Convex polygons. Definition of convex polygon. The diagonals munhu convex polygon

Izvi geometric zvitarisiko vose yakatipoteredza. Convex polygons vari dzinongoitika, akadai wouchi kana chakagadzirwa (murume akaita). Izvi manhamba zvinoshandiswa kugadzira mhando dzakasiyana-siyana coatings mu mifananidzo, mapurani, zvishongo, etc. Convex polygons vane pfuma kuti avo pfungwa okurutivi rumwe rutivi achirurama mutsetse kuti inopfuura nomunyika peya pedyo vertices yacho geometrical nhamba. Pane dzimwe tsanangudzo. It akadana convex polygon, izvo zvakarongwa rimwe hafu-ndege panyaya chero kururamiswa mutsetse ine rimwe kumativi acho.

convex polygons

Nokufamba chepuraimari geometry munogara kurapwa zvikuru nyore polygons. Kuti tinzwisise ehupfumi pamusoro geometric zvitarisiko zvaunofanira kunzwisisa hunhu hwavo. Kutanga kunzwisisa kuti namahwindo ndeuya mutsetse ane miromo zvakafanana. Uye nhamba yacho kuumbwa narwo, vanogona zvakasiyana siyana configurations. Polygon anonzi nyore namahwindo polyline vane zviumbu pedyo haana riri nomumwe paakatarisana mutsetse. Links ayo uye nodes vari ukuwo, mativi uye misoro geometrical nhamba. A nyore polyline ngaarege piringisha pacharo.

vertices of polygon vanonzi vavakidzani, kana vari kumagumo rimwe kumativi acho. A geometric mufananidzo, ine N-fung uwandu vertices, uye saka N-fung uwandu mapato inonzi N-gon. Pacharo akaputsika mutsetse ndiwo muganhu kana gandiwa pamusoro geometric nhamba. Ndege Polygonal kana sandara polygon inonzi okupedzisira chero ndege, avo zvishoma. Pedyo mativi geometric nhamba inonzi polyline zvikamu anobva kubva chete vertex. Vachange vasiri vavakidzani kana vari kwakavakirwa siyana vertices ari polygon.

Other tsanangudzo convex polygons

Chepuraimari geometry, pane dzakawanda rakafanana zvinoreva tsanangudzo, zvichiratidza chii inonzi convex polygon. Uyezve, mashoko aya ose vari zvazvakangoitawo. A convex polygon ndiye ane:

• mumwe chidimbu unobatanidza chero mbiri pfungwa mukati maro, riri zvachose mairi;

• mairi nhema diagonals yayo yose;

• chero angled mukati mukuru 180 °.

Polygon nguva dzose anokamura ndege muzvikamu zviviri. Mumwe wavo - shoma (inogona akanyudzwa mudenderedzwa), uye mumwe - risingagumi. Chokutanga anonzi yomukati nharaunda, uye wechipiri - chokunze nzvimbo geometric nhamba. Ndiyo mharadzano of polygon (nemamwe mazwi - yose zvinoriumba) akawanda hafu-ndege. Saka, mumwe chidimbu chine mativi dzimwe pfungwa iyo nhengo imwe polygon zvachose nderake.

Mhando dzakasiyana-siyana convex polygons

Definition convex polygon hariratidzi kuti kune marudzi mazhinji. Uye mumwe nomumwe wavo ane zvimwe zvinodiwa. Nokudaro, convex polygons, izvo zvine womukati pakona rimwe 180 °, anotaurwa zvishoma convex. The convex geometric mufananidzo kuti ane Nhongonya matatu, anonzi netriangle, mana - quadrilateral, shanu - nePentagon, etc. neimwe convex N-gons asangana zvinotevera zvinokosha zvinodiwa: .. N anofanira kuva akaenzana kana kupfuura 3. imwe neimwe zvokuridza ndiyo convex. The geometric chirungamutauro ichi mhando umo vose vertices dziri pamusoro denderedzwa, rinonzi rakanyorwa denderedzwa. Zvinorondedzerwa convex polygon anonzi kana kumativi ayo ose akapoteredza denderedzwa kumubata. polygons mbiri dzinonzi vakaenzana chete panyaya kana kushandisa kuifukidza dzinogona kubatanidzwa. Flat polygon inonzi polygonal ndege (ndege mugove) kuti ichi shoma geometrical nhamba.

