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64 A ? N ≠ R S J M 1 @ M Q OS C H? B M R ~ A L G A M
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66 A ? N ≠ R S J M 1 @ M Q OS C H? B M R ~ A L G A M
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1 , / 5 M @ H C R M Q G P P C E S J ? P C Q 67
A´KM BG@SH?P • L E S J M Q CL DMPK? GQMKßRPGA?
Cj bg`shm ksjrgtgqr_ _ j_ gxosgcpb_ kscqrp_ rpcq Ålesjmq bc 4.è, Cl sl bg`shm gqmkÑ+
rpgam* lgleslm bc jmq rpcq Ålesjmq qcpÅ bc 4.è,
+ Bg`shc jgecp_kclrc sl_ a_h_ cltmjtclrc sq_lbm j_q bgkcl+qgmlcq b_b_q* _ cvacnagâl bc j_ bg +
kclqgâl U vosc lm qc npmnmpagml_,
+ N_p_ clamlrp_p j_ bgkclqgâl U q bg`shc _ r_k_àm amknjcrm cj rpgÅlesjm ?A> bc j_ tgqr_ qsnc+
pgmp* amkm qc kscqrp_ cl j_ dgesp_,
PR D B OB K @F>@kilnk^]_eáj `a hko Äjcqhko eokiÅpne_ko
N_p_ amltclacpqc _ qá kgqkm bc osc lgleslm bc jmq Çlesjmq qcpÇ bc 4.í* kgb_ a_b_ Çlesjm cl cj bg`shm gqmkÜrpgam bc j_ dgesp_ 1,23 aml sl rp_lqnmpr_bmp m sl ncb_xm bc n_ncj* w _lmrc cj Çlesjm amkn_p_bm aml cj tcpb_bcpm bc 4.í, Lgleslm bc jmq Çlesjmq kmqrp_bmq qml ges_jcq cl cj bg`shm gqmkÜrpgam, Bmq qml kclmpcq w slm cq k_wmp bc 4.í,
Bopei]_eáj `a Äjcqhko `a 1.å
Qg sqrcb cqrÇ `mqosch_lbm cl n_ncj
epÇdgam w lcacqgr_
cqrgk_p
Çlesjmq*
sl Çlesjm bc 1.í qc m`rgclc _npmvgk_b_kclrc _j _t_lx_p 0 as_bpmq w qs`gp /,
PRp_lqdgcp_ j_ bgkclqgâl U _j bg`shm gqmkÑrpgam n_p_ amknjcr_p j_ a_h_ cltmjtclrc,
Clasclrpc j_ bgkclqgâl V kcbg_lrc sl kÉrmbm qgkgj_p w bcqnsÉq rp_lqdgÉp_j_ _j bg`shm gqmkÑrpgam,
Lm cq bc 4 . í @ Lm cq bc 4 .í
Lm cq bc 4 .í
+ Amknjcrc cj bg`shm gqmkÑrpgam _j jma_jgx_p cj
nslrm B sq_lbm j_ bgkclqgâl Hv
amkm qc kscqrp_
cl
j_ dgesp_,
Cl
jmq bg`shmq gqmkÉrpgamq lm nscbc sq_pqc sl rp_lqnmpr_bmp pcesj_p n_p_ kcbgp jmq Ålesjmq, Amltgcpr_ j_q kcbgb_q _lesj_pcq cl kcbgb_q jglc_jcq _ jm j_gem bc j_q jÖlc_q bc jmq chcq gqmkÉrpgamq,
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7. A ? N ≠ R S J M 1 @ M Q OS C H? B M R ~ A L G A M
0, Jma_jgac jmq nslrmq ?* A w B* w _q á qsacqgt_kclrc
3, Mqaspcxa_ j_q jálc_q dgl_jcq
/, Sqc j_q kcbgb_q bc amknclq_agål ^ v _ cl cj bg`shm gqmkÜrpgam n_p_ jma_jgx_p cj nslrm > qm`pc j_ aspt_
1, @mqoschc sl_ aspt_ qs_tc w jgecp_ _ k_lm _jx_b_ _ rp_tÖq
bc jmq nslrmq
1,25 Aspt_q cl sl bg`shm gqmkÜrpgam,
2, Bg`shc sl_ jálc_ tcprga_j bcqbc cj nslrm > n_p_ jma_jgx_p cj nslrm >%* v _qá
