Cov duab geometric puag ncig peb txhua qhov chaw. Convex polygons tuaj yeem yog ntuj, xws li honeycomb, lossis khoom tsim (man-made). Cov duab no yog siv nyob rau hauv kev tsim ntawm ntau hom coatings, painting, architecture, decorations, thiab lwm yam. Convex polygons muaj cov cuab yeej uas tag nrho lawv cov ntsiab lus nyob rau tib sab ntawm ib txoj kab ncaj nraim uas hla ib khub ntawm qhov nyob ib sab ntawm daim duab geometric no. Tseem muaj lwm yam txhais thiab. Lub polygon yog hu ua convex yog tias nws nyob rau hauv ib nrab-dav dav hlau nrog kev hwm rau ib txoj kab ncaj nraim uas muaj ib qho ntawm nws sab.
Convex polygons
Nyob rau hauv chav kawm ntawm theem pib geometry, tsuas yog cov polygons yooj yim ib txwm xav txog. Kom nkag siab tag nrho cov khoom ntawm xws ligeometric duab, nws yog ib qhov tsim nyog los nkag siab txog lawv qhov xwm txheej. Yuav pib nrog, nws yuav tsum to taub hais tias tej kab yog hu ua kaw, qhov kawg ntawm uas coincide. Ntxiv mus, daim duab tsim los ntawm nws muaj peev xwm muaj ntau yam configurations. Ib tug polygon yog ib qho yooj yim kaw cov kab tawg, uas nyob sib ze txuas tsis nyob ntawm tib txoj kab ncaj nraim. Nws cov kev sib txuas thiab cov vertices yog, ntsig txog, sab thiab ntsug ntawm daim duab geometric no. Ib qho yooj yim polyline yuav tsum tsis txhob muaj kev sib tshuam ntawm tus kheej.
Cov vertices ntawm ib tug polygon yog hu ua nyob ib sab yog tias lawv sawv cev rau qhov kawg ntawm ib sab. Ib daim duab geometric uas muaj tus naj npawb nth ntawm vertices, thiab li no tus naj npawb ntawm sab, hu ua n-gon. Cov kab tawg nws tus kheej yog hu ua ciam teb lossis contour ntawm daim duab geometric. Ib lub dav hlau polygonal lossis lub tiaj tus polygon yog hu ua qhov kawg ntawm ib lub dav hlau bounded los ntawm nws. Cov sab uas nyob ib sab ntawm daim duab geometric no yog hu ua ntu ntawm txoj kab tawg tawm ntawm ib qho vertex. Lawv yuav tsis nyob ib sab yog tias lawv los ntawm qhov sib txawv ntawm qhov sib txawv.
Lwm cov ntsiab lus ntawm convex polygons
Nyob hauv geometry theem pib, muaj ntau ntau qhov sib npaug ntxiv uas qhia tias polygon hu ua convex. Tag nrho cov lus no yeej muaj tseeb tiag. Ib tug polygon suav hais tias yog convex yog:
• txhua ntu uas txuas ob lub ntsiab lus hauv nws nyob hauv nws;
• hauv nwstag nrho nws kab pheeb ces kaum dag;
• ib lub kaum sab hauv tsis tshaj 180 °.
A polygon ib txwm faib lub dav hlau ua 2 ntu. Ib qho ntawm lawv yog txwv (nws tuaj yeem muab ntim rau hauv lub voj voog), thiab lwm qhov yog txwv tsis pub. Thawj yog hu ua cheeb tsam sab hauv, thiab qhov thib ob yog thaj tsam sab nrauv ntawm daim duab geometric. Qhov no polygon yog kev sib tshuam (hauv lwm lo lus, ib qho kev tivthaiv) ntawm ob peb lub dav hlau ib nrab. Ntxiv mus, txhua ntu uas xaus rau ntawm cov ntsiab lus uas muaj nyob rau hauv lub polygon kiag li belongs rau nws.
