pHAr{ vAN rruffirrr - nd vAn BAc K+{ I TRT€N* HINH GO Bi€n dich: VO Mai Li - Nguy6n Xtttur Qui NHA xu^{T sAN HAr PHONG rei Bi6[ thiQu Cudn "Khai tridn hinh gb" drtgc bi6n soan tr€n co sd nhflng grdng cia tfic gi6 cho c6ng nhAn kg thuat gb hhn bii NQi dung cudn s6ch kh6ng nhirng gidi thiQu nguy6n lf vd phrrCIrg phdp thr/c hiQn bdn vE khai tridn hinh gb, mh cdn trinh biy nhfrng vd'n dd co bin vd hinh hoc hga hinh cing nhfrng phrrong phdp khai tridn hinh gd cu thd cria nhilng cdu kiQn gd itidn hinh thgc t€', dnc biQt ldm s6ng t6 nguy6n lf vd qui luft cria nhirng phrrong ph6p c16, cudn s6ch niy Ia tai cho c6ng nh6.n, nhdn vi6n tflp vh tham kh6o bd ich thio't thuc thuAt, gido viOn d4y nghd hQu hoc ki CHUONG NquYrN lv cHrv H|NH | - | Ph6p chi6u ttring (trgc €,iao) Phuong phrip ding mQt chim tia s6ng song song vuOng g6c vdi mit phEng, chidu vho vft thd dd a,rgc ttinh chidu cria vQt nhy tr6n mdt ph&ng gqi ld ph6p chidu truc giao; nhrr thd hien d hinh 1-1 Qua didm A kh6ng gian drrng duong vufing g6c v6i mdt ph8ng P; drrdng vuOng g6c ndy cdt met phEng P d didm a; mit phEng P ndy goi ld m4t chidu, drrdng vuOng g6c Aa qua didm A goi ld drrdng chieu; tlidm a g9i Id hinh chidu th8ng cria didm A tr6n mat ph6ng P Sau xdc dinh m6t chidu, ddi vdi m5i didm hodc v4t thd cria khOng gian ddu c6 thd ding phrrong ph6p tr6n tim duoc hinh chidu cfra n6 trOn mdt chi€-u Nhu hinh 1-2 chinh ld loi dqng phuong phrip hinh chidu chinh dd c6 hinh Hinhl -1 Phuong Hinh - Phudng ph6phinhchiduchinh chiSu tam giric abc cfra tari grde ABC phtip hinh chieu c0a tam giac gian didm chinh cria tr6n mit chidu P khOng Do phuong ph6p hinh chidu c6 thd phAn dnh mQt cdch tuong ddi chinh x6c hinh d6ng vir kich thudc cria v{t thd, dgng hinh cfrng trJong cldi tign lgi, vi thd drrgc rlng dqng rOng r5.i kf thuat Dd trinh bdy dugc ng6n gon, cdc chtlong, tidt sau niry ddu goi tdt Id hinh chidu 1-2 Hinh chidu c&a diiim tron$ hp ba m|t chi€u Dd c6 thd chn crl vho hinh chidu crja didm nhhm xdc rlinh vi tri cria didm ndy khOng gian, vE kf thu4t co 6p dung ba m4t chidu vu6ng g6c v6r Nhu thd hiOn d hinh 1-B (a), mpt chieu vi tri thEng dfing chinh diQn gqi ld mdt chidu drlng chinh, bidu thi bbng V, ggi tdt le m4t V ho4c m{t chinh M4t chi6u vi tri ngang bhng ggi lh m4t chidu bbng, bidu thi bdng H, ggi t{t le mdt H ho{c m4t bhng fVfit ctridu vi tri b6n c4nh, goi lh m{t chidr.r canh, bidu thi bhng W, g9i tit mqt W ho{c mht b6n Nhrr thd, ba m4t H - V - W vudng g6c vdi t4o thinh mgt hQ thdng ba mqt chidu Trong d6, dudng giao gitra V vdi H ggi li trqc chidu X, bidu thi bbng X, g9i tdt trr=rc X; dddng giao-nhau giita H vdi W goi lir truc chidu Y, gqi tdt irUc Y; drrgng giao giira V v6i W goi ld tryc chidu Z gqi tdt truc Z Didm giao O cria ba trgc chidu goi li tlidm gdc' l Hinh chiSu bo mgt cfro didm Trong hinh 1-3 (a), cdn crl vd.o phrrong ph6p lQp hinh chidu til tlidm A khdng gian dung dudng thing vdi mdt H, chAn dudng thiirg drlng cria n6 ld hinh chidu cfra didm A tr6n mit H; goi ld hinh chiSu bhng cria didm A, bidu thi bhng a; trl didm A dqng drrdng vu6ng g6c vdi mdt V, chAn dudng vu6ng g6c a', goi ld hinh chidu drlng cria didm A; til didm A dgng dudng vuOng g5c vdi m{t W, chAn drrdng vu6ng g6c {'; gqi ld hinh chi6u canh cria tlidm A Xoay m4t H quanh trr;c X xudng, xoay mit W quanh ttuc Z sang phAi, khidn chfng trirng vdi m{t V dugc bdn v6 hinh chidu ba m6t cfia didm nhu thd