Regular convex polygons

polygons dzose inonzi geometric zvitarisiko zvakaenzana pamakona uye mativi. Inside pavo pane pfungwa 0, inova kure chete kubva nomumwe vertices ayo. Inonzi Nhongonya geometrical nhamba. Lines unobva pakati pamwe vertices ari geometric nhamba inonzi apothem, uye avo kuti kubatanidza pfungwa 0 pamwe mapato - radii.

Rairira rectangle - mativi. Equilateral netriangle anonzi equilateral. For zvitarisiko kwakadaro pane zvinotevera mutemo: mumwe convex polygon angled iri 180 ° * (n-2) / N,

apo N - Nhamba vertices ari convex geometric nhamba.

Nzvimbo ipi dzose polygon iri pakutsvaga nzira:

S = peji * H,

apo peji akaenzana hafu Verengai kumativi ose polygon, uye H nokureba apothem.

Properties convex polygons

Convex polygons vane zvimwe zvivako. Nokudaro, chidimbu unobatanidza chero mbiri dzinokosha munhu geometric nechipi, hazvo iri mariri. uchapupu:

Ngatitii P - the convex polygon. Tora mbiri zvemasanga pfungwa, semuenzaniso, A uye B, izvo ndezvaJehovha P. By mamiriro tsanangudzo convex polygon, pfungwa idzi dziri pane rimwe divi yakarurama mutsetse kuti rine chero mirayiridzo R. Somugumisiro, AB anewo pfuma iyi iri mubhuku R. A convex polygon nguva dzose anogona rakakamurwa zvokuridza akawanda chose zvose diagonals, iyo akabata imwe vertices ayo.

Pamakona convex geometric zvitarisiko

The pamakona munhu convex polygon - ndivo pamakona kuti inoumbwa mapato. Mukati makona vari mukati nzvimbo geometric nhamba. The angled kuti chakaumbwa kumativi aro wekuita panguva vertex, yainzi pakona rimwe convex polygon. Makona yakabatana kune dzakawanda makona geometrical nechipi, inonzi zvekunze. Mumwe pakona convex polygon, akaronga mukati maro, ndiyo:

180 ° - X

apo X - ukoshi kunze mukona. Iyi nzira nyore rinoshanda chero mhando geometric zvitarisiko dzakadaro.

Kazhinji, nokuti kunze pamakona dziripo kutevera mutemo: mumwe convex polygon yorusvingo akaenzana musiyano pakati 180 ° uye kukosha zvokunyaradza yorusvingo. Inogona kuva netsika kubvira -180 ° kusvika 180 °. Somugumisiro, apo yomukati angled iri 120 °, pakaita achava kukosha 60 °.

Verengai pamakona ose convex polygons

Verengai zvokunyaradza pamakona ane convex polygon chinosimbiswa pamusoro nzira:

180 ° * (n-2),

apo N - Nhamba vertices ari N-gon.

Verengai pamakona ane convex polygon rinowanikwa zvikuru nyore. Funga chinhu geometric chimiro chakadaro. Kuti tizive Verengai pamakona ari convex polygon vanofanira kubatanidza mumwe vertices yayo dzimwe vertices. Somugumisiro chiito ichi anotendeuka (n-2) pamusoro Triangle. Zvinozivikanwa kuti vakaverenga pamakona upi netriangle anogara 180 °. Nekuti kuwanda kwavo chero polygon vakaenzana (n-2), vakaverenga mukati pamakona ari nhamba vakaenzana 180 ° X (n-2).