Ntau yam ntawm convex polygons
Lub ntsiab lus ntawm lub convex polygon tsis qhia tias muaj ntau yam ntawm lawv. Thiab txhua tus ntawm lawv muaj qee yam. Yog li, convex polygons uas muaj lub kaum sab xis ntawm 180 ° yog hu ua weakly convex. Lub convex geometric daim duab uas muaj peb vertices yog hu ua ib daim duab peb sab, plaub - plaub-quadrangle, tsib - ib tug pentagon, thiab lwm yam. Txhua lub convex n-gons ua tau raws li qhov tseem ceeb hauv qab no: n yuav tsum sib npaug los yog ntau dua 3. daim duab peb sab yog convex. Ib daim duab geometric ntawm hom no, uas txhua qhov vertices nyob rau tib lub voj voog, hu ua inscribed hauv lub voj voog. Lub convex polygon yog hu ua circumscribed yog tias tag nrho nws sab nyob ze ntawm lub voj voog kov nws. Ob lub polygons tau hais kom sib npaug tsuas yog tias lawv tuaj yeem raug suav los ntawm superposition. Lub dav hlau polygon yog hu ua lub dav hlau polygonal.(ib feem ntawm lub dav hlau), uas yog txwv los ntawm daim duab geometric no.
Nco ntau convex polygons
Txoj polygons tsis tu ncua yog cov duab geometric nrog cov kaum sib npaug thiab sab. Nyob rau hauv lawv muaj ib tug point 0, uas yog nyob rau tib qhov deb ntawm txhua ntawm nws vertices. Nws yog hu ua qhov nruab nrab ntawm daim duab geometric no. Cov ntu txuas nrog qhov nruab nrab nrog cov vertices ntawm daim duab geometric no hu ua apothems, thiab cov uas txuas taw tes 0 nrog rau sab yog hu ua radii.
Ib lub quadrilateral yog ib square. Ib daim duab peb sab sib npaug yog hu ua daim duab peb sab sib npaug. Rau cov duab zoo li no, muaj txoj cai hauv qab no: txhua lub ces kaum ntawm lub convex polygon yog 180 °(n-2) / n, qhov twg n yog tus naj npawb ntawm cov vertices ntawm no convex geometric daim duab.
Thaj chaw ntawm ib qho kev sib koom ua ke yog txiav txim siab los ntawm tus qauv:
S=ph, qhov twg p yog ib nrab ntawm tag nrho cov sab ntawm lub polygon muab thiab h yog qhov ntev ntawm apothem.
Properties ntawm convex polygons
Convex polygons muaj qee yam khoom. Yog li, ib ntu uas txuas ib qho 2 cov ntsiab lus ntawm cov duab geometric yuav tsum nyob hauv nws. Pov thawj:
Xav tias P yog ib lub convex polygon. Peb coj 2 cov ntsiab lus tsis txaus ntseeg, piv txwv li, A, B, uas yog rau P. Raws li cov ntsiab lus uas twb muaj lawm ntawm lub convex polygon, cov ntsiab lus no nyob rau tib sab ntawm kab, uas muaj ib sab ntawm P. Yog li ntawd, AB kuj muaj cov cuab yeej no thiab muaj nyob rau hauv P. Ib tug convex polygon yeej ib txwm muab faib ua ob peb daim duab peb sab los ntawm tag nrho cov diagonals kos los ntawm ib tug ntawm nws vertices.
Lub kaum sab xis ntawm qhov convex geometric duab
Cov ces kaum ntawm lub convex polygon yog cov ces kaum tsim los ntawm nws sab. Sab hauv cov ces kaum yog nyob rau hauv cheeb tsam sab hauv ntawm ib daim duab geometric muab. Lub kaum sab xis uas yog tsim los ntawm nws sab uas converge ntawm ib lub vertex yog hu ua lub kaum sab xis ntawm lub convex polygon. Lub kaum sab xis nyob ib sab ntawm lub kaum sab hauv ntawm ib daim duab geometric muab hu ua sab nraud. Txhua lub ces kaum ntawm lub convex polygon nyob rau hauv nws yog:
180° - x, qhov twg x yog tus nqi ntawm lub kaum sab xis. Cov qauv yooj yim no ua haujlwm rau txhua daim duab geometric ntawm hom no.
