Convex polygons. Definition na wani convex polygon. A diagonals na wani convex polygon

Wadannan lissafi siffofi suna kewaye da mu. Convex polygons ne na halitta, kamar wani saƙar zuma ko wucin gadi (mutumin sanya). Wadannan alkaluma an yi amfani da a samar da daban-daban na coatings a art, gine-gine, da kawa, da dai sauransu Convex polygons da dukiyar da cewa su maki kwanta a gefe daya daga wani mike layin cewa ya wuce ta biyu daga m vertices na geometrical adadi. Akwai sauran ma'anar. Yana kira da convex polygon, wanda aka shirya a guda rabin jirgin sama tare da girmamawa ga wani madaidaiciya line dauke da daya daga sãsanninta.

convex polygons

A cikin shakka daga na farko lissafi an ko da yaushe bi da musamman sauki polygons. Don gane da kaddarorin lissafi siffofi kana bukatar ka fahimci yanayinsu. Don fara gane cewa rufaffiyar ne wani layi wanda iyakar ne guda. Da kuma adadi kafa ta da shi, zai iya samun wani iri-iri jeri. Polygon aka kira sauki rufaffiyar polyline wanda m raka'a ba located a daya mike layin. Its links da nodes ne, bi da bi, da tarnaƙi kuma fi na geometrical adadi. A sauki polyline dole ba ratsa kanta.

vertices na polygon ake kira makwabta, idan sun kasance sũ ne iyakar daya daga sãsanninta. A lissafi adadi, wanda yana da n-th yawan vertices, ya kuma inganta n-th yawan jam'iyyun da ake kira n-gon. Kanta karye line ne da iyaka ko kwane-kwane na lissafi adadi. Polygonal jirgin sama ko lebur polygon kira da karshe na wani jirgin sama, su da iyaka. M bangarorin da lissafi adadi kira polyline segments da suka samo asali daga wannan kokuwa. Ba su zama makwabta, idan sun dogara ne a kan daban-daban vertices na polygon.

Sauran fassarorin convex polygons

A na farko lissafi, akwai da dama daidai a ma'anar ma'anar, na nuna abin da ake kira mai convex polygon. Haka kuma, duk wadannan kalamai ne daidai da gaskiya. A convex polygon ne daya cewa yana da:

• kowane sashi cewa ya haɗu da wani maki biyu a cikin shi, ya ta'allaka ne gaba ɗaya a cikin shi;

• cikinta karya duk da diagonals.

• wani ciki kwana ba mafi girma daga 180 °.

Polygon ko da yaushe ya raba jirgin shiga kashi biyu. Daya daga cikinsu - da iyakance (da shi za a iya kewaye a cikin wani da'irar), da kuma sauran - Unlimited. A farko an kira ciki yankin, da kuma na biyu - da matsanancin yankin na lissafi adadi. Wannan ne mahada daga cikin polygon (a cikin wasu kalmomi - jimlar bangaren) da dama rabin-jirage. Saboda haka, kowane kashi da ciwon iyakar a maki wanda kasance a wani polygon gaba daya da yake da shi.

Iri na convex polygons

Definition convex polygon ba ya nuna cewa akwai mutane da yawa irin na su. Kuma kowane daga cikinsu yana da wasu sharudda. Saboda haka, convex polygons, wanda da wani ciki kwana na 180 °, ake magana a kai dan kadan convex. A convex lissafi adadi cewa yana da uku kololuwa, aka kira alwatika, hudu - quadrilateral, biyar - Pentagon, da dai sauransu Kowace daga cikin convex n-gons ya gana da wadannan muhimmanci bukatun: .. N dole ne daidai ko mafi girma daga 3. Kowace daga cikin triangles ne convex. A lissafi adadi na da irin wannan a cikin abin da duk vertices suna located a kan wani da'irar, da ake kira rubũtacce da'irar. Aka bayyana convex polygon aka kira idan duk sãsanninta a kusa da wani da'irar zuwa taba ta. Biyu polygons ake kira daidaita kawai a cikin yanayin lokacin amfani da mai rufi za a iya hada. Flat polygon kira polygonal jirgin sama (wani jirgin saman rabo) cewa wannan iyaka geometrical adadi.

Regular convex polygons

Regular polygons kira lissafi siffofi da daidaita kusassari da kuma bangarorin biyu. Tu su akwai wani batu 0, wanda shi ne guda nesa daga kowace ta vertices. Shi ne ake kira da cibiyar da geometrical adadi. Lines a haɗa da cibiyar da vertices na lissafi adadi kira apothem, da waɗannan da connect da maki 0 tare da jam'iyyun - radii.

Daidai murabba'i mai dari - square. Equilateral alwatika ake kira equilateral. Domin irin wannan siffofi akwai wadannan mulki: kowane convex polygon kwana ne 180 ° * (n-2) / n,

inda n - yawan vertices na convex lissafi adadi.

A fannin wani yau da kullum polygon ne m da dabara:

S = p * h,

inda p ne daidai da rabin Naira Miliyan Xari da duk bangarorin na polygon, kuma h ne tsawon apothem.

