Convex polygons. Definition o ka convex iieeaii. Nā diagonals o ka convex iieeaii

Mau geometric kinona mea a pau a puni mākou. Convex polygons i maoli, e like me ka meli a me kaʻimi hoʻopunipuni (ke kanaka i). Mau oeeo? I hoʻohana 'ia ma ka hanaʻano o ka paleʻana i loko o ke akamai, kuhikuhipuʻuone, e nani ai, etc. Convex polygons i ka waiwai i kā lākou mau kumu i moe ma kekahiʻaoʻao o ka pololei o ka laina e hele ma ka paʻa o ka pili vertices o ka geometrical huahelu. Aia i nā wehewehena i. Ua kapaʻia ka convex iieeaii, a ua hoonohonoho i loko o ka hookahi hapa-pelane, me ka mahalo i ka pololei i kekahi laina i loaʻa i kekahi o kona aoao.

convex polygons

I ka holo ana o ke kula haʻahaʻaa anahonua e mau hana loa na mea polygons. E maopopo i nā waiwai o ka geometric kinona oe pono e ike i ko lakou ano. E hoʻomaka i ka ike i ka paʻa o kekahi laina o kona welau i ka ia. A me ka huahelu i hanaʻia e ia, hiki i keʻano o nā kŰia ke kau. Iieeaii Ua kapaia na mea paʻa polyline nona e pili huahelu i ole aia ma luna o kekahi pololei laina. Kona nā loulou a me nā aka wahi e, niioaaonoaaiii, o ka aoao a me ka piko o ka geometrical huahelu. A noʻonoʻo polyline pono ole intersect iho.

vertices o ka iieeaii i kapaia hoalauna, ma ka ina ka mea, e na welau o kekahi o kona aoao. A geometric huahelu, a i ka N-la helu o vertices, a nolaila ka N-la helu o ka poe i kapaia ka N-gon. Ia i haki laina o ka palena 'ole contour o ka geometric huahelu. Polygonal pelane a lalo iieeaii kapa i ka hapa hope loa o kekahi pelane, kā lākou i kaupalena '. E pili aoao o ka geometric huahelu kapaia polyline Hoʻohana ka hoʻomakaʻana mai o ka ia vertex. Ka mea, e ole e hoalauna ina ka mea, i ka nānā 'ana ma luna o kekahi mau vertices o ka iieeaii.

Other wehewehe o convex polygons

I kula haʻahaʻaa anahonua, loaʻa nō kekahi mau hua'ōlelo ma ka ano wehewehena, e hoike ana i ka ua kapaia he convex iieeaii. A, i kēia mau māmala'ōlelo e hōʻike ana i like oiaio ia mau mea. A convex iieeaii o ka mea i loaʻa:

• kēlā me kēia Hoʻohana i hoʻohuiʻia kekahi mau heluʻai i loko o ia, moe loa i loko o laila;

• ilaila e moe i kona diagonals;

• kekahi Keena Kalaiaina huina i oi aku mamua o 180 °.

Iieeaii mau i hookaawale ai i ka pelane, i loko o nā māhele. Kekahi o ia mau mea - ka kaupalena '(ka mea hiki ke puni iho la ia ma ka poai nei), a me ka mea' ē aʻe - kona palena. Ka mua ua kapaʻia ka pā'āina, a me ka lua - ka pā wahi o ka geometric huahelu. 'O kēia ke kuʻina o ka iieeaii (i loko o nā hua'ōlelo - ka huina ke keʻena) kekahi mau hapa-Kekoa. Penei, kēlā Hoʻohana ka welau i mea nui i pili ana i ka iieeaii loa no ia ia ia.

ʻano likeʻole o ka convex polygons

Definition convex iieeaii 'aʻole i hōʻike' ana i mea, ua nui ano o ia mau mea. A i kela mea keia mea o ia kekahi mau ana hoʻohālike. Penei, ka convex polygons, a i ka na huina o 180 °, haawiia i iki convex. Ke convex geometric huahelu e loaʻa ekolu kuahiwi, ua kapaia he triangle, eha - quadrilateral, elima - pentagon, etc. kēlā me kēia no ka convex N-gons me ka hahai nui koi: .. N pono e like no paha ka nui o lāua 3. kēlā o ka triangle mauna mea convex. Ke geometric kiʻi o kēia 'ano i loko i nā vertices a pau i aia ma luna o ka pōʻai, i kapaʻia ka kakauia p ÷ ai. Ua kapaʻia ia i ho'ākāka 'convex iieeaii ina mea a pau kona aoao a puni i ka hoa kanaka, e hoopa mai ia ia. Mau polygons i kapaia like wale nō i loko o ka hihia o ka uhi hiki ke hui ka wā hoʻohana. Oeaeo iieeaii kapaia polygonal pelane (he pelane hapa) i keia kaupalena geometrical huahelu.

