nraim. Lub algorithm rau kev daws qhov teeb meem no muaj nyob rau hauv ib theem zuj zus ntawm kev ua:

• nrhiav cov kev tswj hwm ntawm ob lub vectors los ntawm kev sib txuas ua ke paub cov ntsiab lus;
• xam lawv cov khoom hla thiab tau txais vector ib txwm rau lub dav hlau;
• sau qhov sib npaug dav siv qhov pom vector thiabib qho ntawm peb lub ntsiab lus.

Cia peb ua piv txwv. Cov ntsiab lus muab:

R(1; 2; 0), P(0; -3; 4), Q(1; -2; 2)

Cov haujlwm ntawm ob lub vectors yog:

RP¯(-1; -5; 4), PQ¯(1; 1; -2)

Lawv cov khoom hla yuav yog:

n¯=[RP¯PQ¯]=(6; 2; 4)

Kev ua haujlwm ntawm taw tes R, peb tau txais qhov sib npaug:

6x + 2y + 4z -10=0 or

3x + y + 2z -5=0

Nws raug nquahu kom kuaj xyuas qhov tseeb ntawm qhov tshwm sim los ntawm kev hloov cov kev sib koom ua ke ntawm ob lub ntsiab lus ntxiv rau hauv kab lus no:

rau P: 30 + (-3) + 24 -5=0;

rau Q: 31 + (-2) + 22 -5=0

Nco ntsoov tias nws tsis tuaj yeem nrhiav cov khoom vector, tab sis tam sim ntawd sau qhov sib npaug rau lub dav hlau hauv daim ntawv parametric vector.