Convex uas. Lus Txhais ntawm ib tug convex polygon. Lub diagonals ntawm ib tug convex polygon

Cov geometric duab yog tag nrho cov nyob ib ncig ntawm peb. Convex uas yog tej yam ntuj tso, xws li ib tug honeycomb los dag (tus txiv neej ua). Cov nuj nqis yog siv nyob rau hauv ua ntau hom ntawm coatings nyob rau hauv kos duab, architecture, ornaments, etc. Convex uas muaj cov cuab yeej uas lawv cov ntsiab lus dag rau ib sab ntawm ib tug ncaj kab uas kis los ntawm tus khub ntawm ib sab vertices ntawm lub geometrical daim duab. Nws muaj lwm yam lus. Nws hu ua lub convex polygon, uas yog cov txheej txheem ntawm nyob rau hauv ib nrab-lub dav hlau nrog rau kev hwm rau tej txoj kab ncaj nraim muaj ib qho ntawm nws cov sab.

convex uas

Nyob rau hauv lub chav kawm ntawm elementary geometry yeej ib txwm kho tsis tshua muaj neeg yooj yim uas. Yuav kom nkag siab txog cov thaj chaw ntawm geometric duab koj yuav tsum to taub lawv qhov. Yuav pib to taub hais tias kaw yog tej kab uas nws xaus yog tib yam. Thiab daim duab tsim los ntawm nws, yuav muaj ib tug ntau yam ntawm configurations. Polygon yog hu ua yooj yim kaw polyline uas nws nyob ib sab rau lwm cov tsis nyob rau ntawm ib txoj kab ncaj nraim. Nws mus thiab ntshav yog, ntsig txog, lub sab thiab saum ntawm lub geometrical daim duab. Ib tug yooj yim polyline yuav tsum tsis txhob tshuam nws tus kheej.

vertices ntawm lub polygon yog hu ua cov neeg nyob ze, nyob rau hauv cov ntaub ntawv lawv yog cov hnub ntawm ib qho ntawm nws cov sab. Ib tug geometric duab, uas muaj ib tug n-th tooj ntawm vertices, thiab chaw pib lub n-th tooj los ntawm ob tog hu ua cov n-gon. Nws tus kheej lawm kab yog tus ciam los yog contour ntawm lub geometric duab. Polygonal dav hlau los yog tiaj tus polygon hu ua zaum kawg ib feem ntawm tej dav hlau, lawv tsuas siv. Nyob ib sab sab ntawm lub duab daim duab hu ua polyline feem originating los ntawm tib lub kab los sib ntsib. Lawv yuav tsis tau cov neeg nyob ze hais tias lawv yog raws li nyob rau txawv vertices ntawm lub polygon.

Lwm yam ntsiab txhais txog ntawm convex uas

Nyob rau hauv elementary geometry, muaj ntau ntau sib npaug nyob rau hauv lub ntsiab lus txhais cov ntsiab lus, qhia dab tsi yog hu ua ib tug convex polygon. Ntxiv mus, tag nrho cov nqe lus yog Attendance muaj tseeb. Ib tug convex polygon yog ib tug uas muaj:

• txhua ya uas txuas ob cov ntsiab lus nyob hauv nws, cov lus dag nkaus nyob rau hauv nws;

• nyob rau ntawd pw tag nrho nws cov diagonals;

• tej sab hauv lub tsis ntau tshaj 180 °.

Polygon yeej ib txwm faib lub dav hlau mus rau hauv ob qhov chaw. Ib tug ntawm lawv - lub txwv (nws yuav tsum muab ntim rau hauv ib lub voj voog), thiab lwm yam - unlimited. Thawj yog hu ua lub puab cheeb tsam, thiab lub thib ob - tus txheej thaj tsam ntawm lub geometric duab. Qhov no yog cov kev tshuam ntawm lub polygon (nyob rau hauv lwm yam lus - tag nrho cov kev tivthaiv) ob peb ib nrab-dav hlau. Yog li, txhua ya muaj xaus ntawm cov ntsiab lus uas yuav mus rau ib tug polygon kiag belongs rau nws.

