Nā Manaʻo Kumu o ke Geometry Euclidean
ʻO ke geometry Euclidean kahi lālā o ka makemakika e aʻo ana i ke ʻano, ka nui, ke kūlana, a me nā waiwai o ka hakahaka e pili ana i nā manaʻo i hoʻokumu ʻia e Euclid (c. 3rd century BC) ma kāna hana nui, ʻo Elements. No nā kenekulia, ʻo kēia geometry ke kumu nui no ka hoʻomaopopo ʻana i ka hakahaka ʻelua-dimensional (plane) a me ʻekolu-dimensional (space) e like me kā mākou e hālāwai ai i ke ola o kēlā me kēia lā. Ke kahakiʻi mākou i kahi laina pololei me kahi lula, e ana i nā kihi o kahi triangle, a i ʻole e helu i ka ʻāpana o kahi huinahā, ke hoʻohana nei mākou i nā kumumanaʻo o ke geometry Euclidean. Kūkākūkā kēia ʻatikala i nā manaʻo kumu o ke geometry Euclidean, kāna mau mea kumu, nā axioms, a me kekahi o nā theorems koʻikoʻi e hoʻokumu i kona kumu.
1. Nā Kiko, nā Laina, a me nā Papahele: Nā Mea Kumu
Kūkulu ʻia ke ʻano honua Euclidean mai ʻekolu mau mea nui: nā kiko, nā laina, a me nā mokulele.
1. ʻO kahi kiko ka mea maʻalahi loa e hōʻike wale ana i ke kūlana a ʻaʻohe ona ana (ʻaʻohe lōʻihi, laulā, a kiʻekiʻe paha). Hoʻohālikelike pinepine ʻia nā kiko e nā leka nui e like me A, B, a i ʻole C.
2. ʻO ka laina kahi hui o nā kiko e hoʻolōʻihi palena ʻole ʻia ma nā ʻaoʻao ʻelua a he ana hoʻokahi, ʻo ia hoʻi ka lōʻihi. I ke ʻano geometry kūpono, ʻaʻohe mānoanoa o ka laina. Hiki ke wehewehe ʻia kahi laina e ʻelua mau kiko like ʻole, no ka laʻana, ʻo kahi laina e hele ana ma o A a me B ua kapa ʻia ʻo ka laina AB.
3. He ʻili pālahalaha ka mokulele e hohola palena ʻole ʻia ma nā ʻaoʻao a pau, he ʻelua ana (ka lōʻihi a me ka laulā), a ʻaʻohe mānoanoa. Hiki ke wehewehe ʻia kahi mokulele e ʻekolu mau kiko ʻaʻole i loko o kahi laina pololei.
ʻOiai ma nā kiʻi ma ka pepa e ʻike ʻia he mānoanoa nā laina a me nā palena o nā mokulele, ma ka manaʻo makemakika Euclidean, he mau idealizations kēia a pau.
2. Nā Postulates a Euclid a me ke kuleana o nā Axioms
ʻO ke ʻano o ke geometry Euclidean ʻo ia kona ʻano deductive: e hoʻomaka ana mai nā ʻōlelo kumu i ʻae ʻia me ka ʻole o ka hōʻoia (axioms a i ʻole postulates), a laila e hoʻokomo iā lākou i loko o nā theorems ma o ka hōʻoia logical.
Ua hoʻokumu ʻo Euclid i ʻelima mau postulate kaulana. Ma ke ʻano hou a pōkole, hiki ke hoʻomaopopo ʻia kēia mau postulate penei:
1. ʻElua mau kiko like ʻole e hoʻoholo pololei i hoʻokahi laina pololei.
2. Hiki ke hoʻolōʻihi mau ʻia kahi ʻāpana laina e hana i kahi laina pololei.
3. Me kahi kikowaena a me ka radius i koho ʻia, hiki ke hana ʻia kahi pōʻai.
4. Ua like nā kihi ʻākau a pau.
5. Postulate like: Inā e hui kekahi laina i ʻelua mau laina ʻē aʻe i emi ai ka huina o nā kihi o loko ma kekahi ʻaoʻao ma mua o 180°, a laila e hui nā laina ʻelua ma kēlā ʻaoʻao inā e hoʻolōʻihi ʻia.