qsacqgt_kclrc* f_agÖlbmj_q rmb_q ges_jcq _ j_ _jrsp_ bcj `jmosc &a' bcqnsÖq bg`shc sl_ aspt_ jgecp_ _ rp_tÖq bc jmq nslrmq
1,/6 @ROS>P BK RK AF?RGLFPLJBQOF@L
Sqrcb nscbc bg`sh_p aspt_q gqmkÉrpga_q sq_lbm sl_ qcpgc bc kcbgb_q bc amknclq_agâl qgkgj_pcq _ j_q osc qc _l_jgx_pml cl j_ qcaagâl 1,/2, Qcjcaagmlc as_josgcp låkcpm bcqc_bm bc nsl+rmq _j _x_p _ jm j_gem bc j_ aspt_ cl j_ tgqr_ qsncpgmp b_b_* amkm jmq nslrmq > q ? w A bc j_ dgesp_ 1,25, Cjgh_ qsdgagclrcq nslrmq
n_p_ s`ga_p aml npcagqgâl j_ rp_wcarmpg_ bc j_ aspt_ &_ k_wmp a_lrgb_b bc nslrmq* k_wmp npcagqgâl', Bg`shc bcqbc a_b_ nslrm sl_ as_bpÖasj_ aml j_q jÖlc_q bc amknclq_agâl n_p_jcj_q _ jmq chcq gqmkÉrpgamq w åqcj_q n_p_ jma_jgx_p a_b_ nslrm cl cj bg`shm gqmkÑrpgam amkm cl cj chcknjm bc j_ dgesp_ 1,25,
Ncjmr_ bc rclgq &pcaf_x_b_ bc dÇ`pga_', @^of`^qro ̂
ab
Oldbo Mof̀ b, &Amprcqá_ bc Bpmmbjcq* %Rfc Aj_qqga Amjjcargml',
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1,/7 BIFMPBP SBOA>ABO>P BK AF?RGLP FPLJ{QOF@LPQg sl aÖpasjm cqrÅ cl sl nj_lm osc lm cq n_p_jcjm _j nj_lm bc npmwcaagâl* cj aÖpasjm qc npm+wcar_ amkm sl_ cjgnqc* j_ as_j qc nscbc amlqrpsgp cknjc_lbm kcbgb_q bc amknclq_agâl,
1 , / 7 C J G N Q C Q T C P B ? B C P ? Q CL B G @ S H M Q G Q M K ~ R P G A M Q 7/
BG@SHM BC SL? CJGNQC GQMKßRPGA? K C B G ? L R C KCBGB?Q BC AMKNCLQ?AG´L
JÅpk`k
`a
h]o
hÉja]o
]ha]pkne]o
+ Bg`shc jÖlc_q n_p_jc+j_q cqn_ag_b_q bc dmp+
k_ _jc_rmpg_ _ rp_tÉq bcj aÖpasjm,
PRp_lqdgcp_ cqr_q jÖlc_q _j bg`shm gqmkÑ+
rpgam, As_lbm cj mpgdgagm q_je_ bc j_ n_prc gldcpgmp bcj `jmosc* jma_+jgac jmq nslrmq _j kcbgp sl_ bgqr_l+ag_ ges_j _ j_ _jrsp_ a bcj `jmosc bcqbc a_b_ slm
bc jmq
nslrmq
qsncpgmpcq,
Bg`shc
j_ cjgnqc*
n_prc
bc j_ as_j cqr_pÅ masjr_* _ rp_tÉq bc cqmq nslrmq, Mqaspcxa_ j_q jÖlc_q bcj bg`shm dgl_j,
Kgqk_ npmdslbgb_b
JÅpk`k
`a
hko
k_dk
lqjpko
+ Clagcppc cj aÖpasjm b_bm cl sl as_bp_bm w rp_ac bg_eml_jcq, Bg+
`shc mrpm as_bpm _ rp_tÉq bc jmq nslrmq bc glrcpqcaagâl bc j_q bg_eml_jcq w cj aÖpasjm amkm qc kscqrp_,
+ Bg`shc cqr_ kgqk_ amlqrpsa+agâl cl dmpk_ gqmkÑrpga_* rp_lq+dgpgclbm j_q bgqr_lag_q ^ v _, &Qg
qc bcqc_l kÅq nslrmq* _epcesc jÖlc_q n_p_jcj_q _j _x_p* amkm cl cj kÉrm+bm _lrcpgmp', J_q jÖlc_q aclrp_jcq bcj bg`shm gqmkÑrpgam qml jmq bgÅkcrpmq amlhse_bmq bc j_ cjgnqc, J_q bg_eml_+jcq _ 23è amglagbcl aml jmq chcq k_wmp w kclmp bc j_ cjgnqc, Cj chc kclmp rgclc j_ kgqk_ jmleg+rsb osc jmq j_bmq bcj as_bp_bm glqapgrm,
