Ngemapholigoni convex. Definition of a ipholigoni convex. I diagonals we ipholigoni ayizindunduma

Lezi zeJiyomethri yonke indawo. Convex ngemapholigoni kukhona zemvelo, ezinjengomoya lezinyosi noma yokufakelwa (indoda wenza). Lezi zibalo zisetshenziswa ekukhiqizeni izinhlobo ezahlukene ezinengilazi kwezobuciko, ukuklanywa kwezakhiwo, imihlobiso, njll ngemapholigoni convex babe impahla ukuthi amaphuzu abo amanga ngakolunye uhlangothi lwale umugqa oqondile okudlula pair of vertices eduze sibalo Jomethri. Zikhona nezinye izincazelo. Kuthiwa ipholigoni convex, okuyinto ahlelwe nengxenye-indiza olulodwa maqondana nanoma yikuphi okuqukethwe umugqa oqondile equkethe eyodwa ezinhlangothini zalo.

ngemapholigoni ayizindunduma

Ngokuhamba geometry aphansi njalo aphathwe ngemapholigoni ngokwedlulele elula. Ukuze uqonde property ngamajamo weJiyomethri udinga ukuqonda ubunjalo babo. Ukuze ukuthoma ukuzwisisa bonyana avaliwe yiliphi umugqa kabani semikhawulweni ziyafana. Futhi isibalo ezakhiwe ke, ungaba ezihlukahlukene ukucupha. Polygon libizwa polyline elula avaliwe amayunithi kabani eduze ungekho kwenye umugqa oqondile. link Kwalo ISIZINDA kukhona, ngokulandelana, izinhlangothi bese iziqongo sibalo Jomethri. A polyline elula akumelwe aphambana uqobo.

vertices the ipholigoni abizwa ngokuthi omakhelwane, uma kukhona imikhawulo eyodwa ezinhlangothini zalo. Isibalo weJiyomethri, okuyinto has a inombolo n-th of vertices, futhi kungakho inombolo n-th amaqembu ngokuthi i-n-gon. Wona umugqa eziphukile liwumngcele noma ukunquma nentaba sibalo Jomethri. indiza zikanhlangothiningi noma ipholigoni flat ngokuthi engxenyeni yokugcina iyiphi indiza, elinganiselwe yabo. izinhlangothi Eduze sibalo Jomethri ngokuthi izingxenye polyline okusuka vertex efanayo. Ngeke omakhelwane uma zisekelwe vertices ezahlukene ipholigoni.

Okunye izincazelo ngemapholigoni ayizindunduma

Ngo-geometry aphansi, kukhona okulingana eziningana linencazelo izincazelo, okubonisa okubizwa ipholigoni convex. Ngaphezu kwalokho, zonke lezi zitatimende ngokulinganayo kweqiniso. A ipholigoni ayizindunduma nguye has:

• ingxenye ngayinye exhumanisa yimuphi amaphuzu amabili esikhathini ke, amanga ngokuphelele kuwo;

• therein amanga zonke diagonals yayo;

• noma iyiphi engela ingaphakathi asisikhulu kunenkosi 180 °.

Polygon njalo uhlukanisa indiza yaba izingxenye ezimbili. Omunye wabo - the kungagcini (kungaba azungezwe umbuthano), kanye nezinye - angenamkhawulo. Eyokuqala ibizwa ngokuthi esifundeni kwangaphakathi, kanti eyesibili - endaweni elingaphandle sibalo Jomethri. Lena empambana-polygon (ngamanye amagama - ingxenye Imininingwane) eziningana isigamu-ezindizeni. Ngakho, ingxenye ngayinye inezinsuku imikhawulo ezindaweni okungokwendlela ipholigoni kungokwalabo ngokuphelele kuye.