Feem ntau, rau cov ces kaum sab nraud muaj txoj cai hauv qab no: txhua lub kaum sab xis ntawm lub kaum sab xis ntawm lub kaum sab xis yog sib npaug ntawm qhov sib txawv ntawm 180 ° thiab tus nqi ntawm lub kaum sab xis. Nws tuaj yeem muaj txiaj ntsig ntawm -180 ° txog 180 °. Yog li, thaum lub kaum sab hauv yog 120 °, lub kaum sab nraud yuav yog 60 °.
Sum ntawm cov ces kaum ntawm convex polygons
Cov lej ntawm cov kaum sab hauv ntawm lub convex polygon yog teeb tsa los ntawm cov qauv:
180°(n-2), qhov twg n yog tus lej ntawm qhov chaw ntawm n-gon.
Cov lej ntawm cov kaum sab xis ntawm lub convex polygon yog qhov yooj yim los xam. Xav txog tej daim duab geometric li no. Txhawm rau txiav txim siab qhov sib npaug ntawm cov ces kaum hauv lub convex polygon, nws yog qhov tsim nyogtxuas ib qho ntawm nws cov vertices mus rau lwm qhov vertices. Raws li qhov kev txiav txim no, (n-2) daim duab peb sab tau txais. Peb paub tias cov lej ntawm cov kaum sab xis ntawm ib daim duab peb sab yog ib txwm 180 °. Txij li thaum lawv tus lej nyob rau hauv ib lub polygon yog (n-2), qhov sib npaug ntawm cov ces kaum sab hauv ntawm daim duab yog 180 ° x (n-2).
Cov sums ntawm lub kaum sab xis ntawm lub convex polygon, uas yog ob qho tib si sab hauv thiab sab nraud uas nyob ib sab, rau ib qho convex geometric daim duab yuav ib txwm sib npaug rau 180 °. Raws li qhov no, koj tuaj yeem txiav txim siab tag nrho nws cov ces kaum:
180x n.
Cov lej ntawm cov kaum sab hauv yog 180 °(n-2). Raws li qhov no, cov lej ntawm txhua qhov chaw sab nraud ntawm daim duab no yog tsim los ntawm cov qauv:
180°n-180°-(n-2)=360°.
Cov txiaj ntsig ntawm sab nrauv lub kaum sab xis ntawm txhua qhov convex polygon yuav ib txwm yog 360 ° (tsis hais tus naj npawb ntawm sab).
Lub kaum sab nraud ntawm lub convex polygon feem ntau sawv cev los ntawm qhov sib txawv ntawm 180 ° thiab tus nqi ntawm lub kaum sab xis.
Lwm yam khoom ntawm lub ntsej muag convex
Ntxiv rau cov khoom tseem ceeb ntawm cov duab geometric, lawv muaj lwm tus uas tshwm sim thaum siv lawv. Yog li, ib qho ntawm cov polygons tuaj yeem muab faib ua ob peb convex n-gons. Txhawm rau ua qhov no, nws yog qhov yuav tsum tau txuas ntxiv rau txhua sab thiab txiav daim duab geometric raws cov kab ncaj nraim. Nws kuj tseem tuaj yeem faib cov polygon rau hauv ntau qhov convex nyob rau hauv txoj kev uas cov vertices ntawm txhua daim coincide nrog tag nrho nws cov vertices. Los ntawm cov duab geometric zoo li no, daim duab peb sab tuaj yeem ua tau yooj yim los ntawm kos tag nrhodiagonals los ntawm ib tug vertex. Yog li, txhua tus polygon thaum kawg tuaj yeem muab faib ua qee tus lej ntawm daim duab peb sab, uas ua rau muaj txiaj ntsig zoo hauv kev daws teeb meem ntau yam cuam tshuam nrog cov duab geometric.
