Tài liệu Ebook Khai triển hình gò - Phàn Văn Huyên - Hồ Văn Bác

4 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 trong 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 1. 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 trong 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 khi 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 1 - 2 Phudng ph6phinhchiduchinh chiSu tam giric abc cfra tari grde ABC trong 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 trong kf thuat. Dd trinh bdy dugc ng6n gon, trong 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 trong khOng gian, trong vE kf thu4t co khi 6p dung ba m4t chidu vu6ng g6c v6r nhau. Nhu thd hiOn d hinh 1-B (a), mpt chieu 6 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 6 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 6 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 nhau t4o thinh mgt hQ thdng ba mqt chidu. Trong d6, dudng giao nhau 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 nhau giira V v6i W goi ld tryc chidu Z gqi tdt truc Z. Didm giao nhau O cria ba trgc chidu goi li tlidm gdc' .7 l. Hinh chiSu bo mgt cfro didm Trong hinh 1-3 (a), cdn crl vd.o phrrong ph6p lQp hinh chidu til tlidm A trong 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 trong b6n vE hinh chidu ba mit kh6ng cdn vE ranh gidi cfia mdt chidu, md chi vE truc chi6'u. Trong d6, truc Y khi quay theo mrnt H thi bidu thi bing YH , khi quay theo mrnt W thi bidu thi bing Y*, nhu thd hien 6 hinh 1-3 (c). (a) (b) (c) Hinh 1 - 3 Hlnh chieu c0a didm trong 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 khi 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 khi mit W quay trilng vdi mit V thi drrdng ndi cria a'- a" nhdt dinh vu6ng g6c vdi tryc Z. Ngoii ra, trong hai hinh chfr nhdt do Aa' = aax ve Aa' = {'at n6n trong b6n vE hinh chidu ba mdt, aE = {'^, . Cdn crl vdo phdn tich tr6n, c6 thd rrit ra quy lu4t chidu cfia mQt tlidm trong 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. 8 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 1 - 4 Hinh chieu c0a dudng th&ng xi6n Trong hinh 1-4 cdn thd hiCn, khi 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, khi 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 1. Drrdng song song vdi m4t chidu: Drrdng thing song song vdi mQt m{t chidu mi, ch6o vdi hai m6t chiSu kia 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 6 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 1 - 5 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 1 - 6 Hinh chi€u crla dr;dng song song blng (a) (b) YH Hinh 1 - 7 Hinh chidu c0a dr/dng song song cqnh 10 2. 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 1 - 8 Hinh chidu dUdng vuong g6c chinh Hinh 1 - 9 Hinh chidu dt/dng vu6ng gdc dung Hinh 1 : 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 ra. 