hiOn d hinh 1-3 (b) Bdi vi m6t chidu c6 thd md rQng tny y theo y6u cdu, cho ndn b6n vE hinh chidu ba mit kh6ng cdn vE ranh gidi cfia mdt chidu, md chi vE truc chi6'u Trong d6, truc Y quay theo mrnt H thi bidu thi bing YH , quay theo mrnt W thi bidu thi bing Y*, nhu thd hien hinh 1-3 (c) (a) (b) (c) Hinh - Hlnh chieu c0a didm hQ ba m{t chidu ll Quy ludt hinh chiSu c0o didm lrong b6n vE hinh chidu bq mqt Til hinh 1-B (a) c6 thd bidt: Aaa* a' ld hinh chf, nhat a'a* cria n6 vu6ng g6c truc X, aa* vudng g6c truc X, sau mnt H quay tring vdi m{.t V, dudng ndi aa' gif,a a - a' nhdt dinh vu6ng g6c.vdi truc X Vi Aa' az { cfrng ld hinh chf, nhdt, dttgc chfng minh cirng l1i 16 tr6n, sau mit W quay trilng vdi mit V thi drrdng ndi cria a'- a" nhdt dinh vu6ng g6c vdi tryc Z Ngoii ra, hai hinh chfr nhdt Aa' = aax ve Aa' = {'at n6n b6n vE hinh chidu ba mdt, aE = {'^, Cdn crl vdo phdn tich tr6n, c6 thd rrit quy lu4t chidu cfia mQt tlidm bin v6 hinh chidu ba mIt li: cltrdng ndi gifa hinh chidu chinh vdi hinh chidu bhng cfra didm, vudng g6c vdi truc X, ttc a' a vudng g6c truc X; dttdng ndi gitra hinh chidu chinh vdi hinh chidu b€n cfra d.idm, vu6ng g6c vdi truc Z, ttc a'{' vuOng g6c truc Z; khoAng c6ch hinh chidu bhng cria tlidm tldn tryc X bhng kho6ng crich hinh chidu b6n cria didm ddn truc Z,tttc aax = d'ar Ndu xem hQ thdng ba mpt chidu thAnh h0 tqa dQ vu6ng g6c kh6ng BraD, trlc xem m6t chidu li mit tga dQ, qua hinh 1-3 c6 thd bidt, khoing c6ch til didm A tldn ba mlt chidu W - V - H cflng trlc lh ba tga dO Xn Yn - Zoc&a tlidm A Gifra ba hinh chidu d, d' , {' crta didm A vdi tqa tlQ tdn tai quan hQ sau: XA =A{' =8&r=a'&s=OB1 YA = A a' - aE = d'd2 =Oa" Z^ =Aa =a'ax-{'^r=Odz Vi the', ndu rld bidt toa d0 (xa - yA - Z,pi crta didm A thi c6 thd xiic clinh hinh chidu ba m4t (a - a' - {') cfia didm niy; nguQc lai, dte bidt hinh chidu ba m4t cfia didm A thi cflng c5 thd x6c dinh tri sd toa dO cfia didm A 1-5 Hinh chidu c&a dudng thin€ l Hinh chiSu c0o dUdng thdng xi€n I(hi dudng thing kh6ng vu6ng g6c vdi m6t chidu, hinh chidu cria n6 trOn mqt chidu v6n ld dudng th8ng Trong bin vE hinh chidu ba mit, hinh chidu cria dudng thing cfing duoc x6c dinh blng hinh chidu cua hai didm tr6n tlrrdng th8ng Vi dg dudng th8ng AB nhu thd hiQn d hinh 1-4 Dd vE hinh chidu ba mdt cfra n6, trudc tiOn ldn lrrgt vC hinh chidu ba mft cria hai tlidm ddu A, B ld a, d' , d' vd b, b', b-, sau d6 ndi hinh chidu cfia hai didm tr6n cDng mQt mdt chi€'u (trlc hinh chidu ctrng m4t) sE drloc hinh chidu ba n{t ab, d'b', a"b crja dudng th8ng AB Nhung tir hinh (a) c6 thti thdy, vi ABba li hinh thang vudng, cho n6n d0 dei thrlc stl cria doan th8ng AB ldn hon d0 ddi cfra hinh chidu bhng ab Cfrng ctng ly lC d6, tlQ dii cfra hinh chidu chinh a'b' vd hinh chidu b6n a'b" cfing ddu ngin hon dQ ddi thuc srJ cfra doan thfng AB, trt hinh chidu cria doan th8ng ddu khdng phdn rinh d6 dAi thrrc td crja doan thing (b) Hinh - Hinh chieu c0a dudng th&ng xi6n Trong hinh 1-4 cdn thd hiCn, tlidm K nhm tren dudng th&ng AB thi hinh chidu ba m6t K, K' , K' cua tlidm K nhdt tlinh ldn lrrot d tr6n hinh chidu cing m4t ab, a'b' , {V cfra tlrrdng th8ng Ngodi ra, vi AaJ/I*/tBb, A{ lfiW ltBb' , A{ lllff IIBH, cho ndn AK: I(B = ak:kb = a'k': k'b' , {'V: Yb', trlc ti lC dg ddi cfia hai doan th&ng tr€n ttrrdng th8ng bhng ti lQ hinh chidu cfia n6 Ngttgc lai, cdc h)nh chidu cria didm ldn lrrot nlm tr€n c6c hinh