Giáo trình khai triển hình gò

2 Hinh chiSu bbng cd song song vdi truc X; hinh chiSu mf,t canh i, d' songHinh 1 - 5 Hinh chidu c0a dudng song song chinh d'f" song song vdi truc Y*, tlQ dai hinh chidu cria chring ddu n

pHAr{ vAN rruffirrr - nd vAn BAc

rei Bi6[ thiQu

dd co bin vd hinh hoc hga hinh cing nhfrng phrrong phdp khai tridn hinh

cudn s6ch niy Ia tai hQu hoc tflp vh tham kh6o bd ich thio't thuc

CHUONG 1.

| - | Ph6p chi6u ttring (trgc €,iao)

truc giao; nhrr thd hien d hinh 1-1.

tidt sau niry ddu goi tdt Id hinh chidu

Z Didm giao nhau O cria ba trgc chidu goi li tlidm gdc'

.7

chinh cria didm c0a tam giac

l Hinh chiSu bo mgt cfro didm

bidu thi bing YH , khi

minh cirng l1i 16 tr6n, sau khi mit W quay trilng vdi mit V thi drrdng ndi cria a'- a"

Z^ =Aa =a'ax-{'^r=Odz

8

1-5 Hinh chidu c&a dudng thin€

hai tlidm ddu A, B ld a, d' , d' vd b, b', b-, sau d6 ndi hinh chidu cfia hai didm tr6n

b6n a'b" cfing ddu ngin hon dQ ddi thuc srJ cfra doan thfng AB, trt hinh chidu cria

(b) Hinh 1 - 4 Hinh chieu c0a dudng th&ng xi6n

dinh ti tr6n drrdng thing; ti le dO ddi hinh chi6u cfra hai doan trOn cDng mQt dddng

th6ng CD (2) Hinh chiSu bbng cd song song vdi truc X; hinh chiSu mf,t canh i, d' song

Hinh 1 - 5 Hinh chidu c0a dudng song song chinh

d'f" song song vdi truc Y*, tlQ dai hinh chidu cria chring ddu nh6 hdn d0 ddi thuc srr

2 Drrdng vu6ng g6c vdi mit chidu Dudng thing vu6ng g6c vdi m6t mht chieu

11

(1) Ilinh chidu chinh a'(b' ) co lai thdnh mOt didm'

vdi truc Z; abrrl*'U" adu"Ubng aO Jaithrrc cria AB Cdn cr1 vdotinh chdt hinh chi6u

1-4 Phucrr€ phrip dgng dO dd,i thgc cira doqn thin€

phdn rinh dO ddi thrJc td cfia tloatt th&ng ndy, phai thong qua phrlong ph6p d!t'*

tU b', a' ddn t4rc X Trong b6n v6 hinh

dO ddithtjc

12

\

Trong hinh 1-11(:r), ndu qua A dung

thr/c td c0a dudng th&ng (2)

(c)

t3

1-5 Vi tri tucrng ddi cira hai duon$ thin$

a"b" ll c" d .

z

I4

3 Hai dtidng thing ch6o nhau: Hai drrdng thing khOng song song nhau cfrng

(a)

Hinh 1 - 18 Hinh chidu g6c k9p c0a hai dr-ldng thtng giao nhau song song vdi mit chieu

(a)

Gi6i: Bdi vi tludng thing AB vu6ng g6c vdi mit chidu bdng, EF le dudng

"nat dinh phAi

hinh thuc (hinh L-22) (2) Hinh chidu c4nh tam gi6c g"h"k cta m{t vu0ng g6c b6n

t7

tam gi5.c GHK co tu thdnh mQt cltfdng th8ng,

Hinh 1 - 22 Hlnh chieu c0a mdt vu6ng g6c chinh

18

a"b c" co tu thinh mQt clt/dng thing vd lAn lugt song song vdi truc X, truc Yw Cdn

g'}r'k'/l truc Z (hinh ].-26).

r9

ll Didm vo drtdng th6ng tr6n mQt ph6ng.

Hinh 1 -28Ldy dudng thing tr6n mit phing (1).