Mau convex polygons

Mau polygons kapa geometric kinona me ka like pana pua a me nāʻaoʻao. Loko ia he mea kekahi wahi 0, i mea no ka like ka mamao, mai kela a me keia o kona vertices. Ua ua kapaia ka waena konu o ka geometrical huahelu. Laina ai pili aku i ka piko me ka vertices o ka geometric huahelu i kapaia o apothem, a me ka poe e hoʻohui i ka wahi 0 me na aoao elua - Radii.

Pololei rectangle - ahalike. Equilateral triangle ua kapaia equilateral. No ka mea, ano ka mea, o ka mea kēia rula: kēlā convex iieeaii ka makau o 180 ° * (N-2) / n,

kahi N - helu o vertices o ka convex geometric huahelu.

Ke kahi ana o kekahi mau iieeaii me ka nānā 'ana i ka haʻilula:

S = P * m,

kahi pa, ua like no ka hapalua o ka huina o nāʻaoʻao o ka iieeaii, a me ka? h o ka lōʻihi apothem.

Waiwai convex polygons

Convex polygons i kekahi mau waiwai. Pela, ke Hoʻohana i hoʻohuiʻia kekahi mau wahi o ka geometric huahelu, pono ke aupuni 'o ia. maopopo ai:

Ina i P - ka convex iieeaii. E mau ākeʻakeʻa kumu, e laʻa me, A a me B, a no ia P. By ka papa ka wehewehena o ka convex iieeaii, i kēia mau mea nui i ke aupuni 'ma kekahiʻaoʻao o ka pololei o ka laina e paka me kekahi olelo R. No, AB i ua keia waiwai, a ua hookomoia ma ka R. A convex iieeaii mau hiki ke maheleia i mau triangle mauna loa na diagonals a pau, a paʻa i kekahi o kona vertices.

Pana pua convex geometric kinona

I ka pana pua o ka convex iieeaii - i pana pua e i hana ia e na aoao elua. Loko kihi nō i loko o ka loko wahi o ka geometric huahelu. Ka huina i ka hana ma kona aoao a converge ma ka vertex, i kapaia o ka huina o ka convex iieeaii. Kihi e pili ana i ka na kihi o ka geometrical huahelu, kapa mawaho. Kēlā me kēia kihi o ka convex iieeaii, hoonohonoho loko ia, o:

180 ° - m

kahi m - waiwai ma waho kihi. Kēia poe noonoo ole haʻilula mea pili i kekahi 'ano o ka geometric kinona ia.

I mau, no ka waho kihi noho kēia mau rula: kēlā convex iieeaii ka makau like i ka likeʻole ma waena o 180 ° a me ka waiwai o ke Keena Kalaiaina huina. Ua hiki i nā loina HEN, mai -180 ° i ka 180 °. Nolaila, i ka wa a ka loko huina o 120 °, ka helehelena e i ka waiwai o 60 °.

I ka huina o na pana pua o convex polygons

I ka huina o ka Kalaiaina pana pua o ka convex iieeaii ua hoʻokumuʻia ka haʻilula:

180 ° * (N-2),

kahi N - helu o vertices o ka N-gon.

I ka huina o ka pana pua o ka convex iieeaii he pōpilikia loa wale. E noonoo oe i kekahi o ia geometric shape. E kau i ka huina o ka pana pua i loko o ka convex iieeaii Pono e hoʻohui i kekahi o kona vertices i nā vertices. E like me ka hopena o keia hana huli (N-2) o ka triangle. Ua ua ike ia ka huina o na pana pua o kekahi triangle mea mau 180 °. No ka mea, i ko lakou helu ana ma kekahi iieeaii lākou, (N-2), lākou, i ka huina o ka Kalaiaina pana pua o ka huahelu 180 ° m (N-2).