Amount convex polygon makona, kureva chero maviri pedyo zvemukati uye zvimwewo pamakona kwavari, mune ichi convex geometric mufananidzo achagara akaenzana 180 °. On hwaro ichi, tinogona kuona uwandu dzayo pamakona ose:

180 X n.

Verengai mukati pamakona ndiyo 180 ° * (n-2). Saizvozvowo, Verengai vanhu vose dzokunze makona mufananidzo wakasiyiwa fomura:

180 ° * N-180 ° - (n-2) = 360 °.

Verenga chokunze pamakona upi convex polygon achagara akaenzana 360 ° (zvisinei nhamba kumativi acho).

Kunze pakona convex polygon kazhinji dzaimiririrwa musiyano pakati 180 ° uye kukosha zvokunyaradza yorusvingo.

Zvimwe zvivako ane convex polygon

Kunze dzinokosha ehupfumi ose geometric manhamba data, ivo vane zvimwe, izvo kuitika kana kubata navo. Saka, chero polygons anogona rwakakamukana vakawanda convex N-gons. Kuti aite izvi, ramba nomumwe kumativi acho uye kucheka geometric chimiro pamwe izvi ngaatarire mitsetse. Akatsemura chero polygon kupinda convex kumativi akawanda kunobvira uye kuti peimwe neimwe nhindi uwirirane ose vertices dzayo. Kubva geometrical mufananidzo kunogona nyore chaizvo kuti zvokuridza kuburikidza ose diagonals kubva vertex. Saka, chero polygon, pakupedzisira, unogona kukamurwa kuva imwe nhamba zvokuridza, izvo chaizvo kugadzirisa mabasa akasiyana-siyana zvine chokuita geometrical zvitarisiko dzakadaro.

The mukomberedzo of convex polygon

The zvikamu zviri polyline, mapato polygon-inonzi, kazhinji akaratidza pamwe zvinotevera tsamba: AB, bc, CD, de, ea. divi uyu geometrical mufananidzo pamwe vertices A, B, C, D, e. Verengai urefu kumativi ari convex polygon anonzi mukomberedzo kwayo.

The sekamufarenzi of polygon

Convex polygons anogona akapinda uye kurondedzerwa. Circle tangent vose kumativi geometric mufananidzo, akaritumidza yakanyorwa mariri. polygon ichi anonzi inorondedzerwa. Muzinda denderedzwa riri akanyorwa polygon ari kurehwa mharadzano pamusoro bisectors ose pamakona mukati rakapihwa geometric chimiro. Nharaunda polygon akaenzana:

S = peji * R,

apo R - the nharaunda yacho yakanyorwa denderedzwa, uye peji - semiperimeter polygon ichi.

A denderedzwa ine polygon vertices, havanzi zvinorondedzerwa pedo naro. Uyezve, ichi convex geometric anonzi akanyorwa. The denderedzwa centre, iyo inotsanangurwa pamusoro polygon yakadaro inonzi mharadzano nguva midperpendiculars mativi ose.

Diagonal convex geometric zvitarisiko

The diagonals munhu convex polygon - mumwe chidimbu unobatanidza kwete vakavakidzana vertices. Mumwe nomumwe wavo ari mukati ichi geometric nhamba. Kuwanda diagonals ari N-gon wakaisirwa maererano fomura:

N = N (n - 3) / 2.

Kuwanda diagonals ane convex polygon inoita basa rinokosha chepuraimari geometry. Kuwanda zvokuridza (K), izvo zvinogona kutyora ose convex polygon, dzaona kubudikidza mutoo:

K = N - 2.

Kuwanda diagonals ane convex polygon anogara novimba uwandu vertices.

Mupanda ane convex polygon

Mune zvimwe zviitiko, kugadzirisa geometry mabasa zvakafanira kuti kutyora convex polygon kupinda wandei zvokuridza nevasiri intersecting diagonals. dambudziko iri rinogona kugadziriswa nokubvisa imwe nzira.