As_lbm qc pcosgcp_ kÅq npcagqgâl* bgtgb_ cj
aÖpasjm cl
/0 n_prcq
ges_jcq*
amkm
qc kscqrp_
cl j_ dgesp_, KÉrmbm bc jmq /0 nslrmq
IÉja]o jk eokiÅpnÉ_]o
Qg sl_ aspt_ cqrÅ cl sl nj_lm lm gqmkÑrpgam* lm rmb_q j_q kcbgb_q bc amknclq_agâl nscbcl _njga_pqc bgpcar_kclrc, J_ a_p_ cjÖnrga_ osc qc kscqrp_ cl j_ tgqr_ _svgjg_p qc clasclrp_ cl sl
nj_lm glajgl_bm lm gqmkÑrpgam,
+ Bg`shc jÖlc_q cl j_ tgqr_ mprmepÅdga_ n_p_ jma_jgx_p jmq nslrmq,
Clagcppc cj agjglbpm cl sl_/ a_h_ bc amlqrpsaagâl w bg`s+
hc j_ a_h_ cl cj bg`shm gqmkÑrpgam,
Bg`shc
j_ `_qc
sq_lbm
kcbgb_q
bc amknclq_agâl* w amlqrpsw_ j_ cjgnqc glajgl_b_ _j jma_jgx_p jmq
nslrmq w bg`sh_p j_ aspt_ dgl_j _ rp_tÉq bc cjjmq,
Kgb_ j_q bgqr_lag_q n_p_jcj_q _ sl chc gqmkÑrpgam &:̂ * _
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70 A ? N ≠ R S J M 1 @ M Q OS C H? B M R ~ A L G A M
1,0. LOFBKQ>@FßK AB BRMPBP BK AF?RGLP FPLJ{QOF@LP
Cl j_ dgesp_ 1,26 qc kscqrp_l cjgnqcq bc as_rpm aclrpmq amlqrpsgb_q cl j_q rpcq a_p_q tgqg`jcq bc sl as`m, Rcle_ cl asclr_ osc rmb_q j_q bg_eml_jcq qml fmpgxmlr_jcq m cqrÅl _ 4.è bc j_ fmpgxmlr_j, Qg qc rgclc cl asclr_ cqrm* qc d_agjgr_pÅ cj bg`shm bc j_q dmpk_q,
J_q cjgnqcq
_npmvgk_b_q
amkm
Éqr_q*
amlqrpsgb_q
_ n_prgp
bc
as_rpm
_pamq*
qml
`_qr_lrc npcagq_q n_p_ j_ k_wmpÖ_ bc jmq bg`shmq gqmkÉrpgamq, Cj kÉrmbm bc jmq as_rpm aclrpmq nscbc sq_pqc qâjm n_p_ cjgnqcq cl nj_lmq gqmkÉrpgamq, J_q tcpqgmlcq _lrcpgm+pcq bc qmdru_pc bc A?B* amkm ?srmA?B /.* sq_`_l cqrc kÉrmbm n_p_ apc_p dmpk_q cjÖnrga_q _npmvgk_b_q bgqnmlg`jcq cl cj qmdru_pc, J_q tcpqgmlcq _ars_jcq srgjgx_l sl_cjgnqc cv_ar_, Chhnqc bc as_rpm aclrpmq
BG@SHM BC SL? CJGNQC BC A S ? R P M ACLRPMQ
+ Bg`shc m gk_eglc sl as_+bp_bm cltmjtclrc n_p_ cj aÖpasjm osc qc kscqrp_ cl cj bg+
`shm ksjrgtgqr_, Bg`shc j_ tgqr_ gqmkÑrpga_ bcj as_bpm &sl n_p_+ jcjmep_km cosgjÅrcpm aml j_bmq ges_jcq _j bgÅkcrpm bcj aÖpasjm',
K_posc cj nslrm kcbgm bc /a_b_ jÖlc_,
PBg`shc jmq bmq _pamq ep_l+bcq* bc p_bgm O * bcqbc j_q glrcpqcaagmlcq bc j_q ncpncl+
bgasj_pcq osc cqrÅl cl j_q bmq cqosgl_q kÅq acpa_l_q bcj n_p_+ jcjmep_km,
BgÇkcrpm bcj aápasjm
BgÇkcrpm bcj aápasjm
U
PR D B OB K @F>Fc _osá sl_ pcej_ érgj, Cj chc k_wmp bc j_ cjgnqc qgcknpc cq ncpnclbgasj_p _ j_ álc_ aclrp_j bcj agjglbpm* w cj chc kclmp cq ncpnclbgasj_p _j chc k_wmp w amglñagbc aml j_ jálc_ aclrp_j,
t Cj chc kclmp amglagbc aml j_ jálc_ aclrp_j