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 trong song hO thdng hai mlt chidu. Ndu th6ig qua didm .ldu A dung drrdng thang AB, song vdi ab vir cdt iludng chidu Bb 6 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 ra hi cria n6 chinh le dO ddi thrrc td cria doan t thd cfra n6 nhu thd hien 6 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 sao cho bBo = ZB td cfra dudng thing AB. dO ddithtjc Hinh 1 - 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 1 - 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 1 - 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 khi 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, trong hinh 1-13(a) truc quay O0 th6ng qua didm dAu A, khidn vi tri crja tlidm A tnJdc vd sau khi 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 6 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 nhau 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 1 - 13 PhrJdng phdp quay tim dd ddi thUc t6 cOa dudng rh&ng t3 tucrng ddi cira hai duon$ thin$ Vi tri trrong ddi cria hai dudng thing kh6ng gian c6 ba kieiu: song song, c6t ld 1-5 Vi tri mdt, loai sau'goi nhau vd ch6o nhau, hai loai tnJdc goi Id dudng tn?"g cing dudng thing kh6c m[t' song 1. Hai dudng thing song song: Hinh chidu cua hai dtldng thing song L4a)- c|'c nh6m hinh ai song song nhau trong k h chidu cUng tat phii thfrng song s chi6u cirng m thi ab//cd; a'b'llc'd' ' hion 6 hinh song song v6 a"b" ll c" d thi hai tlrrdng Cung cDng Ic d6 hinh chidu cirng mat ddu song song vdi nhau thing trong kh6ng gian cfrng nhdt dinh song song vdi nhau. . z (o, (a) Hinh 1 - 14 Hlnh chieu c0a haidudng thing song song thing cdt nhau: Hinh chieu cira hai tlrrdng thing c6t nhau trong ctrng mdt cua kh6ng gian tat pfral cdt ,.rft"u (xem hinh 1-15a). C6c nh6m hinh chidu ctia 2. ;g,.ai dirdng nhau' Giao ilidm hau' Nhrr thd hien d du ab vdi cd' 6b' vdi h h tq hai dudng th n6 chinh lDr hinh (1-15b) c.d.,a..b.'v6ic"d" cirngtdtpheildnlrrgtc6tnhaudk,k,,k,,,vdphirhQpquiluat chidu cua hinh chieu cua mOt did; K khOng gian. Cgng cilng 16 d5, cdc nh6m hinh phn hgp chidu trai arrung th6ng ddu c6t nhau, *a aic- giao nhau crja cdc nh6m hinh nhy cflng vdi qui lu4t hinh chidu cria mot didm kh6ng gian thi hai drJdng thing z nhdt dinh cdt nhau trong khOng gian' (a) (b) Hinh 1 - 15 Hinh chieu cOa hai dudng thing song song I4 3. Hai dtidng thing ch6o nhau: Hai drrdng thing khOng song song nhau cfrng lihOng cdt nhau goi ld hai dudng thing ch6o nhau. Bin vd hinh chidu cria chfng c6 hai kieju: (1) Nhtr thd hiQn d hinh 1-16, hinh chieu cria hai drrdng thing ch6o nhau c6 thd co m6t nh6m hodc hai nh6m hinh chidu cr)ng mdt song song vdi nhau, nhung nh6m hinh chidu ctng mdt thrl ba tdt ph:ii cdt nhau (kd cri cdt nhau sau khi k6o dei). (2) Nhu thd hi6n o hinh I-I7, cd.c nh6m hinh chidu cing mdt cua hai drrdng (a) (b) Hinh 1 - 16 Hinh chieu c0a haiduong th8ng ch6o nhau (1) thing ch6o nhau deu cdt nhau (kd cA c6t nhau sau khi k6o ddi), nhung giao didm cua n6 nhat dinh khOng phr) hop qui luAt hinh chieu cing mOt didm. Nhrr trong hinh 1-17 (a), (b), giao didm cria ab vd cd tr6n thrJc te ldn luot ld hinh chieu I(2) cua didm I trOn AB vd didm II trOn CD trirng nhau m6t ch6, hai didm I-II goi ld mOt d6i didnr hinh chiSu tring(chAp) ddi vdi mdt H. Do ZrrZ' cho nOn khi nhin thing tir tr6n xudng, didm I c6 thd thdy, didm II khOng thdy Theo tAp qudn, d:inh ddu ngodc don vdi hinh chieu didm kh6ng thd nhin thdy nhtr (2). Cnng lo d6, hinh chieu chinh 3 vd 4 cria hai didm III 'vd IV tring nhau mQt ch6, cho n6n hai didm III, IV la d0i didm hinh chidu chAp ddi vdi m4t V. d6'i (a) (b) Hinh 1 - 17 Hlnh chieu c0a hai dudng tning c5t nhau (2) Khi hai tlrrdng th8ng cdt nhau song song vdi cDng mQt mat chi5u, hinh chieu g6c kep crja hai drrdng thhng trOn mit chidu ndy nhat dinh phrin 6nh dring dO ldn thrJc ctia g6c ndy. Nhu thei hien d hinh 1-18, bdi vi AB vi BC ddu ld drrdng ngang, cho ndn labc = IABC = lO, ndu trong g6c vu6ng c6 mOt canh song song vdi mOt mdt chidu, thi hinh chieiu cua n6 trOn m5,t chiefu d6 vAn phin 6nh g6c vu6ng. Nhtl trong l5 (b) (a) Hinh 1 - 18 Hinh chidu g6c k9p c0a hai dr-ldng g6c nOn t ph rinh 1-19, g AB, r6n bc cfing J6c vdi thtng giao nhau song song vdi mit chieu song vdi m4t H. Bdi vi BC vu6ng at phing ABba, lai vi BC// bc, cho 900. , bc vu6ng g6c ab, ttc labc = lai, n6u hinh chidu cfra mOt g6c vu ti chi6'u tdi thidu c6 mQt cung song song lrrdng thing ch6o nhau bing vdi g6c k9 lUdng song song cria hai drrdng thing ch6 khi vu6ng :hidu n6i tr6n c6 thd md rqng cho trtrdng hop hai dudng thing ch6o nhau song l6n xudng, thi bc vd ab vAn 36c. Ndu di chuydn tlrrdng ,rg"rrg bhng nC .ottg c irudng g6c v6i nhau. u rTgrrQc AI I a (a) song song vdimfi chieiu g6c vu0ng c6 mOt cqnh Hinh 1 '19 Hinh chidu c0a tvi EF drrdng th&ng AB vh hai cira chung g6c dul tim-dudng w6ng e' d. k' Y \ b' agb (b) c0a AB thin3 dfing chuns drrdns Tim 20 *.i1?', 16 - EF l' ftinh 1-20)' Gi6i: Bdi vi tludng thing AB vu6ng g6c vdi mit chidu bdng, EF le dudng th8ng vi tri binh thudng cho n6n dudng vu6ng g6c chung cria chring dinh phAi li drrdng ngang bing, c6 thd loi dung "ddc tinh hinh chidu g6c vu6ng""nat dd drrng hinh. Trlc til tlidm a(b) hinh chidu bXng cria tlrrdng thing AB dr.rng drrdng vu6ng gdc v6i ef, dugc giao didm k, tU k dgng dudng vu6ng g6c vdi truc X cdt e'f 6 rlidm k', tU didm k' dr,lng drrdng song song trr,rc X c6t a'b' 6 ttidm g', thi g'k' va gk chinh ld hinh chidu tr6n hai mdt cira tludng vu6ng g6c chung GK cAn tim. 1 I Hinh - 6 Hinh chidu m+t phin€ chidu cOo coc togi mgt phdng. Trong cdng t6c thrrc td, ding hinh chidu cria hinh vE m4t ph&ng dd bidu thi vi m6t phing trong khdng gian, Khi mdt ph8ng vu6ng g6c vdi mEt chidu, thi hinh chiSu cfra n6 co lai thdnh mQt drrdng thing; khi mat ph&ng kh6ng vuOng g6c vdi mdt chi6u thi hinh chidu cfra n6 ld mQt hinh tdong trr nhd hinh v€ m4t ph8ng. C5'n crl vdo vi tri cria m4t ph8ng trong hQ thdng ba mat chidu, c6 thd chia ra ba loai: mdt vu6ng g6c v6i mbt chidu, mdt song song vdi mit chidu, vd m{t phing vi tri xi6n. tri cfra 1. Met vuOng g6c vdi mdt chidu: Mdt phing vu6ng g6c vdi mQt mdt chidu md xi6n vdi hai mdt chidu khdc thi goi ld mdt vu6ng g6c vdi mit chidu. M4t phEng vu6ng g6c vdi mit H ggi li mat ph8ng vu6ng g6c dring; m+t phing vu6ng gdc vdi m*t V g9i ln mdt vudng g6c chinh; mdt ph8ng vuOng g6c v6i mit W goi li mdt vu6ng g6c canh. Nhu trong hinh l-2L, tam gi6c ABC ld m6t m4t thEng drlng, tinh chdt hinh chidu cira n6 ld hinh chidu bhng abc co tu thhnh mQt drrdng thing; hinh chidu chinh tam gi6c a'b'c' vd hinh chidu b6n tam gi6c a"b" c" tldu ld hinh