Hom ntawm convex uas

Lus Txhais convex polygon tsis qhia hais tias muaj ntau ntau yam ntawm lawv. Thiab txhua yam uas lawv muaj tej yam kev. Yog li, lub convex uas, uas muaj ib tug nrog kaum sab xis ntawm 180 °, raug xa mus rau ib nyuag convex. Lub convex geometric duab uas muaj peb peaks, yog hu ua ib tug daim duab peb sab, plaub - quadrilateral, tsib - pentagon, etc. Txhua lub convex n-gons raws li nram qab no tseem ceeb yuav tsum: .. N yuav tsum sib npaug zos los yog ntau dua 3. Txhua lub duab peb ceg yog convex. Cov duab daim duab ntawm no hom nyob rau hauv uas tag nrho cov vertices muaj nyob rau ntawm ib lub voj voog, hu ua lub inscribed voj voog. Piav convex polygon yog hu ua hais tias tag nrho nws cov sab nyob ib ncig ntawm ib lub voj voog rau nws kov. Ob tug uas yog hu ua sib npaug zos xwb nyob rau hauv rooj plaub thaum uas siv cov overlay yuav ua ke. Pav ca polygon hu ua polygonal dav hlau (ib tug dav hlau feem) hais tias qhov no tsuas geometrical daim duab.

Tsis tu ncua convex uas

Tsis tu ncua uas hu ua geometric duab nrog sib npaug ces kaum thiab sab. Hauv lawv muaj ib tug taw tes 0, uas yog tib yam kev ncua deb ntawm txhua yam ntawm nws vertices. Nws yog hu ua qhov chaw ntawm lub geometrical daim duab. Kab txuas qhov chaw nrog rau cov vertices ntawm lub geometric duab hu ua apothem, thiab cov neeg uas txuas lub point 0 ob tog - radii.

Yog duab plaub - square. Equilateral triangle yog hu ua equilateral. Rau xws li daim muaj cov nram qab no txoj cai: txhua convex polygon lub yog 180 ° * (n-2) / n,

qhov twg n - tus naj npawb ntawm vertices ntawm lub convex geometric duab.

Lub cheeb tsam ntawm tej kev polygon yog txiav txim los ntawm cov mis:

S = p * h,

qhov twg p yog sib npaug zos rau ib nrab ntawm cov sum ntawm tag nrho cov sab ntawm lub polygon, thiab h yog qhov ntev apothem.

Properties convex uas

Convex uas muaj tej yam khoom. Yog li, lub ya uas txuas ob cov ntsiab lus ntawm ib tug geometric duab, tas nyob rau hauv nws. pov thawj:

Piv txwv hais tias P - lub convex polygon. Siv ob arbitrary ntsiab lus, e.g., A thiab B, uas yuav mus rau P. Los ntawm tam sim no cov lus txhais ntawm ib tug convex polygon, cov ntsiab lus yog nyob rau ntawm ib sab ntawm lub txoj kab ncaj nraim uas muaj cov kev taw qhia R. Thiaj li, AB kuj muaj cov cuab yeej no thiab yog muaj nyob rau hauv R. A convex polygon yeej ib txwm tej zaum yuav tsum tau muab faib mus rau hauv ob peb lub duab peb ceg kiag li tag nrho cov diagonals, uas muaj nyob rau ib qho ntawm nws vertices.

Ces kaum convex geometric duab

Lub ces kaum ntawm ib tug convex polygon - yog cov ces kaum uas ua los ntawm cov tog. Hauv fab yog nyob rau hauv lub hauv thaj tsam ntawm lub geometric duab. Lub kaum uas yog tsim los ntawm nws cov sab uas sib nyob rau hauv ib tug kab los sib ntsib, hu ua lub kaum sab xis ntawm lub convex polygon. Fab uas nyob ib sab mus rau lub hauv fab ntawm lub geometrical daim duab, hu ua lwm. Txhua lub ces kaum ntawm ib tug convex polygon, cov txheej txheem ntawm nyob rau hauv nws, yog:

180 ° - x

qhov twg x - tus nqi sab nraum ces kaum. Qhov no yooj yim mis yog muaj feem xyuam rau txhua yam ntawm geometric duab xws li.