Kadarorin convex polygons

Convex polygons da wasu kaddarorin. Saboda haka, kashi cewa ya haɗu da wani maki biyu na wani lissafi da adadi, dole located in shi. hujja:

Yi tsammani cewa P - da convex polygon. A sha biyu sabani maki, msl, A da B, wanda ya kamãta da P. By yanzu definition of a convex polygon, wadannan maki suna located a daya gefen a mike layin cewa ya ƙunshi wani shugabanci R. Saboda haka, AB ma yana da wannan dukiya da kuma yake kunshe a cikin R. A convex polygon ko da yaushe iya raba da dama triangles cikakken duk diagonals, wanda aka gudanar da daya daga ta vertices.

Kusassari convex lissafi siffofi

A kusassari da wani convex polygon - ne kusassari da cewa an kafa ta ne ƙungiyõyin. Tu sasanninta ne a cikin yankin na lissafi adadi. A kwana da cewa an kafa ta sãsanninta wanda converge a wata kokuwa, da ake kira kwana na convex polygon. Sasanninta m zuwa ciki sasanninta na geometrical adadi, da ake kira waje. Kowace kusurwa na wani convex polygon, shirya ciki da shi, shi ne:

180 ° - x

inda x - darajar waje kusurwa. Wannan sauki dabara ne zartar da wani irin lissafi siffofi irin wannan.

A general, domin waje sasanninta da zama na wadannan mulki: kowane convex polygon kwana daidaita da bambanci tsakanin 180 ° da kuma darajar da ciki kwana. Yana iya da dabi'u jere daga -180 ° zuwa 180 °. Saboda haka, a lokacin da ciki kwana ne 120 °, da bayyanar zai yi darajar 60 °.

A Naira Miliyan Xari da kusassari na convex polygons

A Naira Miliyan Xari da ciki kusassari da wani convex polygon aka kafa da dabara:

180 ° * (n-2),

inda n - yawan vertices na n-gon.

Jimlar kusassari da wani convex polygon da aka lasafta quite kawai. Ka yi la'akari da irin wannan lissafi da siffar. Domin sanin Naira Miliyan Xari da kusassari a convex polygon buƙatar haɗi daya daga cikin vertices zuwa wasu vertices. A sakamakon wannan mataki ya jũya (n-2) na alwatika. An sani cewa Naira Miliyan Xari da kusassari da wani alwatika ne ko da yaushe 180 °. Saboda su lambar a wani polygon daidai (n-2), Naira Miliyan Xari da ciki kusassari da adadi daidai 180 ° x (n-2).

Adadin convex polygon sasanninta, wato, kowane biyu m ciki da waje kusassari zuwa gare su, a cikin wannan convex lissafi adadi zai zama ko da yaushe daidaita 180 °. A wannan hasashe, za mu iya sanin Naira Miliyan Xari da dukkan sasanninta:

180 x n.

A Naira Miliyan Xari da ciki kusassari ne 180 ° * (n-2). Haka kuma, da Naira Miliyan Xari duk da matsanancin kusurwa ta adadi kafa ta da dabara:

180 ° * n-180 ° - (n-2) = 360 °.

Sum na waje kusassari da wani convex polygon za su kasance kullum daidai da 360 ° (ko da kuwa da yawan sãsanninta).

Waje kusurwa na wani convex polygon ake kullum wakilta bambanci tsakanin 180 ° da kuma darajar da ciki kwana.

Other Properties na wani convex polygon

Bayan da asali Properties na lissafi Figures data, su ma suna da sauran, wanda ke faruwa a lokacin da tanadin su. Saboda haka, wani daga polygons iya raba cikin mahara convex n-gons. Don yin wannan, ci gaba da kowanne daga sãsanninta, kuma a yanka da na lissafi da siffar tare da wadannan madaidaiciya Lines. Tsattsage wani polygon cikin da dama convex sassa ne yiwu kuma haka cewa saman kowane daga cikin guda zo daidai da duk da vertices. Daga mai geometrical adadi na iya zama mai sauqi qwarai yi triangles ta hanyar duk diagonals daga daya kokuwa. Saboda haka, duk wani polygon, kyakkyawan, za a iya raba wani yawan triangles, wanda shi ne mai amfani wajen warware daban-daban ayyuka da suka shafi wannan geometrical siffofi.

A kewaye da convex polygon

A segments na polyline, polygon-kira jam'iyyun, sau da yawa ya nuna da wadannan haruffa: ab, bc, cd, de, EA. Wannan gefen wani geometrical adadi da vertices wani, b, c, d, e. A Naira Miliyan Xari da tsawo daga cikin bangarorin da wani convex polygon aka kira ta kewaye.

A karkara na polygon

Convex polygons iya shiga da kuma bayyana. Circle tangent to duk bangarorin na lissafi adadi, da ake kira rubũtacce a cikin shi. Wannan polygon aka kira da aka bayyana. A cibiyar da'irar da ke rubuce a polygon ne wani batu na rarrabawa da bisectors na kusassari cikin wani da aka ba lissafi siffar. A yankin na polygon ne daidai to:

S = p * r,

inda r - da radius daga cikin rubũtacce da'irar, da kuma p - semiperimeter wannan polygon.