Izinhlobonhlobo ngemapholigoni ayizindunduma

Incazelo ipholigoni ayizindunduma akubonisi ukuthi kukhona izinhlobo eziningi zazo. Futhi ngamunye wabo izimfuneko ezithile. Ngakho, ngemapholigoni convex, ezithinta engela sangaphakathi 180 °, okukhulunywe kancane convex. I ayizindunduma sibalo beJiyomethri lokho has iziqongo ezintathu, ibizwa ngokuthi unxantathu, ezine - quadrilateral, emihlanu - pentagon, njll ngamunye we ayizindunduma n-gons sihlangabezana nalezi zidingo ezilandelayo ezibalulekile: .. N kufanele alingane noma enkulu kuno 3. Ngamunye aboncantathu kuyinto convex. Lesi sibalo Jomethri yalolu hlobo lapho bonke vertices zitholakala embuthanweni, ngokuthi emjikelezweni olotshiwe. Kuchazwe ipholigoni ayizindunduma ibizwa ngokuthi uma zonke nezinhlangothi zalo nxazonke kumbuthano ukuba umthinte. polygon amabili abizwa ngokuthi alinganayo kuphela icala uma usebenzisa Imbondela zingahlanganiswa. ipholigoni Flat ngokuthi indiza zikanhlangothiningi (ingxenye indiza) ukuthi lesi sibalo elinganiselwe yejeyomethri.

ngemapholigoni avamile ayizindunduma

ngemapholigoni njalo ngokuthi zeJiyomethri ne-engeli alinganayo futhi izinhlangothi. Ngaphakathi kuwo kukhona iphuzu 0, okuyinto elifanayo ukusuka ngamunye vertices zayo. Ibizwa ngokuthi maphakathi sibalo Jomethri. Lines yokuxhuma maphakathi ne vertices sibalo Jomethri ngokuthi apothem nalabo ukuxhuma iphuzu 0 ne amaqembu - radii.

isikwele Lungisa - sikwele. unxantathu Equilateral libizwa equilateral. Ukuze bobunjwa kukhona mithetho elandelayo: ayizindunduma ipholigoni engela ngayinye 180 ° * (n-2) / n,

lapho n - nenani vertices the ayizindunduma sibalo Jomethri.

Le ndawo kwanoma iyiphi ipholigoni njalo kunqunywa ifomula:

S = p * h,

lapho p ilingana isigamu isamba sazo zonke izinhlangothi ipholigoni, futhi h iyona apothem obuphelele.

Izakhiwo ngemapholigoni ayizindunduma

Convex ngemapholigoni babe izakhiwo ezithile. Ngakho, ingxenye exhumanisa yimuphi amaphuzu amabili sibalo Jomethri, ngempela itholakala kuwo. ubufakazi:

Ake sithi P --polygon convex. Thatha amaphuzu amabili kabi amandla akhe, isib, A no-B, okungokwendlela P. ngu yencazelo samanje a ipholigoni convex, la maphuzu zitholakala endaweni eyodwa ohlangothini emgceni locondzile equkethe isiphi isiqondiso R. Ngenxa yalokho, AB Ubuye lokhu impahla kanye kuqukethwe R. A ipholigoni ayizindunduma njalo ingahlukaniswa onxantathu eziningana ngokuphelele zonke diagonals, okuyinto wayebambe esinye vertices zayo.

Angles ayizindunduma ngamajamo weJiyomethri

I-engeli we ipholigoni ayizindunduma - kukhona engeli ukuthi akhiwa amaqembu. emakhoneni Ngaphakathi kukhona endaweni ngaphakathi sibalo Jomethri. I-engeli esenzelwa ngu ezinhlangothini zalo okuyinto bukhomba ngesikhathi vertex, ngokuthi engela ipholigoni convex. Emakhoneni eduze emagumbini sangaphakathi sibalo Jomethri, ngokuthi zangaphandle. ekhoneni ngalinye ipholigoni convex, wahlela ngaphakathi kuwo, ithi:

180 ° - x

lapho x - Inani ngaphandle ekhoneni. Lokhu ifomula elula usebenza noma uhlobo ngamajamo weJiyomethri ezifana.

Ngokuvamile, i-emakhoneni ngaphandle akhona elandelayo umthetho: ngalinye ayizindunduma ipholigoni engela elilingana umehluko phakathi 180 ° futhi inani le-engeli ingaphakathi. Kungaba abe namanani ezisukela -180 ° 180 °. Ngenxa yalokho, lapho engela bangaphakathi 120 °, ukubukeka uzoba ukubaluleka 60 °.

Isamba engele of ngemapholigoni ayizindunduma

Isamba engele angaphakathi a ipholigoni ayizindunduma inqunywe ifomula:

180 ° * (n-2),

lapho n - nenani vertices the n-gon.