^b

tlidm A dung ilrrdng song song bbng AD, trttdc ti6n c6 thd qua a' dung hinh chiSu

20

l-7 Vitri tucmg deii $ifta ducmg thinE, m|t phing

AB

vdi truc X thi tlrrdng th&ng MN Ia mQt dudng th8ng

2l

drJdng vuOng g6c v6i mit chidu

CD

phhng khiic, thi hai m4t phing ney song song vdi nhau Nhrl t;rong hinh 1-34, do

g6c vdi nhau

xi6n, tim giao tlidm cfia n6 Bdi vi drrdng thing EF la drrdng thing drlng, cho nOn

22

Hinh 1 - 36 Giao didm c0a dudng th&ng dfng vdi mdt phing vi trixi6n

23

chi6u bbng mn cita n6 ilt sd trirng vdi hinh chidu bhng eH cira mit th6ng dfng, sau

tuyOn

m6t"bon

Hinh 1 - 38 Giao tuyeln c0a mdt th8ng drrng vdi mit phtng xi6n

crriou.-Hinh 1 - 39 Giao tuydn gi[a m6r phtng bing vdi mqt phlng xi6n

ilt Vu6ng g6c

m'tphing il:# ;??ff',,:.i

a m

a'd' - b'c' crta mdt ph&ng hinh chrl nhAt ABCD, qua didm * aU"s mk song song vdi

nhdt ABCD c6 thd tim ra, cfing c6 thd ]hi ng tim Do drrdng thing MN tiry .! v6 ra

b'c

Hinh 1 - 41 Qua didm da bi6t dung mit ph6ng Hlnh 1 - 42 Qua didm dd bi6t drJng mft th&ng dfng

26

Hinh 1 43 Tim hinh rhgc cOa hinh tam gidc

hinh chidu bing ab cira drrdng thing bhng bhng

Hinh 1 - 44 Tim hinh thr,/c c0a hinh binh hdnh

DE vir

d6y

21

dd v6 ra tam gi6c DFG 6 b6n kh6c cfra tam gi6c DEF, hinh h.gp thdnh tt gt6c DEFG

X

ai"r, ld 0 tr6n drrdng thlng cfia hinh chieu bdng a Hinh 1 - 45 Xoav tron mat th&ns drlns

ir"rt .rre cing 5i a.rurrg thEng vu6ng g6c cria bc qua a t4i didm a, ndi arb vd arc thi

[Vf du 1] Da biet hinh chidu chinh d' e'f' g' vd hinh chidu bing defg cria hinh

Gi6i: B<ii vi e'f ll d' g' , ef il d,g // truc X, cho nen hai canh EF vd DG cfra hinh

didm d, drJng drrdng song song crta e'f' ai auurrl vu6ng g6c thrl hai d ajdm {, Si

Hlnh 1 - 47 Tim hinh thUc c0a hinh thang vi trixi6n Hinh 1 - 48 Tim hinh thr/c c0a tam gidc vi trf xi6n

tVi dr,r 2J De bidt hinh chiSu chinh tam gi6c a'b'c' vh hinh chidu bhng tam

qui tich quay

"Uai

29

tldi, thi drJdng ndi giCta hai tlidm a, vdi d chinh ld phuong hinh chidu bhng cria canh

30

- 49 Nguy6n ly phudng phdp hodn ddi mat

truc mdi bhng vdi khoting crlch tU hinh chidu bi thay thd ddn truc hinh chidu cfr.

hinh cu thd, chi cdn til b' dung drrdng vu6ng jo ad' trgc x, ua do idy brb,,,=uu, trei

ldn thay, ch8ng qua ld l6p lai mQt ldn nffa qud trinh dung trintr Hi;h r-sz tUl tfrd