chidu cing mdt cria dudng th8ng thi didm ndy nhdt dinh ti tr6n drrdng thing; ti le dO ddi hinh chi6u cfra hai doan trOn cDng mQt dddng thEng bhng ti lC dO ddi th{c td cria hai doan th8ng ll Hinh chiSu cOo dudng th6ng co v! tri dgc biQt Khi dudng th&ng c6 quan hQ vudng g6c, song song vdi mdt chidu dudng th&ng vi tri ddc biQt thi ggi ld Drrdng song song vdi m4t chidu: Drrdng thing song song vdi mQt m{t chidu mi, ch6o vdi hai m6t chiSu thi goi ld drrdng song song vdi mit chidu Drrdng thing song song vdi mit V gpi Id clrrdng song song chinh; tltrdng thing song song vdi m4t H goi ld iludng song song bhng; tlrrdng thing song song vdi mit W thi goi Id dudng song song canh Drrdng thing CD thd hiQn hinh 1-5 Id dudng song song chinh Hinh chidu cira n6 c6 c6c tinh chdt sau: (1) Hinh chidu chinh id bbng dO dai th{c eria dudng th6ng CD (2) Hinh chiSu bbng cd song song vdi truc X; hinh chiSu mf,t canh i, d' song song vdi truc Z, tlQ dai hinh chiSu cua chring ddu nh6 hon dg ddi thrrc cria CD (b) (a) Hinh - Hinh chidu c0a dudng song song chinh Cdn crl vdo tinh chdt hinh chidu ctia drrdng song song chinh c6 thd suy dod,n tinh chat hinh chidu cria dtrdng song song bhng EF vd drrdng song song canh KL ld: (1) Hinh chidu bbng ef cria tlrrdng song song bhng EF bing d0 dei th{c sr/ ctia drldng thdng EF; hinh chi6'u chfnh e'f'song song vdi truc X; hinh chi6-u b6n d'f" song song vdi truc Y*, tlQ dai hinh chidu cria chring ddu nh6 hdn d0 ddi thuc srr cira EF (Xem hinh L-6) (2) Hinh chidu canh V'1" crta clrrdng song song canh KL bang dO dai thrlc srJ cfra tlttdng th8ng KL; hinh chidu chinh k'l' song song vdi trucZ, hinh chieu bing kl song song vdi truc Y", dg dai hinh chidu cria chring ddu nh6 hon itQ dii thrJc srJ cfra KL (xem hinh 1-7) z (a) (b) Hinh - Hinh chi€u crla dr;dng song song blng (a) (b) YH Hinh - Hinh chidu c0a dr/dng song song cqnh 10 Drrdng vu6ng g6c vdi mit chidu Dudng thing vu6ng g6c vdi m6t mht chieu vi tldu song song vdi hai mdt chidu kh6c thi goi ld drrdng vuong g6c v6i mdt chidu Drrdng thing vu6ng g6c vdi mdt V thi ggi ld clt/dng vu6ng g6c chinh; drrdng thing vudng g6c vdi m5t H goi ld drrdng th8ng dr1ng, tlttdng thing vudng g6c vdi mat W thi goi ld dudng vudng g6c canh AB thd hiOn ti hinh 1-8 le drrdng vudng g6c chinh Hinh chidu ciia n6 c6 cd.c tinh chdt nhtr sau: Hinh - Hinh chidu dUdng vuong g6c chinh Hinh - Hinh chidu dt/dng vu6ng gdc dung Hinh : 10 Hinh chidu dr/dng vuOng goc b6n 11 ) co lai thdnh mOt didm' (2) Hinh chidu bhng ab vu6ng g6c v6i truc X, hinh chidu bOn a"b " vuong g6c vdi truc Z; abrrl*'U" adu"Ubng aO Jaithrrc cria AB Cdn cr1 vdotinh chdt hinh chi6u (1) Ilinh chidu chinh a'(b' ria ttttdng th&ng -df"g c(d) cfia tlrrdng th8ng g6c vdi t.,rc X, hinh cria CD chidu b6n i'd' vu6ng g6c vdi truc YW, c'd' vd i'd' ddu bhng dO dai thr.rc sg (xem hinh 1-9) (2) m"n chidu b€n e"(f") cria drrdng vu6ng g6c b6n EF co I4i thhnh vu6ng g6c mOt didm; hinh chidu chinh e'f ' vufing g6c vdi truc Z, hinh thld-u bbng ef ,rOi ti": YH, e'f ' vh ef ddu bhng dO dei thrlc sr.l cria EF (xem hinh 1-10)' cfra CD drln th ng g6c c4nh EF d"[ mQt didm g6c chinh, c6 1-4 Phucrr€ phrip dgng dO dd,i thgc cira doqn thin€ Hinh khai tridn Ih bdn vE hinh dq hinh d?ng thuc td cria bd md.t, trttdc thing be mat Hinh chi6u cria dudng t dO dei thrrc td cria doan th8ng ndy, cbn hi phdn rinh dO ddi thrJc td cfia tloatt th&ng ndy, phai thong qua phrlong ph6p d!t'* hinh rld 6m Tim dO dai thrrc td c,ia drrOng tnEng xi€n thddng :ip dung hai phrrong ph6p: phrrong ph6p tam gi6c vu6ng vd phuong ph6p xoay trbn l Phudng ph6P tom gioc vu6ng Hinh 1-11(a) Id dudng thang xien AB vh hinh chidu ab vd a'b' c&a n6 song hO thdng hai mlt chidu Ndu th6ig qua didm ldu A dung drrdng thang AB, song vdi ab vir cdt iludng chidu Bb did clrrdng song song vu6ng g6c v6i nhau; cho gi5.c vuOng Canh huydn AB ctia tam gi vu6ng AB, bhng c4nh ddi ab cfia hinh c BB.