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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 tr6n mit chidu P.
didm
chinh cria
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'

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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.
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