+ Bg`shc jmq bmq _pamq nc+oscàmq* aml p_bgm p* bcqbc j_q glrcpqcaagmlcq bc j_q ncpncl+
bgasj_pcq osc cqrÅl bclrpm bcj n_p_j cj m ep_km* n_p_ amknjcr_p j_ cjgnqc,
PRDBOBK@F>Amkm sl_ amknpm`_agål bc j_ s`ga_agål cv_ar_ bc cqrmq aclrpmq* nscbc rp_x_p sl_ bg_eml_j j_pe_ bcj n_p_jcjmep_km amkm qc kscqrp_ cl cj n_qm 2, Jmq nslrmq kcbgmq bc jmq j_bmq bcj n_p_jcjmep_km qml jmq nslrmq bc r_leclag_ n_p_ jmq as_rpm _pamq,
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1 , 0 . M P G C L R ? A G ± L BC C J G N Q C Q CL B G @ S H M Q G Q M K ~ R P G A M Q 71
Bifmpbp
j|p
mob`fp^p
J_ cjgnqc bc as_rpm aclrpmq qc bcqtÖ_ aml+qgbcp_`jckclrc bc sl_ cjgnqc tcpb_bcp_, Amkm qc kscqrp_ cl j_ dgesp_ 1,27_* sl_ cjgnqc bc as_rpm aclrpmq cq _jem kÅq amp+ r_ w °empb_¢ osc sl_ cjgnqc tcpb_bcp_, As_lbm j_ cjgnqc bc as_rpm aclrpmq lm
cq jm `_qr_lrc npcagq_* nscbc sq_pqc sl_ _npmvgk_agâl kÅq acpa_l_ jj_k_b_ cjgn+qc loqldoÄcf` ̂bc as_rpm aclrpmq n_p_ ec +lcp_p sl bg`shm kÅq cv_arm,
1,27 Glcv_argrsb bc j_ cjgnqc bc as_rpm aclrpmq,
tcpb_bcp_
bc as_rpm aclrpmq
BG@SHM BC SL? CJGNQC MPRMEP•DGA? BC AS?RPM ACLRPMQ
JÅpk`k
`ah
na_pÄjcqhk
ajrkhrajpaN_p_ apc_p
sl_
cjgnqc
_npmvgk_b_
kÅq npcagq_ cknjc_lbm cj kÉrmbm mprmepÅdgam* qge_ jmq n_qmq bc jmq kÉrmbmq qgesgclrcq, As_lbm qc glgag_ bcqbc sl _eshcpm m sl ag+jglbpm* cq kÅq aâkmbm qcesgp cj kÉrmbm bc j_ jÖlc_ aclrp_j,
JÅpk`k
`a
h]
hÉja]
_ajpn]h
+ Bg`shc j_q jÖlc_q aclrp_+jcq gqmkÉrpga_q, Bcqbc cj aclrpm* bg`shc sl aÖpasjm bc
amlqrpsaagâl ges_j _j bgÅkcrpm pc_j bcj _eshcpm m agjglbpm, Cj aÖpasjm glrcpqca_pÅ j_q jÖlc_q aclrp_jcq cl as_rpm nslrmq* >,* ? v@ v
A,
P? n_prgp bc jmq bmq nsl+rmq bc glrcpqcaagâl qm`pc sl_ jÖlc_ aclrp_j* bg`shc jÖlc_q
ncpnclbgasj_pcq
f_qr_
j_
mrp_
jÖlc_ aclrp_j, BcqnsÉq* bg`shc jÖlc_q ncpnclbgasj_pcq bcqbc jmq bmq nslrmq bc glrcpqcaagâl bc j_ mrp_ jÖlc_ aclrp_j f_qr_ j_
npgkcp_ bc cjj_q,
PAml j_q glrcpqcaagmlcq bc j_q ncpnclbgasj_pcq amkm aclrpmq* bg`shc bmq _p+
amq ncoscàmq w bmq ep_lbcq,
\Aápasjm bc
amlqrpsaagål B ges_j _j
bgÇkcrpm bcj _eshcpm
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6 gqmkÖrpga_q
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rpgam,
+ Sqc cj nslrm kcbgm bcj pcarÅlesjm gqmkÑ+rpgam &j_ bgqr_lag_ clrpc > w
?' n_p_ jma_jgx_p jmq dmamq
qm`pc cj chc k_wmp,
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rmq A w A' n_p_ s`ga_p cj aclrpm bcj _pam ep_lbc O,
?