Isamba engeli we ipholigoni ayizindunduma ibalwa impela nje. Cabanga noma iyiphi ukuma okunjalo weJiyomethri. Ukuze sithole isamba engeli endaweni ipholigoni ayizindunduma kudingeka ukuba uxhume omunye vertices baso kwezinye vertices. Ngenxa lesi senzo uphendulela (n-2) unxantathu. Kuyaziwa ukuthi isamba engeli kwanoma iyiphi unxantathu uhlale 180 °. Ngenxa yokuthi isibalo sabo kunoma iyiphi ipholigoni kulingana (n-2), isamba engeli ingaphakathi sibalo kulingana 180 ° x (n-2).

Kwinani ayizindunduma ipholigoni emakhoneni, okungukuthi, noma yimuphi eduze engeli amabili zangaphakathi nezangaphandle kubo, kule sibalo ayizindunduma weJiyomethri uyohlale elilingana 180 °. Kulesi sisekelo, singakwazi ukuthola isamba sazo zonke emagumbini alo:

180 x n.

Isamba engele elingaphakathi 180 ° * (n-2). Ngakho, isamba wonke amakhona elingaphandle sibalo esabekwa ifomula:

180 ° * n-180 ° - (n-2) = 360 °.

Sum engele yangaphandle kwanoma iyiphi ipholigoni ayizindunduma uyohlale elilingana 360 ° (kungakhathaliseki inani ezinhlangothini zalo).

ekhoneni ngaphandle ipholigoni ayizindunduma ngokuvamile emelelwa umehluko phakathi 180 ° futhi inani le-engeli ingaphakathi.

Ezinye izimpahla we ipholigoni ayizindunduma

Ngaphandle izakhiwo ezisemqoka zedatha Jomethri izibalo, babe nezinye, okuyinto zenzeka lapho usingatha nabo. Ngakho, noma iyiphi ngemapholigoni zingase zihlukaniswe zibe ayizindunduma amaningi n-gons. Ukuze wenze lokhu, qhubeka ngalunye lwaso wanquma ukuma weJiyomethri okuphathelene nalezi zimo ngqo. Hlukanisa iyiphi ipholigoni izingxenye eziningana ayizindunduma kungenzeka futhi ukuze phezulu ngamunye izingcezu uqondane bonke emavetheksi yayo. Kusukela sibalo Jomethri kungaba silula ukwenza onxantathu ngokusebenzisa zonke diagonals komunye vertex. Ngakho, noma iyiphi ipholigoni, ekugcineni, zingahlukaniswa zibe inani elithile onxantathu, okuwusizo kakhulu ekuxazululeni imisebenzi ehlukahlukene ehlobene enjalo beJiyomethri nobujamo okungasibo beJiyomethri.

Azungezwe we ipholigoni ayizindunduma

I izingxenye polyline, amaqembu ipholigoni okuthiwa, ngokuvamile wabonisa izincwadi ezilandelayo: ab, bc, cd, de, ea. Lokhu uhlangothi sibalo Jomethri nge vertices a, b, c, d, e. Isamba ubude izinhlangothi a ipholigoni ayizindunduma libizwa ipherimitha yayo.

Obungazungeza ipholigoni

Convex ngemapholigoni angafakwa futhi kuchazwe. Circle tangent zonke izinhlangothi sibalo Jomethri, ebizwa ngokuthi elalibhalwe kuwo. Lokhu ipholigoni ibizwa ngokuthi kuchazwe. Umbuthano isikhungo eqoshwe ku-polygon kuyinto iphuzu empambana bisectors ka-engeli ngaphakathi elinikeziwe ukuma weJiyomethri. Indawo ipholigoni ilingana:

S = p * r,

ukuphi - lo engaba umbuthano olotshiwe, futhi p - semiperimeter lokhu ipholigoni.

A ngombuthano oqukethe vertices ipholigoni, elibizwa echazwe eduze kwazo. Ngaphezu kwalokho, lesi sibalo ayizindunduma weJiyomethri elibizwa olotshiwe. Inombolo yomhlaba wonke yesikhungo mbuthano, elichazwa mayelana ipholigoni esinjalo okuthiwa empambana iphuzu midperpendiculars zonke izinhlangothi.

Ovundlile Umumo ayizindunduma weJiyomethri

I diagonals we ipholigoni ayizindunduma - ingxenye exhumanisa hhayi engumakhelwane vertices. Ngamunye wabo ongaphakathi kulesi sibalo Jomethri. Isibalo diagonals we n-gon isethwe ngokuvumelana ifomula:

N = n (n - 3) / 2.

Isibalo diagonals we ipholigoni ayizindunduma linendima ebalulekile geometry aphansi. Isibalo onxantathu (K), okungase lephule lonke ipholigoni convex, ebalwe fomula elandelayo:

K = n - 2.