Nokududza dambudziko: dana Kodzero partition ane convex N-gon dzikapinda zvokuridza akawanda nokuda diagonals kuti piringisha chete panguva vertices ane geometric nhamba.

Solution: Ngatiti p1, p2, P3, ..., PN - kumusoro N-gon. Number Xn - nhamba partitions ayo. Kunyatsoongorora zvichiguma diagonal geometric mufananidzo Pi PN. In chero dzose partitions p1 PN ndorwevaya imwe, triangle p1 Pi PN, umo 1

Regai Ini = 2 pane boka dzose partitions, nguva dzose ane diagonal p2 PN. Kuwanda partitions kuti vanobatanidzwa mairi, zvakaenzana nokuwanda partitions (n-1) -gon p2 P3 P4 ... PN. Mune mamwe manzwi, ndiye akaenzana Xn-1.

Kana ini = 3, ipapo vamwe boka partitions vachagara iine diagonal P3 p1 uye P3 PN. Kuwanda zvakarurama partitions kuti huri boka, zvichange zvakafanana nhamba partitions (n-2) -gon P3, P4 ... PN. Mune mamwe manzwi, zvichatyisa Xn-2.

Ngaaiswe Ini = 4, ipapo zvokuridza pakati zvakarurama partition wakasungwa iine netriangle p1 PN P4, izvo zvichaita adjoin ari quadrangle p1 p2 P3 P4, (n-3) -gon P5 P4 ... PN. Kuwanda zvakarurama partitions quadrilateral kwakadaro akaenzana X4, uye uwandu partitions (n-3) -gon vakaenzana Xn-3. Pachishandiswa zvataurwa, tinogona kutaura kuti uwandu dzose partitions kuti huri boka iri vakaenzana Xn-3 X4. Other mapoka, umo ini = 4, 5, 6, 7 ... ichange iine 4 Xn-X5, Xn-5 X6, Xn-6 ... X7 dzose partitions.

Regai Ini = N-2, nhamba akarurama partitions vari kupiwa boka zvichange zvakafanana nhamba partitions vari boka, chandisina = 2 (nemamwe mazwi, zvinotipa Xn-1).

Sezvo X1 = X2 = 0, X3 = 1 uye X4 = 2, ..., nhamba partitions ose convex polygon ndiko:

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

muenzaniso:

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

Kuwanda zvakarurama partitions intersecting mukati imwe diagonal

Patinenge nomumwe nguva zvinogona vaifungidzira kuti uwandu diagonals ose convex N-gon akaenzana zvibereko partitions zvose izvi chati muenzaniso (n-3).

Uchapupu kubvuma ichi: ngatitii P1n = Xn * (n-3), zvino chero N-gon anogona akakamurwa (n-2) ndiyo netriangle. Muchiitiko ichi mumwe wavo anogona ukapisa murwi (n-3) -chetyrehugolnik. Panguva iyoyowo, mumwe quadrangle ndiyo diagonal. Sezvo izvi convex geometric mufananidzo diagonals maviri anogona kuitwa, izvo zvinoreva kuti chero (n-3) -chetyrehugolnikah anogona kuitisa mamwe diagonal (n-3). On hwaro ichi, tinogona kugumisa kuti chero partition yakakodzera ane mukana (n-3) -diagonali musangano zvinodiwa basa iri.

Area convex polygons

Kazhinji, kugadzirisa matambudziko akasiyana-siyana pamusoro chepuraimari geometry panodikanwa kuti kuona nzvimbo ine convex polygon. Kufunga kuti (Xi. Yi), i = 1,2,3 ... N anomirira kutevedzana kwezviitiko Coordonatele ose pedyo vertices ari polygon, vasina kuzvidzora panosangana. Muchiitiko chino, nharaunda yayo rinowanikwa kubudikidza mutoo:

_{S = ½ (Σ (X Ini} + X Ini + ₁₎ (Y _{Ini +} Y _Ini + _1)),

sezvamakaropafadzwa (X _1, Y _{1) = (X N +1, Y} N + _1).