=Bb-Aa= ZB - ZA, mb tron bin v6 hi tU b', a' ddn t4rc X Trong b6n v6 hinh gi6c vu6ng ddu de bidt, vi thd vE hi cria n6 chinh le dO ddi thrrc td cria doan t thd cfra n6 nhu thd hien hinh 1-11(b) T song song cria truc X cdt b'b A bo, clo ldy dhi thrrc td ctia dudng thing AB C6ch hai th&ng d(tng bBo cria ab cho bBo = ZB td cfra dudng thing AB dO ddithtjc Hinh - 11 Phudng ph6p tam gi6c vu6ng tim d9 ddi thtJc c0a drrong thing (1) 12 \ Trong hinh 1-11(:r), ndu qua A dung AB2ll :ib' thi c6 thd drroc mQt tam gidc vu6ng khric AB2B Hai cung g6c vu6ng cria n6 lAn lt4o,t le ABz=6b' -YA=bb*-aa" phtrong phdp Hai thd crja chring nhu thd hien I h Trong bdn v6 hinh khai tridn gb kim loai, dd ldm cho b6n vE 16 rAng th6ng thudng 6p dung phrrong phdp mOt, vi k6o tam giric vudng b'b0A0 sang ph6i (hoec sang trrii) m6t dog.n, nhrr thd hien d hinh I - L2 Hinh - 12 Phudng phdp tam gi6c vu6ng tim dQ dAi thr/c td c0a dudng th&ng (2) ll Phudng ph6p quoy Girl nguy€n mit chidu, xoay drrdng thEng xiOn quanh truc ld dudng th{ng vudng g6c vdi mdt chidu ndo d6 thdnh dudng song song vdi mdt chidu d6, thi hinh chidu cria drtdng thdng tr6n mit chidu song song cfra n6 sE phdn 6nh dQ dni thrlc td crla n6 Nhrr thd hi6n d hinh - lS(a), AB la dudng thing xi€n, ldy iludng thing drlng OO vu6ng g6c vdi mdt H ld.m trvc, xoay trdn n6 thdnh drrdng song song chinh AB1 hinh chidu bhng ab, sau xoay song song vdi truc X, hinh chidu chinh :ib, phdn dnh dQ ddi thrrc cfra drrdng th8ng AB Dd tlon gi6n h6a viQc dUng hinh, hinh 1-13(a) truc quay O0 th6ng qua didm dAu A, khidn vi tri crja tlidm A tnJdc vd sau quay li kh6ng thay ddi, chi cAn quay didm tldu B li drroc Lfc ndy, qu! tfch crja didm B ld hinh trdn bHng, hinh chieu tr6n md.t V lA drrdng thing song song vdi truc X; hinh chidu tr6n rndt H ph6n 6nh hinh thrrc, trlc hinh trbn ldy O ldm tAm drrdng trbn, ob le Khi didm b quay ddn didm b, (abrl/truc X) thi ttg ddi cria ribi bhng dQ dai t B Citc btrdc dung hinh cu thd cria n6 (xem hinh 1-13(b)), qua didm A (a,ri) OO vu6ng g6c vdi m5.t H, ldy didm O (trilng vdi a) ldm tAm, Ob ldm brin kinh v6 cung (theo chidu hodc ngrroc chidu kim ddng hd tldu drrgc); qua a d{ng drrdng song song vdi truc X, cdt cung trbn b,, duqc ab,, sau d6 qua b' dUng dudng song song ctia truc X vd qua b, dgng drrdng vu6ng g6c oiua truc X, hai dttdng cdt d b;,thi db; le dq ddi thrJc td crja a,ra"g thdng AB Hinh 1-13(c) le ldy tlrrdng vudng g6c chinh OO lim truc, quay doan th8ng CD thAnh drrdng nhm ngang, cdch cl{ng hinh cfra n6 giong nhrr hinh 1-1g(b) (a) (b) (c) Hinh - 13 PhrJdng phdp quay tim dd ddi thUc t6 cOa dudng rh&ng t3 u xu6'ng cua n6 Sau d6 chia nfia hinh trbn ng phdp tinh torin tinh d6 ldn nh6 g6c 6n vd d6 dei chu vi hinh trdn sau khai inh khai tridn - 58 Ngodi ra, hinh khai tridn cria b6n cdu cfrng c6 t ph6ng xa khai tridn mdt hinh n6n, dirng phuong ph6p mit hinh tru dd dat duoc i6n d hinh g ph6p tia khai