Pib ib puag ncig ntawm lub convex polygon
Seg ntawm kab tawg, hu ua ob sab ntawm lub polygon, feem ntau txhais tau los ntawm cov ntawv hauv qab no: ab, bc, cd, de, ea. Cov no yog ob sab ntawm daim duab geometric nrog vertices a, b, c, d, e. Qhov sib npaug ntawm qhov ntev ntawm txhua sab ntawm lub convex polygon no yog hu ua nws ib puag ncig.
polygon ncig
Convex polygons tuaj yeem sau thiab hla. Lub voj voog uas kov txhua sab ntawm daim duab geometric no hu ua inscribed hauv nws. Xws li ib tug polygon yog hu ua circumscribed. Qhov nruab nrab ntawm lub voj voog uas tau sau rau hauv ib lub polygon yog qhov kev sib tshuam ntawm bisectors ntawm txhua lub ces kaum hauv ib daim duab geometric. Cheeb tsam ntawm xws li ib tug polygon yog:
S=pr, qhov twg r yog lub vojvoog ntawm lub voj voog inscribed thiab p yog lub semiperimeter ntawm tus muab polygon.
Ib lub voj voog uas muaj cov vertices ntawm ib tug polygon yog hu ua circumscribed nyob ib ncig ntawm nws. Ntxiv mus, daim duab geometric convex no hu ua inscribed. Qhov nruab nrab ntawm lub voj voog, uas yog circumscribed txog xws li ib tug polygon, yog qhov kev sib tshuam point ntawm lub thiaj li hu ua perpendicular bisectors ntawm tag nrho cov sab.
kab pheeb ces kaum ntawm convex geometric duab
Cov kab pheeb ces kaum ntawm lub convex polygon yog ntu uastxuas cov tsis nyob ib sab vertices. Txhua ntawm lawv nyob hauv daim duab geometric no. Tus naj npawb ntawm kab pheeb ces kaum ntawm xws li n-gon yog teem los ntawm tus qauv:
N=n (n – 3)/ 2.
Tus naj npawb ntawm kab pheeb ces kaum ntawm lub convex polygon plays lub luag haujlwm tseem ceeb hauv qib qis geometry. Tus naj npawb ntawm daim duab peb sab (K) uas nws muaj peev xwm faib tau txhua qhov convex polygon yog xam los ntawm cov qauv hauv qab no:
K=n – 2.
Tus naj npawb ntawm kab pheeb ces kaum ntawm lub convex polygon ib txwm nyob ntawm tus naj npawb ntawm nws cov vertices.
Decomposition of a convex polygon
Qee zaum, txhawm rau daws cov teeb meem geometric, nws yog ib qho tsim nyog yuav tsum tau faib ib lub convex polygon rau hauv ob peb daim duab peb sab nrog cov kab pheeb ces kaum tsis sib tshuam. Qhov teeb meem no tuaj yeem daws tau los ntawm kev muab cov qauv qhia tshwj xeeb.
Kev txhais ntawm qhov teeb meem: cia peb hu ib qho kev faib ua kom zoo ntawm lub convex n-gon rau hauv ob peb daim duab peb sab los ntawm kab pheeb ces kaum uas sib tshuam tsuas yog nyob rau sab saum toj ntawm daim duab geometric.
Kev daws teeb meem: Piv txwv tias Р1, Р2, Р3 …, Pn yog vertices ntawm no n-gon. Tus lej Xn yog tus naj npawb ntawm nws cov partitions. Cia peb ua tib zoo xav txog qhov tau txais kab pheeb ces kaum ntawm daim duab geometric Pi Pn. Nyob rau hauv ib qho ntawm cov partitions tsis tu ncua P1 Pn belongs rau ib daim duab peb sab P1 Pi Pn, uas muaj 1<i<n. Ua raws li qhov no thiab xav tias kuv=2, 3, 4 …, n-1, peb tau txais (n-2) pawg ntawm cov partitions, uas suav nrog txhua qhov xwm txheej tshwj xeeb.
Cia kuv=2 yog ib pawg ntawm kev faib ua ntu zus, ib txwm muaj kab pheeb ces kaum Р2 Pn. Tus naj npawb ntawm cov partitions uas nkag rau nws yog tib yam li tus naj npawb ntawm partitions(n-1)-gon P2 P3 P4… Pn. Hauv lwm lo lus, nws sib npaug Xn-1.