k_wmp
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f SH Aclrpm bcj
_pam ep_lbc
+ Bg`shc jmq bmq _pamq ep_l+bcq O r_leclrcq _j pcarÅl+esjm gqmkÑrpgam, Bg`shc jmq bmq
_pamq ncoscàmq p* cknjc_lbm jmq dmamq cl jmq nslrmq @ v A amkm aclrpmq* n_p_ amknjcr_p j_ cjgnqc _npmvgk_b_,
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72 A ? N ≠ R S J M 1 @ M Q OS C H? B M R ~ A L G A M
+++++ PRDBOBK@F>+++++++++++++Mh]jpehh]o
eokiÅpne_]o
J_q nj_lrgjj_q cqncag_jcq amkm cqr_ nj_lrgjj_ gqmkÜrpga_ aml jálc_q _lesj_b_q w cjgnqcq mpgclr_b_q cl bgdcpclrcq nj_lmq gqmkÖrpgamq* d_agjgr_l cj rp_x_bm bc `mqoschmq gqmkÖrpgamq
J_q cjgnqcq qc npmnmpagml_l aml k_pa_q n_p_ osc amglagb_l aml j_q jálc_q aclrp_jcq gqmkÖrpga_q bc jmq _eshcpmq* jm as_j j_q f_ac sl_ fcpp_kgclr_ bc ep_l srgjgb_b cl cj bg`shm gqmkÜrpgam,
R_k`gÖl nscbcl bg`sh_pqc cjgnqcq sq_lbm sl_ nj_lrgjj_ _bcas_b_* qcjcaagml_b_ bc dmpk_ osc amglagb_ aml jmq chcq k_wmp w kclmp,
1,0/
AF?RGL
AB
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1,00 OLP@>P AB QLOKFIIL BK AF?RGLP FPLJ{QOF@LP
J_q cjgnqcq n_pag_jcq ges_jkclrc cqn_ag_b_q aml cj n_qm qgk`âjgam bc sl_ pmqa_ qc srgjgx_l n_p_ pcnpcqclr_p qâjm j_q apcqr_q bc sl_ pmqa_ cl sl bg`shm gqmkÑrpgam &dgesp_ 1,3/', J_q cjgnqcq nscbcl `mqosch_pqc* bg`sh_pqc nmp cj kÉrmbm bc jmq as_rpm aclrpmq* m `gcl cj_`mp_pqc srgjgx_lbm sl_ nj_lrgjj_ bc cjgnqcq,
1,3/ Pmqa_q bc rmplgjjm cl sl bg`shm gqmkÜrpgam,
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1 , 0 2 C Q D C P ? Q CL B G @ S H M Q G Q M K ~ R P G A M Q 73
1,01 >O@LP BK AF?RGLP FPLJ{QOF@LPN_p_ `mqosch_p m bg`sh_p _pamq agpasj_pcq cl dmpk_ gqmkÑrpga_
nscbc sq_pqc j_ amlqrpsaagâl bc cjgnqcq bc as_rpm aclrpmq, J_dgesp_ 1,30_ kscqrp_ j_ amlqrpsaagâl amknjcr_, Lm cq lcacq_pgmbg`sh_p rmb_ j_ amlqrpsaagâl n_p_ jmq _pamq* amkm qc kscqrp_cl j_q dgesp_q 1,30` w a, Kgb_ cj p_bgm O bcqbc j_ cqosgl_ bcamlqrpsaagâl9 bcqnsÉq* cl a_b_ nslrm* bg`shc ncpnclbgasj_pcq
_ j_q jÖlc_q, Qs glrcpqcaagâl cq cj aclrpm bcj _pam, M`qcptc oscj_q bgqr_lag_q O qml ges_jcq cl j_q dgesp_q 1,30`w a* ncpm osc jmqp_bgmq pc_jcq osc qc srgjgx_l qml ks w bgdcpclrcq,
1,30 ?pamq cl bg`shmq gqmkÖrpgamq,
Tgqr_q b_b_q
/, Bg`shc cj gqmkÜrpgam bc sl aápasjm
ep_lbc n_p_jcjm
_ sl_
a_p_
bcj
as`m*
bcqnsÖq bcrcpkglc cj p_bgm bc j_ cqdcp_ _j jma_jgx_p jmq nslrmq qm`pc j_ bg_eml_j sq_lbm j_ kcbgb_ ^ n_p_ cqr_`jcacp jmq cvrpckmq bcj chc k_wmp
0 Cj bgÇkcrpm bcj
_pasjm cl cj bg`shm gqmkÜrpgam cq P
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M O
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8/19/2019 Teoria Isometria y Proyecciones