trrong trJ cfra tam giric ABC nhung nh6 hon hinh thr/c. CIn crl vdo tinh chdt hinh chidu cria m6t thinB dtlng c6 thd suy ra tinh chdt hinh chidu cfra mdf, vu6ng g6c chinh vd m4t vudng g6c b6n lA: (1) Hinh chidu chinh tam gi6c d'e'f cfia mdt vu6ng g6c chinh tam gidc DEF co tu thdnh mQt drrdng th&ng; cbn hinh chidu bing tam gi6c def vd hinh chidu canh tam gi6c d"e"f 'tldu ln hinh trJong trJ cfra tam giSc DEF nhung nh6 hon hinh thuc (hinh L-22). (2) Hinh chidu c4nh tam gi6c g"h"k cta m{t vu0ng g6c b6n Hinh 1 - 21 Hinh chidu c0a mft th&ng drlng t7 tam gi5.c GHK co tu thdnh mQt cltfdng th8ng, cdn hinh chi6u chinh tam gi6c g'h'k' vi hinh chidu bing tam gi6c ghk ddu ld hinh trlong tU cria tam gi6c GHK nhung nh6 hon hinh thuc (hinh 1-23). Hinh 1 - 22 Hlnh chieu c0a mdt vu6ng g6c chinh Hinh 1 - 23 Hinh chidu cOa mdt vuOng g6c ben 2, Met song song vdi mdt chidu: Mdt phing song song vdi mQt mdt chidu vd vdi hai mdt chidu kh6c goi ld md,t song song vdi mdt chidu, m{t phing vdi mit V gqi lh mit chinh, m4t ph8ng song song vdi mdt H goi.ld mdt bing, mdt phing song song vdi mdt W ggi td m5.t canh. Tam giric ABC thd hiCn d hinh I-24 lb, mit bXng, tinh chdt hinh chidu cfia n6 li: hinh chidu bhng tam gi:ic abc bhng hinh thuc cria tam giSc ABC, hinh chidu chinh a'b'c' vd hinh chidu bOn vuOng g6c song song Hinh 1 - 24 Hinh chieu c0a mat phtng b6ng 18 a"b c" co tu thinh mQt clt/dng thing vd lAn lugt song song vdi truc X, truc Yw. Cdn chdt hinh chidu cria m{t bhng c6 thd suy do6n ra tinh chdt hinh chidu crl vdo tinh cfra mqt chinh vh m{t b6n ld (1) Hinh chidu chinh tam gi6c d,'e'f'cfra m{t chinh tam giric DEF bhng hinh thrJc cria tam giric DEF; hinh chi6'u bing co tu thdnh tludng th8ng def song song vdi truc X, hinh chidu canh co tu thanh dtrdng th8ng d e" f " song song vdi truc Z (hinh l-25) (2) Hinh chidu canh tam gi6c r.'h,'k.. crta Hinh 1 - 25 Hinh chieu cOa mdt ph8ng chinh m6.t canh tam giric GHK bhng thdnh tlrrdng thing ghUl truc g'}r'k'/l truc Z (hinh hinh thUc cfia tam gi6c GHK; hinh chidu bhng co tu Y"; hinh chidu chinh co tu thdnh mQt dtrdng thing ].-26). Hinh 1- 26 Hlnh chidu cOa mf,t phEng b6n 3. Mdt phing (vi tri) xi6n. Mdt phlng xi6n vdi cd ba mit chidu thi goi m4t phing (vi tri) xi6n. Nhu thd hiQn d hinh I-27 , d.o tam gidc ABC ddu xi6n vdi ba'mqt chidu do d6 bahinh chidutamgiric abc, tam gi6,ca'b'c', tam gi6ca"bc" crtan6 ddu ld hinh tr/ong tu nh6 hon hinh thdc. Hinh 1 - 27 Hinh chi0'u c0a mdt phEng vi tri xi6n r9 ll. Didm vo drtdng th6ng tr6n mQt ph6ng. 1. Ddng tlrrdng th&ng tr6n mit phing: DUng dtrdng th8ng tr6n mdt phing da bidt thudng ep dvng hai phuong ph6p sau: (1) Mqt dddng thfng di qua hai didm tr6n mqt phing thi tlrrdng thing d6 tdt ph6i nhm tron md,t ph6ng ndy. Nhrr thd hien 6f hinh t-28, tam gi6c ABC quydt clinh m4t phing P; bdi vi hai didm M-N IAn lrrgt d Hinh 1 -28Ldy dudng thing tr6n mit phing (1). tr€n hai cung AB vd AC cfia m[t phang tam giric, thi dudng ndi MN tdt phii nim tr€n mat phing ndy. (2) MOt drldng th8ng di qua mQt didm tr6n m5.t phing vi song song vdi mQt drrdng thing kh6c tr6n m4t phing thi itrldng flring ndy tdt ph6i nhm tr6n m{t phing ndy. Nhu thd