Nyob rau hauv kev, rau sab nraum ntawm cov ces kaum nyob ua ib ke raws li nram no txoj cai: txhua convex polygon lub sib npaug zos rau cov sib txawv ntawm 180 ° thiab tus nqi ntawm lub sab hauv lub. Nws yuav muaj qhov tseem ceeb xws li los ntawm -180 ° rau 180 °. Thiaj li, thaum lub puab lub yog 120 °, qhov tsos yuav muaj ib tug nqi ntawm 60 °.

Lub sum ntawm cov ces kaum ntawm convex uas

Lub sum ntawm lub sab hauv ces kaum ntawm ib tug convex polygon yog tsim los ntawm cov mis:

180 ° * (n-2),

qhov twg n - tus naj npawb ntawm vertices ntawm lub n-gon.

Lub sum ntawm cov ces kaum ntawm ib tug convex polygon yog xam heev tsuas. Xav txog tej duab zoo lawm. Yuav kom txiav txim seb lub sum ntawm cov ces kaum nyob rau hauv ib tug convex polygon yuav tsum tau txuas ib qho ntawm nws vertices mus rau lwm cov vertices. Raws li ib tug tshwm sim ntawm qhov kev txiav txim puv (n-2) ntawm daim duab peb sab. Nws yog lub npe hu hais tias lub sum ntawm cov ces kaum ntawm tej daim duab peb sab yog yeej ib txwm 180 °. Vim hais tias lawv muaj pes tsawg nyob rau hauv tej polygon sib npaug (n-2), lub sum ntawm lub sab hauv ces kaum ntawm daim duab sib npaug 180 ° x (n-2).

Li convex polygon fab, namely, ob uas nyob ib sab hauv thiab lwm cov ces kaum rau lawv, nyob rau hauv no convex geometric duab yuav yeej ib txwm ua ncaj rau 180 °. Nyob rau qhov no hauv paus, peb yuav txiav txim seb lub sum ntawm tag nrho nws cov ces kaum:

180 x n.

Lub sum ntawm lub sab hauv ces kaum yog 180 ° * (n-2). Raws li, qhov sum ntawm tag nrho cov txheej ntawm cov ces kaum ntawm daim duab teem los ntawm cov mis:

180 ° * n-180 ° - (n-2) = 360 °.

Tawm ntawm lub ces kaum sab nraud ntawm tej convex polygon yuav yeej ib txwm ua ncaj rau 360 ° (tsis hais txog ntawm lub xov tooj ntawm nws cov sab).

Sab nraum ces kaum ntawm ib tug convex polygon yog feem ntau sawv cev los ntawm qhov sib txawv ntawm 180 ° thiab tus nqi ntawm lub sab hauv lub.

Lwm yam thaj chaw ntawm ib tug convex polygon

Dhau li ntawm tus yooj yim zog ntawm geometric nuj nqis cov ntaub ntawv, lawv kuj muaj lwm yam, uas tshwm sim thaum kov lawv. Li no, tej ntawm uas tej zaum yuav phua rau hauv ntau yam convex n-gons. Ua li no, mus ntxiv rau txhua yam ntawm nws sab thiab txiav lub geometric zoo raws cov kab ncaj nraim. Phua tej polygon mus rau hauv ob peb lub convex qhov chaw yog tau thiab thiaj li hais tias cov saum toj kawg nkaus ntawm txhua tus ntawm daim coincide nrog tag nrho nws cov vertices. Los ntawm ib tug geometrical daim duab yuav ua tau heev yooj yim mus ua cov vajvoos peb sab ntawm tag nrho cov diagonals los ntawm ib tug kab los sib ntsib. Li no, tej polygon, thaum kawg, muaj peev xwm tau muab faib mus rau hauv ib tug tej yam muaj pes tsawg tus ntawm cov vajvoos peb sab, uas yog heev pab tau nyob rau hauv kev daws ntau yam kev pab raws qib hais txog xws geometrical nrhiav.