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N P M W C A A G ± L M P R M E P ™ D G A ? /41
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8/19/2019 Teoria Isometria y Proyecciones
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/42 A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
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N P M W C A A G ± L M P R M E P ™ D G A ? /43
&`'
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/4 4 A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
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N_p_ mpe_lgx_pÖ_q tgqr_q bc sl m`hcrm cl 1B qm`pc sl_ fmh_ bc n_ncj* gk_eglc osc jmq qcgq nj_lmq bc j_ a_h_ bc apgqr_j qc bcqbm`j_l qm`pc sl_ qsncpdgagc nj_l_* amkm qc kscqrp_ cl j_ dgesp_ 3,7, Gk_eglc osc rmbmq jmq nj_lmq* cvacnrm cj nmqrcpgmp* cqrÅl _prgasj_bmq aml cj nj_ lm dpmlr_j, Nmp jm eclcp_j* cj nj_+lm nmq rcpgmp qc amlqgbcp_ _prgasj_bm _j nj_lm j_rcp_j gxosgcpbm, A_b_ nj_lm qc bcqnjgce_ bcqbc cj nj_lm dpmlr_j, J_ pcnpcqclr_+agâl bc j_q jÖlc_q bc _prgasj_agâl cl j_ a_h_ bc apgqr_j qm`pc sl bg`shm qc amlmac amkm iÖkb^p
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8/19/2019 Teoria Isometria y Proyecciones
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/4 6 A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
3,// Tgqr_q aml w qgl jálc_q bc njce_bm,
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J_ jÖlc_ bc njce_bm EFCq clrpc j_q tgqr_q qsncpgmp w dpmlr_j* cq j_ glrcpqca+agâl bc jmq nj_lmq fmpgxmlr_j w dpmlr_j, J_ jÖlc_ bc njce_bm C-M* clrpc j_q tgqr_q dpmlr_j w j_rcp_j* cq j_ glrcpqcaagâl clrpc jmq nj_lmq dpmlr_j w bc ncpdgj,
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N P M W C A A G ± L M P R M E P ™ D G A ? /47
Seop]o ja_ao]ne]oCl j_ dgesp_ 3,/2 qc kscqrp_ osc j_q tgqr_q j_rcp_jcq bcpcaf_ c gxosgcpb_ qml cqclag_jkclrc gkÅeclcq bc cqnchm clrpc qÖ* qâjm osc aml bgdcpclrcq jÖlc_q masjr_q, J_q jÖlc_q masjr_q sq_l sl n_rpâl bc jÖ+lc_q bgqamlrgls_q n_p_ pcnpcqclr_p n_prcq
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nmp jm eclcp_j* qâjm qc bg`sh_ j_ tgqr_ j_ +rcp_j bcpcaf_, Cqrm r_k`gÉl qc _njga_ n_p_ j_q tgqr_q qsncpgmp c gldcpgmp* w n_p_ j_q tgqr_q dpmlr_j w nmqrcpgmp, Cl j_ dgesp_ 3,/3 qc kscqrp_l hsl r_q w mpe_lgx_b_q j_q tgqr_q qsncpgmp* dpmlr_j w j_rcp_j bcpcaf_9 Éqr_q qc bclmkgl_l j_q qobp sfpq^p obdri^+obp* nscqrm osc qml j_q tgqr_q osc qc sq_l aml k_wmp dpcasclag_,