hien 6 hinh I-29, hai dtrdng th8ng cdt nhau DE vd EF quydt dinh mdt phing P. Ndu tr€n EF ldy tlidm M, qua M drrng MN/iDE thi MN tdt phai nXm tr6n mdt ph8ng nay. 2. Drlng didm tr6n m{t phing: Ndu didm nim tr6n tlrrdng th8ng bdt ky trong mit phing thi didm ndy tdt ph6i nhm tr6n mit phing d6. Ndu mudn didm trOn mit ph8n g fid6c ti€n phdi ldy drrdng thing trOn m{t phing. Nhu thd hiQn 6 hinh L-zg, do tlidm N nhm tr6n dudng th8ng MN trong mit phfrng P, thi didm N nhdt dinh nhm tr6n m{t ph8ng P. ^b Hinh I - 29 Ldy dudng th8ng tr6n m{t phEng Hinh 1- 30 Dudng song song c(Ja mit chi6u tren mqt phing [Vi du]: TrOn m+t ph&ng tam giric ABC d{ng tludng song song chinh song song bhng (hinh 1-30) vi clUdng Gidi: Khi dr;ng drrdng song song trOn m4t ph&ng cAn phdi phi hop hai didu ki6n: M6t, ph6i phn hgp tinh chdt hinh chidu cria drrdng song song. Hai, ph6i phir hop didu ki6n tludng th8ng trOn mi.t ph8ng. Nhu tr6n m{t ph8ng tam giric ABC qua tlidm A dung ilrrdng song song bbng AD, trttdc ti6n c6 thd qua a' dung hinh chiSu chinh a'd' cira tludng bXng, song song vdi tr\rc X, rdi dga vAo tlidu kiOn d{ng tludng th&ng trOn m{t phing dd tim ra hinh chidu bhng ad thi a'd' vd ad trlc ld hinh chidu 0 hai m{t cria tlrrdng ngang bing AD tr6n mht phing tam gi6c ABC. Cfrng cDng 16 d6, c6 thd qua tlidm C dgng ttudng ngang bhng chinh CE. 20 tucmg deii $ifta ducmg thinE, m|t phing Vi tri trrong ildi giita tludng thing vd mdt ph8ng, gifla mdt phing v6i l-7 Vitri mAt phing chia ra ba trt/dng hop: song song, c6t nhau v vudng g6c. Cdn ct1 vdo y6u cdu dUttg hinh gb khai tridn chi gidi thiQu tinh chdt hinh chidu vh qui luat drJng hinh khi mQt drrdng th8ng, mQt met phing vu6ng g6c vdi m4t chidu. l. Song song. 1. Drldng th8ng song song vdi mpt ph&ng: Khi mQt tlrrdng th8ng song song vdi mQt tlrrdng th8ng Uft f.y tr6n mdt phing, thi tttrdng thing d6 vd m4t phing 6t phfli song song vdi nhau. Nhrr trong hinh 1-31, do iludng thing AB song song vdi ttudng thing CD trong mit phing P, cho nOn drldng thing AB song song v6i mdt phing. Ngodi ra, khi m4t phEng ney vu6ng g6c vdi m4t chidu thi hinh chidu cria m+t ph8ng tr€n m{t chidu mh n6 vuOng g6c co tu thdnh mQt drJdng th8ng. Cdn crl vho tinh chdt cfra hinh chidu ndy, khi drrdng thing song song vdi m{t vu6ng g6c cfia m4t chidu, thi hinh chidu cfia drrdng th&ng trOn m{t chidu ndy tdt phAi song song vdi tlttdng thing co tu cria m{t vu6ng g6c. Cbn phrrong c6c hinh chidu kh6c cria dudng thing nlry c6 thd tny y. 6r-t' AB CD Hinh 1 - 31 Duong thtng song song vdi mit ph&ng Hinh 1 - 32 Dudng thEng song song vdi mft vu6ng g6c vdi m{t chi6u Trong hinh L-32, khi hinh chfr nhQtt EFGH vu6ng g6c vdi m5t H, n6'u ttudng thEng MN song song vdi hinh chfi nh4t EFGH thi hinh chidu bhng cria n6 cfrng song song vdi nhau (trlc mn song song vdi efgh). Cbn phuong cfra hinh chidu chinh m'n' cria tlttdng thing MN c6 thd tty y. Ifti m'n' song song vdi truc X thi tlrrdng th&ng MN Ia mQt dudng th8ng bhng; khi m'n' vudng g6c'vdi truc X, m vi n trirng chap nhau mOt didm, thi tlddng thing MN ld mQt tlrrdng thing drlng. D6 ddng thdy rhng, m4t ph8ng song song vdi iludng vudng g6c cria m6t chidu, 6t Id mlt vu6ng