ʻO kēia postulate ʻelima ka mea hoʻopaʻapaʻa loa, ʻoiai ʻaʻole i "maʻalahi" ma mua o nā ʻehā ʻē aʻe. Ua hāʻule nā hoʻāʻo e hōʻoia iā ia mai nā postulate ʻē aʻe no nā kenekulia, a ma hope iho ua hoʻomākaukau i ke ala no ka hānau ʻana o ke geometry non-Euclidean. Akā inā ʻae ʻia ka postulate ʻelima, noho mākou i loko o ke ʻano Euclidean.
3. Ke Kumumanaʻo o nā Laina Parallel a me Perpendicular
I ke ʻano honua Euclidean, ʻōlelo ʻia ʻelua mau laina i loko o kahi mokulele he like inā ʻaʻole lākou e hui pū ʻoiai inā e hoʻolōʻihi ʻia lākou no ka manawa pau ʻole. He waiwai koʻikoʻi: ma o kahi kiko ma waho o kahi laina, hoʻokahi wale nō laina e like ana me kēlā laina (e like me ka postulate like).
I kēia manawa, ua ʻōlelo ʻia ʻelua mau laina he perpendicular inā e hui lākou ma kahi kihi 90°. ʻO ke kumumanaʻo o ka perpendicularity kahi kumu koʻikoʻi no ka hoʻokumu ʻana i nā ʻōnaehana hoʻonohonoho, ke kūkulu ʻana i nā kiʻi mokulele, a me ke ana ʻana i nā kihi.
4. Nā Huina a me kā lākou Ana
Hoʻokumu ʻia kahi kihi e nā kukuna ʻelua e hui ana ma kahi hoʻomaka (ka piko). Ana ʻia nā kihi ma nā kekelē (°) a i ʻole nā radians. Ma ke ʻano geometry Euclidean kumu, ʻo kekahi o nā ʻano kihi i kūkākūkā pinepine ʻia:
– Huina ʻākea: 0° < kihi < 90° - Huina ʻākau: kihi = 90° - Huina ʻōkuʻu: 90° < kihi < 180° - Huina pololei: kihi = 180° He mea nui nō hoʻi ka pilina ma waena o nā kihi, no ka laʻana, nā kihi hoʻohui (huina 180°), nā kihi hoʻohui (huina 90°), a me nā kihi ʻē aʻe (like). 5. Nā ʻAno Papahele: Nā Huikolu, Nā ʻĀpana ʻEhā, a me nā Pōʻai a. Nā Huikolu He ʻano papahele ka huikolu i hoʻopaʻa ʻia e ʻekolu ʻaoʻao. I loko o ke ʻano honua Euclidean, he ʻano kumu ko ka huikolu: ʻo ka huina o nā kihi i loko o kahi huikolu he 180°. He ʻokoʻa kēia i ka ʻano honua ʻaʻole Euclidean. Hiki ke hoʻokaʻawale ʻia nā huinakolu ma muli o nā ʻaoʻao: - Equilateral: ua like nā ʻaoʻao ʻekolu - Isosceles: ua like nā ʻaoʻao ʻelua - Kekahi: ua ʻokoʻa nā ʻaoʻao a pau A ma muli o nā kihi: - ʻĀkiʻoki, ʻākau, obtuse ʻO kahi theorem kaulana i nā huinakolu ʻo ia ka Pythagorean Theorem, e pili ana i nā huinakolu ʻākau: \(a^2 + b^2 = c^2\) kahi \(c\) ka hypotenuse. b. Nā ʻĀpana ʻEhā He ʻehā ʻaoʻao o ka ʻāpana ʻehā. Kekahi mau ʻāpana ʻehā koʻikoʻi: - Huinahā: ua like nā ʻaoʻao a pau, he 90° nā kihi a pau - Huinahā: he 90° nā kihi, ua like nā ʻaoʻao ʻē aʻe - Parallelogram: ua like a like nā ʻaoʻao ʻē aʻe - Rhombus: ua like nā ʻaoʻao a pau - Trapezoid: hoʻokahi pālua o nā ʻaoʻao like Loaʻa i kēlā me kēia kona mau ʻano ponoʻī o nā kihi a me nā diagonals, hiki ke hōʻoia ʻia e ke ʻano Euclidean. c. Pōʻai ʻO ka pōʻai kahi hoʻonohonoho o nā kiko e like ka mamao mai kahi kiko waena. ʻO nā manaʻo koʻikoʻi i loko o nā pōʻai: - Radius (r), diameter (2r) - Circumference: \(K = 2\pi r\) - Area: \(L = \pi r^2\) Eia kekahi, aia nā manaʻo o nā arcs, chords, sectors, segments, a me nā kihi kikowaena a me nā kihi circumference. 6. ʻO ke ʻano like a me ke kūlike ʻana Ua ʻōlelo ʻia ʻelua mau kinona he kūlike inā like loa ko lākou ʻano a me ka nui (hiki ke hoʻohui ʻia ma o ka unuhi ʻana, ka hoʻohuli ʻana, a i ʻole ka reflection). No ka laʻana, ʻelua mau huinakolu kūlike i loaʻa nā ʻaoʻao a me nā kihi like.