A da'irar dauke da polygon vertices, da ake kira da aka bayyana a kusa da shi. Bugu da ƙari, wannan convex lissafi adadi da ake kira rubũtacce. A da'irar cibiyar, wanda aka bayyana game da irin wannan polygon ne ake kira mahada batu midperpendiculars dukkan bangarorin.

Diagonal convex lissafi siffofi

A diagonals na wani convex polygon - wani sashi cewa ya haɗu ba makwabtaka da vertices. Kowane daga cikinsu shi ne a cikin wannan lissafi adadi. Yawan diagonals na n-gon da aka kafa bisa ga dabara:

N = n (n - 3) / 2.

Yawan diagonals na wani convex polygon taka muhimmiyar rawa a cikin na farko lissafi. Yawan triangles (K), wanda na iya karya kowane convex polygon, lasafta ta da wadannan dabara:

K = n - 2.

Yawan diagonals na wani convex polygon ne ko da yaushe dogara a kan yawan vertices.

Bangare na wani convex polygon

A wasu lokuta, don warware lissafi ayyuka zama dole ya karya wani convex polygon cikin da dama triangles da wadanda ba intersecting diagonals. Wannan matsala za a iya warware ta cire wani dabara.

Ma'ana matsalar: kira dama irin bangare na wani convex n-gon cikin da dama triangles da diagonals cewa rarraba kawai a vertices na wani lissafi da adadi.

Magani: A ce P1, P2, P3, ..., Pn - saman n-gon. Number Xn - da yawan ta partitions. Hankali la'akari da sakamakon diagonal lissafi adadi PI Pn. A wani na yau da kullum partitions P1 Pn nasa ne da wani alwatika P1 PI Pn, a cikin abin da 1

Bari i = 2 ne a rukuni na yau da kullum da partitions, ko da yaushe dauke da diagonal P2 Pn. Yawan partitions da suke kunshe a cikin ta, daidai da adadin partitions (n-1) -gon P2 P3 P4 ... Pn. A wasu kalmomin, shi ne daidaita Xn-1.

Idan i = 3, to, da sauran kungiyar partitions zai ko da yaushe dauke da wani diagonal P3 P1 da P3 Pn. Yawan daidai partitions cewa suna kunshe a cikin kungiyar, zai zo daidai da yawan partitions (n-2) -gon P3, P4 ... Pn. A wasu kalmomin, zai zama Xn-2.

Bari i = 4, sa'an nan da triangles daga cikin daidai bangare aka daure su dauke da wani alwatika P1 Pn P4, wanda zai adjoin da quadrangle P1 P2 P3 P4, (n-3) -gon P5 P4 ... Pn. Yawan daidai partitions irin quadrilateral daidai X4, da kuma yawan partitions (n-3) -gon daidai Xn-3. Bisa ta gabatar ba, zamu iya cewa da total number na yau da kullum partitions cewa suna kunshe a cikin wannan kungiya daidai Xn-3 X4. Sauran kungiyoyin, a cikin abin da i = 4, 5, 6, 7 ... zai dauke 4 Xn-X5, Xn-5 X6, Xn-6 ... X7 yau da kullum partitions.

Bari i = n-2, yawan daidai partitions a ba kungiyar za ta zo daidai da yawan partitions a cikin ƙungiyar, a cikin abin da i = 2 (a cikin wasu kalmomi, daidai Xn-1).

Tun x1 = x2 = 0, X3 = 1 da kuma X4 = 2, ..., yawan partitions na convex polygon ne:

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

misali:

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

Yawan daidai partitions intersecting cikin daya diagonal

Lokacin da dubawa mutum lokuta, shi za a iya zaci cewa yawan diagonals na convex n-gon ne daidaita da samfurin dukkan partitions wannan ginshiƙi juna (n-3).

The hujja da wannan zato: zaton cewa P1n = Xn * (n-3), sa'an nan wani n-gon iya raba (n-2) ne a alwatika. A wannan yanayin da daya daga cikinsu za a iya suya (n-3) -chetyrehugolnik. A daidai wannan lokaci, kowane quadrangle ne diagonal. Tun da yake wannan convex lissafi adadi biyu diagonals za a iya za'ayi, wanda ke nufin cewa a wani (n-3) -chetyrehugolnikah iya gudanar da ƙarin diagonal (n-3). A wannan hasashe, za mu iya cewa a wani bangare dace yana da wata damar (n-3) -diagonali taron da bukatun wannan aiki.

Area convex polygons

Sau da yawa, a warware matsaloli daban-daban na firamare lissafi akwai bukatar sanin fannin wani convex polygon. Ɗauka cewa (Xi. Yi), i = 1,2,3 ... n wakiltar wani jerin tsarawa daga duk makwabta vertices na polygon, da ciwon ba kai-intersections. A wannan yanayin, ta yanki da aka lasafta ta da wadannan dabara:

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

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