Yog kuv=3, qhov no lwm pab pawg ntawm kev faib yuav ib txwm muaj cov kab pheeb ces kaum Р3 Р1 thiab Р3 Pn. Nyob rau hauv cov ntaub ntawv no, tus naj npawb ntawm cov partitions tsis tu ncua uas muaj nyob rau hauv cov pab pawg neeg no yuav coincide nrog cov naj npawb ntawm partitions ntawm lub (n-2)-gon P3 P4 … Pn. Hauv lwm lo lus, nws yuav sib npaug Xn-2.
Cia kuv=4, ces ntawm daim duab peb sab ib txwm muab faib yuav yeej muaj ib daim duab peb sab P1 P4 Pn, uas lub quadrangle P1 P2 P3 P4, (n-3)-gon P4 P5 … Pn yuav adjoin. Tus naj npawb ntawm cov partitions tsis tu ncua ntawm xws li plaub-quadrilateral yog X4, thiab tus naj npawb ntawm partitions ntawm ib (n-3)-gon yog Xn-3. Raws li cov lus hais saum toj no, peb tuaj yeem hais tias tag nrho cov kev faib kom raug muaj nyob hauv pab pawg no yog Xn-3 X4. Lwm pab pawg nrog kuv=4, 5, 6, 7… yuav muaj Xn-4 X5, Xn-5 X6, Xn-6 X7 … tsis tu ncua partitions.
Cia kuv=n-2, ces tus naj npawb ntawm qhov raug cais hauv pab pawg no yuav zoo ib yam li cov lej sib cais hauv pab pawg uas i=2 (hauv lwm lo lus, sib npaug Xn-1).
Since X1=X2=0, X3=1, X4=2…, ces tus naj npawb ntawm tag nrho cov partitions ntawm ib tug convex polygon yog:
Xn=Xn-1 + Xn-2 + Xn-3 X4 + Xn-4 X5 + … + X 5 Xn-4 + X4 Xn-3 + Xn-2 + Xn-1.
Example:
X5=X4 + X3 + X4=5
X6=X5 + X4 + X4 + X5=14
X7=X6 + X5 + X4X4 + X5 + X6=42
X8=X7 + X6 + X5X4 + X4X5 + X6 + X7=132
Tus lej ntawm qhov raug muab faib sib tshuam ib kab pheeb suab sab hauv
Thaum kuaj xyuas cov xwm txheej tshwj xeeb, ib tus tuaj yeem tuaj txog ntawmqhov kev xav tias tus naj npawb ntawm kab pheeb ces kaum ntawm convex n-gons yog sib npaug rau cov khoom ntawm tag nrho cov partitions ntawm daim duab no los ntawm (n-3).
Pov thawj ntawm qhov kev xav no: xav txog tias P1n=Xn(n-3), ces ib qho n-gon tuaj yeem muab faib ua (n-2)-daim duab peb sab. Ntxiv mus, ib qho (n-3)-quadrilateral tuaj yeem tsim los ntawm lawv. Nrog rau qhov no, txhua lub quadrilateral yuav muaj kab pheeb ces kaum. Txij li ob kab pheeb ces kaum tuaj yeem kos rau hauv daim duab geometric convex, qhov no txhais tau hais tias ntxiv (n-3) kab pheeb ces kaum tuaj yeem kos rau hauv ib qho (n-3)-quadrilaterals. Raws li qhov no, peb tuaj yeem xaus tias hauv txhua qhov kev faib tawm tsis tu ncua nws tuaj yeem kos (n-3)-kab pheeb suab uas ua tau raws li cov xwm txheej ntawm qhov teeb meem no.
Tej thaj chaw convex polygons
Feem ntau, thaum daws tau ntau yam teeb meem ntawm theem pib geometry, nws yuav tsum tau txiav txim siab txog thaj tsam ntawm lub convex polygon. Piv txwv tias (Xi. Yi), i=1, 2, 3… n yog qhov sib lawv liag ntawm kev sib koom ua ke ntawm txhua qhov chaw nyob sib ze ntawm ib lub polygon uas tsis muaj kev sib tshuam ntawm tus kheej. Nyob rau hauv cov ntaub ntawv no, nws cheeb tsam yog xam raws li nram no:
S=½ (∑ (Xi + Xi + 1) (Yi + Yi + 1), where (X1, Y1)=(Xn +1, Yn + 1).