Ua ʻōlelo ʻia ʻelua mau kinona he like inā he ʻano like ko lākou akā hiki ke ʻokoʻa ka nui; ʻo ka lakio o nā ʻaoʻao pili he mau. He mea nui loa ka like ʻana i ke kaha kiʻi ʻana, ke kaha kiʻi ʻana i nā unahi, ke kuhikuhipuʻuone, a me ke ana ʻana pololei ʻole (e.g. ke ana ʻana i ke kiʻekiʻe o kahi lāʻau me ka hoʻohana ʻana i kona aka). 7. Nā Hoʻololi Geometric ma ka Euclidean Space Ke aʻo pū nei ke geometry Euclidean i nā hoʻololi e mālama i kekahi mau waiwai. ʻO nā hoʻololi kumu e pili ana i: - Translation (shift): ka hoʻoneʻe ʻana i nā kiko āpau e ka vector like - Rotation (rotate): ka hoʻohuli ʻana i kahi ʻano e pili ana i kahi kikowaena o ka hoʻohuli - Reflection (aniani): ke hōʻike ʻana i kahi ʻano ma kahi laina (ma kahi mokulele) a i ʻole ka mokulele (ma kahi ākea) - Dilation (zoom in/out): ka hoʻololi ʻana i ka nui e kahi kumu unahi ʻO nā hoʻololi e like me ka unuhi ʻana, ka hoʻohuli ʻana, a me ka noʻonoʻo ʻana e mālama i nā mamao a me nā kihi (isometries), ʻoiai ʻo ka dilation e mālama i ke ʻano akā hoʻololi i ka nui. 8. No ke aha he mea nui ka Euclidean Geometry? ʻAʻole wale ka geometry Euclidean he mea nui ma ke ʻano he kumumanaʻo makemakika, akā he mea hana pono hoʻi i nā ʻano like ʻole: ʻenekinia kīwila, kuhikuhipuʻuone, hoʻolālā huahana, kiʻi kamepiula, palapala ʻāina, a me ka physics maʻamau. Hiki ke hoʻohālikelike maikaʻi ʻia nā hakahaka a mākou e manaʻo nei he "maʻamau" ma kahi pae o kēlā me kēia lā e ka geometry Euclidean. ʻOiai ma kahi pae cosmic a i ʻole ma ke kumumanaʻo o ka relativity, hiki ke piʻo ka hakahaka (ʻaʻole Euclidean), ʻo ka geometry Euclidean ka mea maʻalahi loa e hoʻomaopopo a hoʻohana pinepine ʻia. Hopena Hoʻomaka nā manaʻo kumu o ka geometry Euclidean me nā mea kumu—nā kiko, nā laina, a me nā mokulele—a laila hoʻomohala ma o nā postulates a me nā hōʻoia e hoʻokumu ana i nā theorems koʻikoʻi e pili ana i nā kihi, nā laina like, a me nā kiʻi mokulele like ʻole e like me nā triangles, quadrilaterals, a me nā pōʻai. Me kona ʻōnaehana logical a me ka hoʻonohonoho ʻia, ʻo ka geometry Euclidean kekahi o nā hoʻokō noʻonoʻo nui loa i ka mōʻaukala o ka makemakika, a me kahi mea hana pono e pili mau ana i kēia lā. ʻO ka hoʻomaopopo ʻana i nā kumu he ʻanuʻu mua ikaika ia no ke aʻo ʻana i ka makemakika holomua, me ka geometry analytic, trigonometry, a me ka geometry non-Euclidean.