(b) Hinh 1 - 52 Hai ldn thay Odic0a oidm

ho[c nhidu ldn ildi mat chidu, do mht chidu mdi phAi vudng g6cvdi mflt chidu cfr khOng

ddi kia, cho nOn sau khi ddi mgt m[t chidu, trong hQ thdng hai m[t mdi lai phAi thay

hinir ctridu mdi a,, b; (3) ding dudng thEng

32

cd

vuOng goc

g6c v6i m{t V \n thd mdt chidu mdi li m{t H, vu6ng g6c vdi m{t V, trlc ph6i dDng

dfng thi ph6i ldy m4t H, thay mit H; giir nguyOn mlt V Cdn rtidu kign bd tri p4t H,

ld: VUa ph6i vu6ng g6c vdi tam gi6c ABC, vila phAi vudng g6c vdi mit V glr nguy6n

m{t song song

Hinh 1 - 57 Tim hinh thr,lc c0a tam gi6c mdt ph&ng vi trixiOn

33

mat Hl mdi c6 thd cing 16c ctrng vuorig g6c voi mat V Cho nOn, brrdc dgng hinh ddu

tam gtac DEF Cdc btro-c dqng hinh cu thd la: (1) Dung truc X, song song vdi d {t' Q)Can crf vdo qui lu4t bien ddi crja didm tim ra tam gi6c hinh chidu dr"rf, cfra tam gi6c

sau mot ldn bidn ddi ldm alidu kiOn girii dd rdi drrng hinh ld.n nfra

Girii: c[n crl vio hinh 1-55 vd 1-56 c6 thd bidt khi bidn ddi mqt ph6ng vi tri

Gizii: Khi giao tuydn MN cria hai mat phing vudng g6c vdi mrt chidu thi hinh chidu

gi:ic MNG tr6n mdt H,, chrlng ld.n luot co tu thdnh cdc doan thing (nr) mrf, vd (nr)

H' rdi lai bidn ludng vudng g6c cfia mit v, cdc brrdc cfia

35

CHUOT',JG 2

"ay ta hinh elip (hinh tdong tr,r cfia hinh trbn)'

hiendhinh2-t,d,

37

mh n6 vu6ng g6c sE eo t9 thdnh mQt drldng th&ng, cbn hai hinh chidu kia Ih drrdng elip.

cria hinh bAu dgc nhy lh hinh chidu c'd' cfia dudng kinh th&ng dwrg CD cira tludng trbn,

Sau khi tim drloc tryc diri, t4rc ngSn thi c6 thd dung ttrrgc hinh bAu dqc.

Hinh 2 - 2 Hlnh chidu c0a hinh tron khi vuOng g6c vdi mqt chieu

Hinh 2 - 3 Hinh chidu cOa dr.tong trdn khi nbm xiBn vdi m{t chidu

38

thay H, thi trong h0 thong V/H, drrdng trdn vu6ng g6c v6i H' ltic ndy c6 thd cdn cf

hinh tru trbn, m6t canh g6c vudng khdc li budc S crja tlttdng xoSn dc D0 dhi cria brrdc

39

.t rid1 dong cfra dQng didm tr6n mit hinh n6n goi ld drldng xoSn dc hinh n6n

(S'1' , S'2' S'L2' vit 51, S2 S12 v'v.'.) giao

didm {0, {r, ar arL gifra hinh chidu chinh cria

chinh cfra cdc di6m tr6n d.ttdng xo6n dc n6n, lAn

dudng xo6n tr6n oc hlnh n6n

Hinh chidu cira Cludng cong khOng gian c6 drlgc bdi ndi tron lidn tudn tr/ hinh

40

(a) (b)

40 50 bo 60 70

4l

ivi du 1l: Tim vh dung tidp tuydn cria dttdng xo6n oc tru

trbn-Gi6i: Til tinh chdt cfia dddng xodn oc tru trdn vd hinh 2 - 9(a), ta c6 thd bidt

ir+t fr ac1,"Uhng gOc ttrang a(g6c I6n, g6c ngdng) cria drrdng xo6n oc, cl6 dii cira tidp

l)

2 - 5 M+t dudrn€ thing

dinh, tlttdng trbn d6y vd mit tru trdn ddu qui tr,r thdnh mgt dudng trbn q Sau cr)ng

"'7, d.'d"

43

kh6c Nhrr hinh 2 - 11(b), de biSt hinh chidu chinh m' cria didm M trOn m[t tru trbn,