Ntawm lub perimeter ntawm lub convex polygon

Cov theem ntawm lub polyline, polygon-hu ua ob tog, feem ntau qhia nrog rau cov nram qab no cov tsiaj ntawv: ab, bc, cd, de, ea. Qhov no sab ntawm ib tug geometrical daim duab nrog vertices ib tug, b, c, d, e. Lub sum ntawm cov lengths ntawm lub sab ntawm ib tug convex polygon yog hu ua nws cov perimeter.

Lub ib ncig ntawm lub polygon

Convex uas tej zaum yuav nkag mus rau thiab piav. Vajvoog tangent rau tag nrho cov sab ntawm cov duab daim duab, hu ua lub inscribed rau hauv nws. Qhov no polygon yog hu ua piav. Qhov chaw vajvoog uas yog inscribed nyob rau hauv lub polygon yog ib tug taw tes ntawm kev tshuam ntawm bisectors ntawm cov ces kaum hauv ib tug muab duab zoo lawm. Lub cheeb tsam ntawm lub polygon yog sib npaug zos rau:

S = p * r,

qhov twg r - lub voos kheej-kheej ntawm lub inscribed vajvoog, thiab p - semiperimeter no polygon.

Ib tug vajvoog uas muaj lub polygon vertices, hu tau piav nyob ze nws. Tsis tas li ntawd, qhov no convex geometric duab hu ua inscribed. Lub voj voog center, uas yog tau piav txog xws li ib tug polygon yog ib tug thiaj li hu ua kev tshuam taw tes midperpendiculars tag nrho cov sab.

Kab pheeb ces kaum convex geometric duab

Lub diagonals ntawm ib tug convex polygon - ib ya uas txuas tsis nyob sib ze vertices. Txhua tus ntawm lawv yog nyob rau hauv no geometric duab. Cov nab npawb ntawm cov diagonals ntawm lub n-gon yog teem raws li cov mis:

N = n (n - 3) / 2.

Cov nab npawb ntawm cov diagonals ntawm ib tug convex polygon plays ib qho tseem ceeb luag hauj lwm nyob rau hauv elementary geometry. Tus nab npawb ntawm cov vajvoos peb sab (K), uas tej zaum yuav lov txhua convex polygon, xam los ntawm cov nram qab no mis:

K = n - 2.

Cov nab npawb ntawm cov diagonals ntawm ib tug convex polygon yog ib txwm nyob rau ntawm lub xov tooj ntawm vertices.

Muab faib rau ntawm ib tug convex polygon

Nyob rau hauv tej rooj plaub, los daws kom tau geometry paub tab tsim nyog mus ua txhaum ib tug convex polygon mus rau hauv ob peb lub duab peb ceg uas tsis-sib tshuam diagonals. Qhov teeb meem no yuav daws tau los ntawm tshem ib tug tej yam mis.

Txhais qhov teeb meem: hu rau txoj kev zoo ntawm muab faib rau ntawm ib tug convex n-gon rau hauv ob peb cov vajvoos peb sab los ntawm diagonals hais tias tshuam tsuas yog nyob rau hauv lub vertices ntawm ib tug geometric duab.

Tshuaj: Piv txwv hais tias P1, P2, P3, ..., Pn - rau sab saum toj ntawm lub n-gon. Number xn - lub xov tooj ntawm nws cov partitions. Ua tib zoo xav txog cov uas ua kab pheeb ces kaum duab daim duab Pi Pn. Nyob rau hauv ib yam ntawm cov kev partitions P1 Pn belongs rau ib qho kev daim duab peb sab P1 Pi Pn, nyob rau hauv uas 1

Cia kuv = 2 yog ib tug pab pawg neeg ntawm kev partitions, yeej ib txwm muaj kab pheeb ces kaum P2 Pn. Cov nab npawb ntawm cov partitions uas muaj nyob rau hauv nws, sib npaug zos rau cov xov tooj ntawm partitions (n-1) -gon P2 P3 P4 ... Pn. Nyob rau hauv lwm yam lus, nws yog sib npaug zos rau xn-1.

Yog hais tias kuv = 3, ces cov lwm pab lwm pawg partitions yuav yeej ib txwm muaj ib tug kab pheeb ces kaum P3 P1 thiab P3 Pn. Cov nab npawb ntawm cov tseeb partitions uas yog muaj nyob rau hauv cov pab pawg neeg, yuav coincide nrog lub xov tooj ntawm partitions (n-2) -gon P3, P4 ... Pn. Nyob rau hauv lwm yam lus, nws yuav tsum xn-2.