Huina i convex iieeaii kihi, oia, i kekahi mau pili kūloko ma, a mawaho pana pua ia lakou, ma keia convex geometric huahelu, e mau e like i ka 180 °. Ma keia kumu, ua hiki ke hoʻoholo i ka huina o nā mea a pau kona kihi:

180 m n.

I ka huina o ka Kalaiaina pana pua, he 180 ° * (N-2). Alaila, o ka huina o nā mea a pau i ka pā kihi o ka huahelu i ma ka haʻilula:

180 ° * N-180 ° - (N-2) = 360 °.

Huina o ka mea mawaho pana pua o kekahi convex iieeaii e mau e like me 360 ° (nānā 'ole i ka helu o kona aoao).

Waho kihi o ka convex iieeaii e nui na poe ma ka likeʻole ma waena o 180 °, a me ka waiwai o ke Keena Kalaiaina huina.

Other waiwai o ka convex iieeaii

Ma waho o ka walaʻauʻana waiwai o geometric huaheluʻikepili, ka mea hoi i'ē aʻe, a hoʻomaka ai ka wā e lawelawe ana ia lakou. Ke'ōlelo mai, i kekahi o ko polygons i ke Wāwahi i mau convex N-gons. E hana i kēia, e hoʻomau a hiki i kela mea keia mea o kona aoao, aʻokiʻoki aʻela i ka geometric shape a pololei mau laina. Wāwahi kekahi iieeaii i loko o kekahi mau convex māhele mea hiki, a no laila, i ka luna o kela a me keia o na apana coincide a me nā mea a pau o kona vertices. Mai ka geometrical huahelu hiki e loa na mea e e triangle mauna ma nā mea a pau i ka diagonals mai kekahi vertex. Penei, i kekahi iieeaii, ultimately, hiki ke mahele ia i loko o kekahi o kekahi helu o triangle mauna, i ka loa ma ka pono ka hoʻoponopono nā hana pili i ka geometrical kinona.

Ke anapuni o ka convex iieeaii

Ka Hoʻohana o ka polyline, iieeaii-kapaia aoao, pinepine hōʻike me ke kēia palapala: ab, BC, CD, De,ʻeā. Kēiaʻaoʻao o ka geometrical aka me vertices ka, e, pela aku, b, e. I ka huina o ka lōʻihi o nāʻaoʻao o ka convex iieeaii ua kapaia kona anapuni.

Ke anapuni o ka iieeaii

e komo Convex polygons i a me ka hoakakaia. E kahalina tangent i nāʻaoʻao a pau o ka geometric huahelu, i kapaʻia ka kākauʻia i loko o ia. Kēia iieeaii ua kapaia i ho'ākāka 'ia. Ke kikowaena kaiapili i ua kākauʻia i loko o ka iieeaii mea he wahi o ke kuʻina o bisectors o ka pana pua i loko o ka haawi geometric shape. Ka wahi o ka iieeaii Ua like ia:

S = P * R,

kahi R - ke kahahńnai o ka kākauʻia a puni, a me ka pa - semiperimeter keia iieeaii.

A p ÷ ai i loaʻa i ka iieeaii vertices, kapa i ho'ākāka 'ia ma kahi o ia. Eia kekahi, ua convex geometric huahelu kapa kākauʻia. Ke kaiapili oaio, a ua ho'ākāka 'ia e pili ana ia i ka iieeaii mea he ai-kapa kuʻina wahi midperpendiculars a pauʻaoʻao.

Diagonal convex geometric kinona

Nā diagonals o ka convex iieeaii - he Hoʻohana i hoʻohuiʻia ole kokoke vertices. Kēlā me kēia no ia mea i loko o kēia geometric huahelu. I ka helu ana o diagonals o ka N-gon ua hoonoho e like me ka haʻilula:

N = N (N - 3) / 2.

Radio ka helu ana o diagonals o ka convex iieeaii nui i kūlana ma ke kula haʻahaʻaa anahonua. Ka helu o nā triangle mauna (K), a e uhai i na convex iieeaii, ho omaulia ma ka kēia haʻilula:

K = N - 2.

Ka helu o nā diagonals o ka convex iieeaii mea mau e kaukaʻi ma o ka helu ana o vertices.