Sinh thing, ctrrdng chudn (cl6n) Q ld tlrrdng trbn, drrdng chudn AB la drrdng thing

Dudng chudn Q

44

Dudng chudn Q

Oudng sinh

tlrrdng trbn dlnh Q lhm drrdng trdn d6y cfra n6, dudng ndi tam OO cria dudng trdn dinh vd dudng trbn ddy, d6 la dudng trr,rc cria m{t tru elip nAy Bdi vi dudn! trr,rc

elip, c6 thd tir mot hinh ctridu cria ,iidm tr6n mdt tru elip, tim ra hinh chidu kh6c

45

Hinh 2 - 13 Hinh thUc mdt cit chinh c0a mir tru etip vir tdy didni rren bd mdr

mat tru elip, c6 thd dp d,.rng phrrong phdp dddng

hai ong trq tron

46

Cing trr; elip

(cluoc) dqa ugo

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(1) Qua drrdng truc ctta hinh tru trdn dung m[t phu, tim ra didm tieip xric E, 6, 7,

cfr_a hai mdt cong - Hinh 3-3 thd hiQn dp dung phuong phdp m6t phing drrdng sinh

m4t phr,r mdi c6 thd cdt hinh tru trbn, hinh n6n t4i dudng sinh Ndu mu6'n m6t phu

mqt phq

61

hai drrdng sinh nhy Hinh chi6u canh cria m5.t phu vd hinh chiSu canh cria hinh n6n,

hinh chidu chinh cria n6 cft nhau tai bdn didm L' ,2', 3' vi 4',b6n didm ndy ld hinh

tinh tldc thr) cria hiQn trrong hinh chidu tr6n, vi thd

vdi mdt cdu

68

c0a hai hinh n6n giao nhau

6 thd ein cr1 vio tinh hinh cg thd tld 6p

cria mdt cong, cho n6n d{ng hinh tiQn loi Nhtrng nhrloc didm cua n6 le khi hai giao

dinh mQt hinh chidu cria dttdng kdt hop, loi

10

phtng vu6ng goc vdi dudng truc mdt tru lron

Khi mat ph&ng cdt xiOn vdi dudng trUc cria mbt tru trbn thi d.udng kdt hgp cria

ddu sE c6 drroc hinh chidu canh cria drrdng kdt hop Trong hinh 3-9, 0" 6" vd 3''9" Idn

song song vdi dudng trgc mqt hinh trq

phhng vuOng 96c voi dUdng irgc m{t n6n

cdt xiCn dudng truc mit hinh tru

Hinh 3 - 11 Hinh chidu drJdng fet nqp khi mbt ph8ng

di qua dinh non c0a mft n6n

ha

7l

ph6ng, ddu ld dudng thing Hinh chidu bXng cria drrdng kdt hop la dudng trdn; hinh

dinh n6n, thi drrdng kdt hop cria n6 lh dudng cong phing (gdm ba tnrdng hop: dttdng

tr6n dudng thf,ng niy Hai hinh chidu cbn lai cria dudng kdt hop c6 thd tim duoc bbng

th6ng vudng g6c vdi mit chidu d'lng chinh, giao didm I vd II gilta drrdng trbn vdi

Hinh 3 - 12MAlph8ng c6t xiCn m{t n6n Hinh 3 13 Mat phtng c6t xien mAt n6n

72

c6c drrdng sinh vdi hinh chi6u chinh criam6t phing, sau d6 dung hinh chidu bbng

0,1,,2 vd hinh chidu canh 0" , f ' ,2"

n6 khi mdt phing vd drrdng ti't"rc cua

song song vdidudng truc crla mat hinh non hinh, hinh chidu b[ng crja drldng kdt

aldu li drrdng cong ph8yrg, nhr/ thd hign d hinh 3-15(c) vd 3-16(c) Drrdng kdt hgp cria

(a) mit phEng song song vdi (b) mit phing song song (c) trr-long hQp khdc

tidt dien tron c0a trq elip vdi dudng trqrc c(la trr,r elip

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