triein mrldi hai phAn bing nhau, coi mdi phAn Trong hinh, nilra triii cfra hai hinh chidu fia phrii ldy brin kinh binh quAn cria nfra rinh dung hinh khai tridn crja n6 ld: tnJdc xu6'ng thAnh 12 phdn bing nhau, duoc trdn, dDng drrdng th6ng ndi tam hinh chidu chinh cria cdc drrdng th6ng kh6ng hinh chidu, hinh chidu chfnh cria n6 ld tam gidc 0981, d0 dei cfra 83, bhng d6 dt phia bing cfra brin hi;h cdu ld ia hinh chidu chinh thinh ba phAn bhng , 2' , 3' vd tim hinh chidu bhng LLt,22t, ra, dung drrdng vu6ng g6c ch6 v6 hinir cAt ldy d9 dei, cho n6 bbng d9 dai cung nsans, dQ ddi cria cdc doan dudng #":i-'ff-Ti;:,t:,::r{:;"lr.t1T Hinh - 59 Bdn cdu vd cdch khai tridn c0a no 185 III vd ns dddns cong ndi c6'c didm cudi drrdng phing ngang O' I' II' Jt'lllJt^llll phuong phdp niy khai tri6n crc hinh khai tridn mOt phdn bhng Ap dgng n cdu thi viQc hdn ndi d ch6 didm o ttrong tldi kh6 khen, thi c6ng pheli c14c t chfi y Trong hinh - 60 ld phrrong ph6p vE gdn dring hinh khai tridn sfi'dung idu th\rc td c6ng trinh Phuong ph6p dgng hinh cria n6 hoi kh6c vdi nguyOn cua c6c vi dq thrrc td khdc C6ch vE niry trdnh tlugc khidm khuydt kh6 hdn ndi rng vi dr,r tren N6 chia brin cAu thinh bdy bO phan g6m mQt tdm hinh trbn vd u id- b6n tao thinh Ddi vdi hinh khai tridn cria n6 mh n6i, chi cdn tim hinh cria tdm hinh trdn vd mQt tdm b6n li drrgc Quri trinh d{ttg hinh cua n6 ar ':dnti6n cdn crl vho drrdng kinh binh qudn ctra b6n cAu v€ hinh chidu chinh, trddc chidu xudng cira n6, vA tr6n hinh chidu chinh, hinh chidu xudng trudc tiOn vE tdm hinh trbn phAn dinh (tludng kinh D cria n6 rldc bhng 1/3 ddn 2/5 iludng kinh a b6n hinh cAu, dudng kinh D, cria hinh khai tridn bhng hai ldn ddy cung a'1') sau tr6n hinh chidu chinh chia bdn phdn bbng cung 1'5', dtrgc didm phdn bhng L' ,2' , B' , 4', b' tldng thdi d{ng hinh chidu bbng cria didm ulc phdn bhng nhau: , 2, 3,4, rdi qua c6c tlidm phdn bbng tr6n b6n hinh cAu drrng hinh trdn hing ngang Tidp tt6 chia hinh chidu xudng thdnh phAn bhng nhau, dddng phAn hng cit cilc hinh trdn phfrng ngang L.,2r,31,41,5r TrOn hinh chiSu chinh il ding dtrdng thing ndi tAm hinh cdu 0' vdi 3', eua didm 3' dttng drrdng vuOng g6c 'c' cta tludng th8ng 0'3' cdt vdi tludng trung tAm vu6ng g6c ttidm c' Dgng rQt drrdng thEng bdt ky, ldy tlidm c trOn dudng thing lh.m trung tAm ldy t10 ddi ' ldm brin kinh vE cung cdt vdi tlrrdng th8ng d tlidm III, rdi lay didm III ldm hudn tr{i phing cung trdn 1'5', ldy cdc ttidm I, II, fV, V tr6n tlrrdng thing Ldy idm c ldm tAm, vE cung qua cdc tlidm I, II, III, IV, V rdi ldy d0 dei cung tr6n g trdn, cho II, - 11r, II II1 = 22r , sau cing dirng drrdng cong ndi didm Il, II1, IIIr c c6 thd dtroc hinh khai tridn mOt mat bOn Hinh - 60 Bdn cdu vd c6ch khaitridn no iVf du 2l DA d6ng i 1rl Oan fn, lu 3l ( g ntn rg hinh godi ;odi Ila ; Nhrrn6 Vhrrng ing, 'dng, vvi ctia tia t6' tan tridn I i t kin hinh cdu elip (hinh _ 61) du ndy co brin sjd'ng nhrr vf du tr6n, chi c6 hai ch6 hoi khric: trlc diem eua B', kh6ng phei lt;;;;" vrrAno oAn,,a! ,B t.-: ,a il ; ;;;'d;;:Jffi j:x;f, :2 l, :""'H fi1xff :: ;t; mdnh trti""fit6n mit xodn dc 1900 mdt cdt hinh vu6ng (hinh _ 62) tg mdt xodn oc b_orr,m1t cong tao thdnh Tam b€n vA daiv rd mdt xodn oc 6c nrnrr {'"1khai LT:: hinh "u'i'a'' bon ngodi i ?.i trong, tridn ti""li,,_::"o" / crudn' -L dudng er'ia dudng r*r^ xodn ,,^!