Isibalo diagonals we ipholigoni ayizindunduma uhlale kuncike inani vertices.

Hlukanisa we ipholigoni ayizindunduma

Kwezinye izimo, ukuxazulula geometry imisebenzi edingekayo ukuze ngephule ipholigoni ayizindunduma ku onxantathu eziningana nge diagonals non-intersecting. Lokhu inkinga ingaxazululwa ngokususa ifomula ezithile.

Kuchazwa inkinga: shayela uhlobo olufanele ukwahlukanisa a ayizindunduma n-gon ku onxantathu eziningana diagonals aphambana kuphela vertices of a sibalo Jomethri.

Isixazululo: Ake sithi P1, P2, P3, ..., PN - phezulu n-gon. Inombolo XN - isibalo ngodonga yayo. Zicabangele ngokucophelela okuholela idayagonali sibalo Jomethri Pi PN. Noma we ngodonga njalo P1 PN ngowomnikazi unxantathu ethile P1 Pi PN, lapho 1

Ake i = 2 iqembu ngodonga njalo, equkethe njalo idayagonali P2 PN. Isibalo ngodonga ezifakwe kulo, ilingana nenani ngodonga (n-1) -gon P2 P3 P4 ... PN. Ngamanye amazwi, kuba elilingana XN-1.

Uma i = 3, bese omunye ngodonga iqembu uyohlale aqukethe idayagonali P3 P1 futhi P3 PN. Isibalo ngodonga elifanele eziqukethwe iqembu, zizofana inani ngodonga (n-2) -gon P3, P4 ... PN. Ngamanye amazwi, kuyoba XN-2.

Vumela i = 4, ke onxantathu phakathi ukwahlukanisa lesifanele nakanjani aqukethe unxantathu P1 PN P4, okuzokwenza adjoin le quadrangle P1 P2 P3 P4, (n-3) -gon P5 P4 ... PN. Isibalo ngodonga lesifanele quadrilateral enjalo kulingana X4, futhi inani ngodonga (n-3) -gon kulingana XN-3. Ngokusekelwe osekushiwo, singasho ukuthi inani eliphelele ngodonga njalo ukuthi iqukethwe kuleli qembu kulingana XN-3 X4. Amanye amaqembu, lapho i = 4, 5, 6, 7 ... sizoqukatha 4 XN-X5, XN-5 X6, XN-6 ... X7 ngodonga njalo.

Ake i = n-2, inani ngodonga elungile iqembu inikezwe zizofana inani ngodonga eqenjini, lapho i = 2 (ngamanye amagama, kulingana XN-1).

Njengoba X1 = X2 = 0, X3 = 1 futhi X4 = 2, ..., inani ngodonga of ipholigoni ayizindunduma kuyinto:

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

Ngokwesibonelo:

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

Isibalo ngodonga lesifanele intersecting ngaphakathi eyodwa idayagonali

Lapho ehlola amacala ngabanye, kungaba bazitshela ukuthi isibalo diagonals lika-n-gon ayizindunduma ilingana umkhiqizo zonke ngodonga walesi sifanekiso ishadi (n-3).

Ubufakazi lokhu nokucabanga: ake sithi P1n = XN * (n-3), bese wonke n-gon ingahlukaniswa (n-2) iyi-unxantathu. Kulokhu omunye wabo kungaba embondelene (n-3) -chetyrehugolnik. Ngesikhathi esifanayo, quadrangle ngamunye idayagonali. Njengoba lesi sibalo ayizindunduma weJiyomethri diagonals amabili kungaba kwenziwe, okusho ukuthi noma iyiphi (n-3) -chetyrehugolnikah angase aqhube ezengeziwe idayagonali (n-3). Kulesi sisekelo, singaphetha ngokuthi ngasiphi ukwahlukanisa efanele has ithuba (n-3) Umhlangano -diagonali nezidingo lo msebenzi.

Area ngemapholigoni ayizindunduma

Ngokuvamile, ekuxazululeni izinkinga ezihlukahlukene geometry aphansi kunesidingo ukunquma kwendawo ipholigoni convex. Ucabange ukuthi (Xi. Yi), i = 1,2,3 ... n kujamele ukulandelana izixhumanisi bonke emavetheksi angomakhelwane we ipholigoni, abangenalo self-zemigwaqo. Kulokhu, endaweni yayo ibalwa le ndlela elandelayo:

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

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