g6c cria mit chi€fu ndy. Trong hinh 1-33, tludng th8ng AB vu6ng g6c vdi mdt phing hinh thang CDEF, nhrr khi tludng thing AB vu6ng g6c vdi mdt H, thi m4t phEng hinh thang CDEF nhdt tlinh ld m4t thing drlng. Hinh 1 - 33 MAt Ph&ng song song vdi drJdng vuOng g6c v6i mit chidu 2. M4t phing song song vdi m4t phing: Ndu hai tludng thing c6t nhau trOn mQt m[t mQt mqt ph&ni, song song trldng rtng vei hai drrdng thing c{t nhau trOn 2l phhng khiic, thi hai m4t phing ney song song vdi nhau. Nhrl t;rong hinh 1-34, do drrdng thing AB song song vdi drrdng th8ng A,B,, dudng thing BC song song v6i dttdng th&ng BrC' cho nOn mdt phing P song sorlg vdi m6t phing Q. Ngodi ra, khi mit vuOng g6c cria hai mdt chidu song song vdi nhau thi hinh chidu co tg thdnh drrdng th8ng cria chring sE song song vdi nhau. Trong hinh 1-35, khi hai m[t ph8ng chinh ABCD vd EFGH song song vdi nhau thi hinh chidu chinh a'b'c' d' vh e'(g'h' cfrng 5t sc song song vdi nhau. Hlnh 1 - 34 MAt phlng song song vdi m{t phAng Hlnh 1 - 35 Hai mf,t vu6ng g6c c0a mf,t chidu vu6ng g6c vdi nhau ll. Cdt nhou 1. Drrdng thing cdt nhau vdi mdt phing. Khi tludng thing kh6ng song song vdi m6t ph6ng thi 6t s6 cdt nhau, didm giao nhau cira n6 ld didm chung cria drrdng thing vd mdt phing. N6i chung, c5n crir vdo tinh chdt niy dd tim ra hinh chidu crla giao didm. Nhu trong hinh 1-36, tlrrdng thing drlng EF cdt nhau vdi tam gidc ABC xi6n, tim giao tlidm cfia n6. Bdi vi drrdng thing EF la drrdng thing drlng, cho nOn hinh chidu bhng cria n6 co tg thinh mQt didm e(0. Do giao tlidm K nhm tr6n drrdng thing EF cho n6n hinh chidu bhng k cria giao didm f 5t s6 trr)ng vdi e(0. Lai vi giao didm K cbn d trOn tam gi6c ABC, cho n6n c6 thd loi dung tinh chdt didm tr6n mdt phing, til k tim ra hinh chidu chinh k' cfra giao didm K. Khi d{ng hinh cu thd, c6 thd trr/dc ti6n thdng qua didm k d trOn hinh chiSu bhng abc cria hinh tam gidc, dUng m6t dfdng thf,ng phu ad, lai tim ra hinh chidu chinh a'b' crja n6 thi giao didm k' cira a' d' vdi e'{ tftc lA hinh chidu chinh cria giao didm K. Hinh 22 1 - 36 Giao didm c0a dudng th&ng dfng vdi mdt phing vi trixi6n hin thdy va Giao Phdn c6 thd uei uu c4nh didm phdn k'fl' bi drrdng thlng ne thhng xi6n a didm K coa n6 erg. L4i m{t thing ddn , ;?: chirng' Do l?dliT#'; a n6 cttng K giiraefg vdi qti; K tr( vbo tinh .hdt d;;;; Ja I {trung Totg tim' didm can ,h ffi",,il"il?1;:#',1;:,,*:U,il{:, dnhin,hdvcrra II (2''2) tren duoirl n'g ia* a'b'.Do didml(t"l) chinh 'dm toan t^rdc;"'ffi;ffie* I c6 ' - ;A^'ntTi*"J;tif,l;; :*1ffi,1i?;ffi'-atqrianglh tinh co t9' trrJdc ffi_.,# ;;;*-+;-Oa"fc6 mit vu.n s g6cv6i m4t chidu, , r'i&og-uai n6' riii lgi'dung ,"J*"* ttttat cria giao didm ti€n tlinh ra mQt hinh chidu. tfti6" dgng ;il;i dd tinh chdt cfia tlidm tron tludng th&ng dudng FG, khi nhin tu jlJS+J,'lnT* d^ ^d. Hinhl.3TGiaodidmgiuadUdngtn&ngvitrixi€nV6idudngth&ngdrlng 2. M4t phing giao nhau vdi m4t P thi tdt ph6i giao nhau; dttdng giao nhau c Vi thd, chi cdn tim ra hai didm chung h cria hai mat phing thi c6 thd tim ra giao hinh 1-38, m4t thfng dri'ng Q grao nhau v tuydn MN cfra n6. G g.o t"yA" MN nhm tr6n mdt thing drlng Q, cho nOn hinh