- 6,c z' sau cria rrh tru hrip v6 rrinrr khai i"ia" ,i" n h6a driy giong rulau, nhau, cnr chi can ^, cdn VE ve L 'E -.rvu ru^,'r.r ,*rrar pnudng phrip drrdng tridn, tam gidc dng kinh khoring B m6t p# Jtlff T:l.,T::il ! ? f :I * I - Ot Ddu dong kin hinh cdu elip vd cdch khai tridn no a t, Hinh - 62 mdt xodn dc 1g0o mat cdt hinh vu6ng vd c6ch khai tridn no 187 ngoii lh: trrrdc ti6n car crl vio bdn kinh eu6 trinh khai tridn tdm ben trong, kl:i"q H cirng chi6u cao d::l ldp trung tinh R vd r cria d1s.m9t :?:1 iy: hinh chidu xudng cria dng md.t xofn dc Sau ffi ;;;?r;; n, "u hinh rtid" chinh, hinh chinh lAn luot vE drrdng phing #r,* #; EiAtir 0', 0;, 8' vh 8i tr6n ph&ng "chidu ngang' cho chrinl ldn lrrot bing dQ ;;r* l4i tlo c6t 1dy khb6ng c6tch httdng vir chu vi nfta hinh trdn $ikh^i tridn nr vd rcR cria chu vi nira hinh trdn tldu khoAng ch c6t ldy ngolr,i dng mdt xo5.n dc, dr,rng tlrrdng vuong g6c E hai dat rldgc trlc ld ,rii a.tU"g ph&ng ngang Hai hinh tt giSc song song KLMN vd hinh khai tridn cria tdm bon vd tdm ben ngoii (tdm driy) lb: trddc tie tr6n hinh chidu Qu6 trinh khai tridn cfia t16y dinh phAn bing xudng irir td* tlinh thinh t6m phdn bhng nhau, sau d6 chia mQ vd oabrihanh thinh hai hinh tam giSc gab vd 0b1, dr/ng hinh chi u chinh O'a'b' dQ 0'b'L' cfra hai hinh tam gi6c Lai dirng phrrong phdp hinh tam gir vudng tim h AB, OB, OI IA dii thr,rc cla c6c canh hai hinh tam gi5.c D0 dei thuc ctia ba c4 idu bbng 0a vh lb ddi hinh ct 4.080, OoBo, O0I0, dO ddi thuc cfia OA vh IB chinh Ia d0 dei thr,ro cfia 6,lf i"g S;; cing cdn c(r vdo phuong ph6p drrdng tam gi6c dirr g dQ d6ng ndY lai vdo inh cdc canh v6 gOp thdnh hinh tam gi6c OAB vir OBI, rdi cen crl li6n tuc vE b6y cdi nfra s6 tlrrgc hinh khai tridn cira tdm dinh i"ii I{ri hinh khai tridn tinh cfia tdm dinh l:: li tim khai tridn: I + (2nr)2 duoc bing tfc: tod.n, DO dei cria d.Udng xo6n dc sau DO hinh Qu?t, cfrng c6 = dai cira drrdng xo6n dc ngodi sau khai tridn: L = +(2nR)2 r, = f"l, Bdn kfnh cung ngod.i: Rr = rr + B B6n kinh cung trong: G6c tAm hinh trbn = 1800 x ( 1-* fiRr I cing, ding r1, Rl vd vE hinh khai tridn' [Vi du aJ dng mdt xo{n dc hinh n6n tidt diQn hinh vu6 Vi dq ld m4t xo6n dc hinh tru, hinh ddng khai tridn cira ddu gidng nhau, cdn phAi dung tUng hinh khai tridn cfra m5i p trinh cU thd cfra n6 lDr: tlJdc ti6n c!,n crl vdo dQ cao dd cho vh dudnl d6y v6 hinh chidu chinh, hinh chidu xudng cria hinh n6n, rdi thdnh 12 phdn bhng nhau, v6 hinh chidu cfia drrdng sinh c6c tl6 ldy dAn trinh cria m4t xodn dc theo ilrrdng truc mlt hinh chinh, rdi chia dAn trinh thdnh 12 phAn bbng nhau, drrng dtrdng tlidm phdn bhng Cdc dudng thing ngang ndy ldn lrrot c6t Sau (hinh - 63) phAn bhng bhng Qu6 kinh cria hinh trdn ia m[t hinh n6n bbng nhar,r Sau n tr6n hinh tdong ftng, duoc c6c d.idm V , !', 2' TrOn hinh chidu cric drrdng sinh chidu xudng dqng hinh chidu bbng 0, 1,2 cria 0' , l' ,2' rdi dtng ndi c6c didm 0', 1' , 2' vdi 0, !, s8 drrgc hinh chi€iu chinh, hi tlttdng xodn dc hinh n6n C[n crl vho dQ rOng cria tidt diQn dng m dUng 0a hinh chidu chi4h, hinh chidu xudng Url'12; vA 0,1r2r cria Lai cin cir vdo dQ cao cfia tidt diQn dng m{t xoin dc hinh n6n ld chidu chinh cria hai dudng cong cdn lai r 188 chiSu ng ngang qua cac au vdi dudng sinh tudng rlng cria hinh g cong ldn lugt chidu xudng cria xodn dc hinh n6n {udng cong 0rIrIIr thd dung hinh 9o 10 20 30 40 [o [o logogo1oo11012o 11r 0r 9r Ol ,tl +tl rtl Z, I al lt I I 0rt 2? 