Cia kuv = 4, ces lub duab peb ceg ntawm cov tseeb muab faib yog ua txhua yam kom muaj ib tug daim duab peb sab P1 Pn P4, uas yuav adjoin lub quadrangle P1 P2 P3 P4, (n-3) -gon P5 P4 ... Pn. Cov nab npawb ntawm cov tseeb partitions xws quadrilateral sib npaug X4, thiab tus naj npawb ntawm partitions (n-3) -gon sib npaug xn-3. Raws li cov foregoing, peb yuav hais tias tag nrho cov xov tooj ntawm cov kev partitions uas yog muaj nyob rau hauv pab pawg neeg no sib npaug xn-3 X4. Lwm yam pab pawg, nyob rau hauv uas kuv = 4, 5, 6, 7 ... yuav muaj 4 xn-X5, xn-5 X6, xn-6 ... X7 kev partitions.

Cia kuv = n-2, tus xov tooj ntawm yog partitions nyob rau hauv ib tug muab pab pawg neeg yuav coincide nrog lub xov tooj ntawm partitions nyob rau hauv cov pab pawg neeg, nyob rau hauv uas kuv = 2 (nyob rau hauv lwm yam lus, sib npaug xn-1).

Txij li thaum X 1 = X 2 = 0, X3 = 1 thiab X4 = 2, ..., tus xov tooj ntawm partitions ntawm convex polygon yog:

Xn = xn-1 + xn-2 + xn-3, xn-X4 + X5 + 4 ... + X 5 + 4 xn-xn-X 4 + 3 + 2 xn-xn-1.

Piv txwv li:

X5 = X4 + X3 + X4 = 5

X6 = X4 + X5 + X4 + X5 = 14

X7 + X5 = X6 + X4 * X4 + X5 + X6 = 42

X7 = X8 + X6 + X4 * X5 + X4 * X5 + X6 + X7 = 132

Cov nab npawb ntawm cov tseeb partitions kev sib tshuam nyob rau ib kab pheeb ces kaum

Thaum kuaj tau tus neeg mob, nws yuav lam xav tias lub xov tooj ntawm diagonals ntawm convex n-gon yog sib npaug zos rau cov khoom ntawm tag nrho cov partitions ntawm no daim ntawv qauv (n-3).

Lub pov thawj ntawm no assumption: xav hais tias P1n = xn * (n-3), ces tej n-gon yuav tsum tau muab faib mus rau hauv (n-2) yog ib tug daim duab peb sab. Nyob rau hauv cov ntaub ntawv no yog ib tug ntawm lawv muaj peev xwm yuav stacked (n-3) -chetyrehugolnik. Nyob rau tib lub sij hawm, txhua quadrangle yog kab pheeb ces kaum. Vim qhov no convex geometric duab ob diagonals yuav nqa tawm, uas txhais tau tias nyob rau hauv tej (n-3) -chetyrehugolnikah tej zaum tus ntxiv kab pheeb ces kaum (n-3). Nyob rau qhov no hauv paus, peb yuav xaus uas nyob rau ntawm tej kom muab faib muaj ib lub sij hawm mus rau (n-3) -diagonali lub rooj sib tham yuav tsum ua hauj lwm no.

Cheeb Tsam convex uas

Feem ntau, nyob rau hauv kev daws ntau yam teeb meem ntawm elementary geometry muaj ib tug xav los mus txiav txim qhov chaw ntawm ib tug convex polygon. Xav hais tias hais tias (soj. Yi), i = 1,2,3 ... n sawv cev ib tug sib lawv liag ntawm coordinates ntawm tag nrho cov neighboring vertices ntawm lub polygon, tsis muaj self-kev sib tshuam. Nyob rau hauv cov ntaub ntawv no, nws cheeb tsam yog xam los ntawm cov nram qab no mis:

_{S = ½ (Σ (X i} + X i + ₁₎ (Y _{kuv +} Y _kuv + _1)),

nyob (X _1, Y _{1) = (X n +1, Y} n + _1).