Paku o ka convex iieeaii

I kekahi manawa, e hoʻoponopono i anahonua hana pono e uhai i kekahi convex iieeaii i loko o kekahi mau triangle mauna me ka 'ole-intersecting diagonals. Kēia pilikia hiki ke Wehewehe i nā hāʻina ma ka wehe 'ana i kekahi kekahi haʻilula.

Wā o ka pilikia: kapa pono ano o ka paku o ka convex N-gon i loko o kekahi mau triangle mauna ma diagonals e intersect wale ma na vertices o ka geometric huahelu.

Pāʻoihana: Ina i P1, P2, p3, ..., Pn - i ka luna o ka N-gon. Helu Xn - i ka helu o kona i pohā. Pono noonoo i ka kūpono diagonal geometric huahelu Pi Pn. Ma kekahi o ka ahakanaka ku i pohā P1 Pn no i ka kekahi triangle P1 Pi Pn, ma i 1

E I = 2 o ka pūʻulu o regular i pohā, mau i loaʻa ka diagonal P2 Pn. I ka helu o ka i pohā i e komo i loko o ka mea, e like me ka helu ana o i pohā (N-1) -gon P2 p3 P4 ... Pn. Ma na olelo e ae, ka mea, ua like pu ia Xn-1.

Inā i = 3, alaila, na kekahi pūʻulu i pohā, e mau no ka diagonal p3 P1 a me p3 Pn. I ka helu o ka pololei i pohā i i na i loko o ka pae, e coincide me ka helu o ka i pohā (N-2) -gon p3, P4 ... Pn. Ma na olelo e ae, ka mea, e ia Xn-2.

E I = 4, alaila, na triangle mauna i waena o ka mea pololei paku ua nakinaki aku la ia ia iloko o ka triangle P1 Pn P4, a e pili ana ma ka quadrangle P1 P2 p3 P4, (N-3) -gon P5 P4 ... Pn. I ka helu o ka pololei i pohā lākou, X4 ia quadrilateral, a me ka helu o ka i pohā (N-3) -gon lākou, Xn-3. Ma muli o ka keia, ua hiki ke olelo aku ia lākou, i ka huina helu o ka pololei, i pohā i i na i loko o kēia hui Xn-3 X4. Other pūʻulu, ma ka a i = 4, 5, 6, 7 ... e komo 4 Xn-X5, Xn-5 X6, Xn-6 ... X7 mau i pohā.

E I = N-2, i ka helu o ka pololei i pohā i loko o ka haawi pūʻulu e coincide me ka helu ana o i pohā i loko o ka pae, ma ka a i = 2 (i loko o nā hua'ōlelo, kākou Xn-1).

Mai X1 = X2 = 0, X3 = 1 a me ka X4 = 2, ..., i ka helu o ka i pohā o convex iieeaii nei:

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

Eia kekahi laʻana:

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

I ka helu o ka pololei i pohā intersecting i loko o kekahi diagonal

Ka wā o kéu kanaka hihia, ka mea hiki ke ea i ka helu ana o diagonals o convex N-gon mea like no ka huina hoonui o nā mea a pau i pohā o keia pakuhi kumu (N-3).

Ka maopopo o keia mea mahuʻi: manao i P1n = Xn * (N-3), laila kekahi N-gon hiki ke maheleia i (N-2) mea he triangle. Ma keia hihia i kekahi o ia mau mea hiki e noae (N-3) -chetyrehugolnik. I ka Ia manawa, kēlā quadrangle mea diagonal. No keia convex geometric huahelu elua diagonals hiki ke lawe mai, a 'o ia hoʻi i loko o kekahi (N-3) -chetyrehugolnikah e hoʻomaka hou diagonal (N-3). Ma keia kumu, ua hiki ke manao nei makou, ma kekahi kupono paku mai he manawa i (N-3) -diagonali halawai i nā koi o keia hana.

Area convex polygons

Pinepine, i ka hoʻoponopono nā pilikia o kula haʻahaʻaa anahonua, he mea i pono e hooholo i ka wahi o ka convex iieeaii. Kuhi i (Xi, Yi), i = 1,2,3 ... N ho i ke kaʻina o nā wahi a pau i ke kokoke vertices o ka iieeaii, ka ole ka hoʻoponopono-Huina ma. Ma keia hihia, kona wahi he pōpilikia ma ka kēia haʻilula:

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

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