3?4 59 6? 7? eq l? 10? 11? 129 il, 5r VI' 61 V[' r xl r^l - 63 6ng mat xodn oc hinh non va cdch khailridn no Quri trinh khai tridn tam b.n trong, ngodi ld trudc ti6n chieu bang cua tdm trrii thdng hinh n trons vA ngodi Vi thd, qua didm ngang ldn lrrot c6t \dv d; JU* U;;"* thdng cdc cung o't,l drrdng uquu5 -'v' thdng L,IIaIrg ngang nay, '^ 1,?,,.?,{;n jirOc ca,c didm 0r, r,r, thi ndy, ,'^0,'r.r-" 0rt2l la dO aai khai tridn ,i" frirrt chidu bing tam bOn e "'goal r dun nhu thd c6 thd tai v6 d6 ddi khai tridn }oI}octia td: b.n bAn :t :r,on5 -^v's, r:il' uql chinh diern cric phdn o""niin":rFi Ia aita"g '"u 20 d{ng dudng thdne tong ;i ;ia_ 0,, oo, 10, g6c cdt :i:'t11*:9,, vdi dudng prra"s ;g." cric diem 0,, f,, tt, vJO r'r"";; i"r, ;"1; cao tiet diQn ong md.t : sau cr)ng cdn cr, vdo dQ odn ,t;jt#trl":":y:oc di chuydn".ll dudng cong khric c6 hin q,El';"u;";5o;il""11f"'T T";:Ll? thAnh trlc ld hinh khai l,$lX-rf'l;: "n,,, hinh h;i ,r,0* d"d"; tao , : ill ;llilli.;? prJ;;;;* lj,-r; liq g;;;i;, ;; Quri trinh khai Ldn tdm dinh (tdm driv) hinh chi6u chinh, hinh ridu xu6'ng cria O'"S -it 90,1,1, IIptz crla tam linh, lgi chia m6i crii th ddi thuc ba canh cia c6 dat duoc hinh chid OI, I II, II III vA O,I,, trong, b6n ngodi, cdn d6 phrip hinh tam giric vu6 cr)ng, ding d6 ddi thuc giSc vO gOp cric hinh rm sidc, duoc hinh khai tridn il"rr?TTjfuo drrdng tam 189 BAI TAP (Ddi vdi ild H6y v6 hinh khai tridn c6c cdu kiQn sau: xo{n dc) 190 o m4t 6n th6p hinh ,'^l]Tq j6'i da sd udn cong m6t dJn sirin, tridn fvr du 1r rh64 Cin crl vdo rr.;;;;;; ;;.;lr'rlmff;i?cdt b6 rhi ddi khai dQ thudc mdr th6p JAITI^'l"Tj:cho "u""'l*i,".Jl""iffrllllJl',?Jl'11',i::H g6c L + H it, aig_ phan cfria e 'ffi J?l I.',Llki;:$-";;:;1'{;;;;T:Hiil:;':'"#"ri:f; rf : A;:;, ;, ;"r";;t;ltilJi,Tftff;rttT,?JI,:i5I uv squu rurlrr qallg vaf, ll il:"-l.,"'1,"fl D v6 hai b6n ,5r fVi du 2l Th6p g6c udn thdnh g6c tiry y (hinh _ 65) ti;""*' liQu khai tridn ndv giong nhu vi du 1, chi can dlng kich L, thay:5t a ld duoc Hinh - 64 Thdp goc u6 goc vuong fVi du SJ Th6p D0 dei vdt li6u hinh thang Cdn crl kic - vAt liOu khai tridn fVi du al Th6p Hle C_ kinh hinh 900 Hinh - 65 Thep goc uon thdnh g6c t0y thrrdc f u6'n thdnh hinh thang (hinh _ 66) Lai tridn bhng tdng A B, C - D kfch thr/dc mit thudc 8",C, vd D, c c6.t c6,c miQng kh*d;'i;iur."t,"n cria dring thdnh g6c trdn 900 (hinh - 6T) D0 dai cua vAt li u khai ,-.]dl nAy bdng tdng kich thd6c L, C, H Trong d6 kich thudc m6 tror cla dO dei cung ldp trung tinh bQ phan g6c trbn,.trlc: L, 7t d _ (b- ;) LAy hai idm A, B ldm tAm hinh trdn, ldy zz kich thrJdc b _ d lim brin vO cu4g cdt ,u i drrdng ch6o 450 qua hai didm A, B & hai didm C, D s6 duoc dring vAt liQu khai tridn lVi 4, 5l Th6p Cach ldm v4t li6u kfch thu6'c C vd g6c kep uO'n uon thdnh g6c trbn cd g6c tDy y (hinh _ 6g) tridn co bd.n grdng nhr-r vi d\r B, chi cdn thay ddi dO dei g:fra hai dudng th8ng AC, BD vdi canh thep g;, ha_i ;ong d6: c = l89t n' (b 9), 1800 \" - 2" e" =- l9l thirnh hinh thang Hinh - 66 Th6p g6c u6n Hinh 6'67 goc tron 900 Thep g6c ucjn thdnh Hinh 6'68 TheP goc u6n c6 goc tuY Y gi6c co goc Hinh - 69 Thep gdc u6n thdnh hinh tam thirnh goc lVi du 61 Th6p g6c udn thAnh hinh tam giric c6 g6c trdn (hinh - 69) Khi che tao cdu kiOn ndy cdn tr] gitra m6t canh ndo d6 cria hinh tam gi:ic dd chdt vat lieu Tdng tlO dai v{t li6u bdng tdng kich thrrdc canh A, B, C hinh tam gi|.c cOng vdi d0 diti2nfu-*1 Kich thud'c cdc phan doan nhuthd hiQn o L hinh - 69, d6 kich thd6c: Lt= l80o-cr d Tt(b -) lg0" lgoo-F ,, Ei fJ = - ;-1-'It(b ) lg0, F 2' d 2' - \ru d;t g6c kep gifra dudng xiOn bO phAn cdt b6 vdi canh th6p g6c ldn lrrgt bhng d/2, W2 vh 4b0 BAi Hiy vE hinh dring vat lieu khai I t{p tridn crta c6c cdu ki6n -ffi 193 I I I I Phu BAng ddi chidu drrdng kinh, chu vi hinh trdn cria hinh trdn Chu vi hlnh Dudng kinh trdn vi DO dai cung phdn t2 l6 d0 dei cung phdn bhng bing 1A 20 32 48 0.0981747 0.0654498 0.130900 I 3.t4r59 0.392699 0.261799 0.196350 l 6.28318 0.785398 0.523600 0.392699 0.314 r 59 0.26r'799 0.785398 0.589049 0.471239 0.392699 ^ 9.42478 t2.5663 r 17810 a 1.57080 t.04'120 0,785398 0.628318 0523600 15.7080 1.96350 1.30900 0.98 r 748 0.78s398 0.654498 r96350 0.294524 0.392699 0.490874 r 8.8496 2.356t9 r.57080 17810 0.942478 0.785398 0.589049 2t 99t I 2.74889 r.83260 1.37445 1.09956 0.687223 25.t32'7 3.14159 2.09440 1.57080 |.25663 0.916298 04120 0.392699 0.458149 0.785398 0.523600 28.2743 2.35619 | 767 15 1.41372 17810 0.883573 0.589049 2.61799 1.96350 1.57080 30900 0.98 1748 2.87919 t.72788 43990 07992 3.t4t59 2.r5984 2.356t9 0.654498 0.719948 1.88500 57080 17810 0.785398 z 3't.6991 3.53429 9.92699 4.31969 4.71239 31.4159 34.5575 57080 l 30900 0.196350 0.26t799 0.327249 40.8407 5.10509 3.40339 2.55254 2.04204 70170 27627 0.850848 43.9823 5.49779 3.66519 '2.r99t1 83260 37445 0.916298 47.t239 5.89049 3.92699 2.74889 2.94524 2.35619 96350 472622 0.981748 2.09440 57080 t.04720 2.22529 6689',7 1.r1265 2.356t9 767 t5 l.l78l0 2.48709 2.61799 2.74889 50.2655 6.28319 4.18879 3.14t59 2.5rr33 '7 53.407 | 6.67588 4.45059 2.67035 2.82743 2.98451 3.45575 3.61283 2.87979 3.0 l 069 86532 96349 2.0616't 2.15984 2.25802 3.7699r 3.r4159 2.35619 22 69 r 150 8.63938 s.75959 z5 9.03208 6.02t39 LA 72.2566 75.3982 3.33794 3.53429 3.73064 3.92699 4.12334 4.31969 4.51604 9.42478 6.28318 4.7 t239 25 78.s398 9.8 r 748 6.54498 4.90874 3.92699 3.27249 2.45437 r.63625 26 8r 6.80678 5.10509 4.08407 3.40339 1.70170 a1 LI 84.8230 7.06858 5.30 r44 4.24r15 3.53429 5.49779 5.69413 4.39823 3.66519 2.55254 2.65072 2.74889 4.55531 3.79609 2.84707 r.89805 5:89049 6.08683 4.7t239 2.94524 4.86947 3.92699 4.05789 3.0434'.;i r.96350 2.02895 6.28318 5.0265s 4.t8879 3.t4t59 2.C9440 56.5487 7.06858 4.7 1239 59.6903 7.46128 4.974r9 20 62.8318 7.85398 5.23600 2l 65.9734 8.24668 s.49799 AA 6814 0.2102 0.6029 28 81.9646 0.9956 7.33038 29 9r.t062 l.3883 7.59218 30 94.2478 97.3894 1.7810 7.85398 2.t737 8.1 l 578 32 100.53 r 2.5664 8.37758 194 3.t4159 3.29867 1.24355 1.30900 r.37445 r.43990 1.50535 1.57080 t;767 t5 1.83260 Dddng kinh Chu vi hinh tron 33 03.613 t 050r 8.63938 6.47953 r 8363 4.31969 34 06.814 3.35 l8 8.901 r 6.67588 5.3407 | 4.45059 35 3.7445 9.t6298 6.87223 5.497'19 36 09.9s6 t3.097 4.1372 9.42478 7.068s8 37 16.239 4.5300 9.68658 7.26493 38 r9.38 r 4.9226 9.94838 7.46128 5.65487 5.8r 195 5.96902 39 22.522 5.31 53 t0.2102 7.65763 6.12610 l 40 25.664 5.7080 10.4720 7.85398 6.28318 4l 28.085 I 007 0.7338 8.05033 A1 3t.947 6.4934 0.99s6 8.24668 43 3s.088 6.8861 L2574 8.44303 6.7 5442 5.62869 4.22152 2.8r434 44 38.230 7.2788 t.5t92 8.63938 6.91150 5.7 5959 4.31969 45 4l | 7.67 15 r.7810 8.83573 7.06858 5.89049 4.4t786 2.87979 2.94524 46 44.J I J 4.51604 3.0r069 48 50.796 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