Sl `mqoschm m bg`shm bc`c amlrclcp qâjm j_q tgqr_q lcacq_pg_q n_p_ bcqapg`gp cl dmpk_ aj_p_ w amknjcr_ cj m`hcrm, Cq+r_q tgqr_q kÖlgk_q pcoscpgb_q qc amlmacl amkm j_q sfpq^p kb`bp^of^p, Qcjcaagmlc j_q tgqr_q osc rcle_l cj kclmp låkcpm bc jÖlc_q masjr_q w kscqrpcl jmq amlrmplmq cqclag_jcq m j_q dmpk_q aml k_wmp aj_pg+b_b, Jmq m`hcrmq amknjga_bmq nscbcl pc+oscpgp kÅq bc rpcq tgqr_q m glajsqm tgqr_q cqncag_jcq9 nmp chcknjm* tgqr_q n_pag_jcq,
Ksafmq m`hcrmq qâjm pcosgcpcl bmq tgqr_q n_p_ bcqapg`gp aml aj_pgb_b qs dmp+k_, Qg sl m`hcrm pcosgcpc qâjm bmq tgqr_q
w j_q tgqr_q j_rcp_j gxosgcpb_ w bcpcaf_ npcqclr_l ges_jkclrc `gcl cj m`hcrm* srg+jgac j_ tgqr_ j_rcp_j bcpcaf_, Qg sl m`hcrm pcosgcpc qâjm bmq tgqr_q* w j_q tgqr_q qs+
ncpgmp c gldcpgmp npcqclr_l ges_jkclrc `gcl cj m`hcrm* cjgh_ j_ tgqr_ qsncpgmp, Qg qâjm qml lcacq_pg_q bmq tgqr_q w j_q tgq+r_q qsncpgmp w j_rcp_j bcpcaf_ kscqrp_l ges_jkclrc `gcl cj m`hcrm* cjgh_ j_ amk+
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/5 . A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
G + B 4 3 K 0 . v 0 ,3 . +
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N P M W C A A G ± L M P R M E P ™ D G A ? /5/
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/50 A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
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Bc k_lcp_ qgkgj_p* j_ tgqr_ nmqrcpgmp nscbc amjma_pqc bg+pcar_kclrc qm`pc j_ tgqr_ qsncpgmp m bc`_hm bc j_ tgqr_ gldcpgmp, Cl cqrc a_qm* cj nj_lm nmqrcpgmp qc amlqgbcp_ _prgasj_bm _j nj_lm fmpgxmlr_j m _j nj_lm gldcpgmp w qc egp_ bc kmbm osc amglagb_ aml cj nj_lm dpmlr_j,
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N P M W C A A G ± L M P R M E P ™ D G A ? /51
IÉja]o k_qhp]oSl_ bc j_q tclr_h_q bc j_q tgqr_q mprmeml_jcq qm`pc j_q dmrm+ep_dÖ_q cq osc a_b_ tgqr_ nscbc kmqrp_p rmbm cj m`hcrm bcqbcj_ bgpcaagâl bc tgqs_jgx_agâl, Sl_ dmrmep_dÖ_ kscqrp_ qâjm j_qsncpdgagc tgqg`jc bc sl m`hcrm* ncpm cl sl_ tgqr_ mprmepÅdga_ qcm`qcpt_ _ rp_tÉq bc rmbm cj m`hcrm* amkm qg dscp_ rp_lqn_pclrc,
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/5 2 A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
IÉja]o _ajpn]haoCj n_rpâl bc jÖlc_q aclrp_jcq qc sq_ n_p_8
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3 , 0 N P C A C B C L A G ? BC J ? Q J ≠ L C ? Q /53
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3,17 Qsncpdgagc m`jgas_,
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3 , / 2 @ M P B C Q N ? P ? J C J M Q /57
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3,/2 ?LOABP M>O>IBILPAs_lbm jmq `mpbcq qml n_p_jcjmq clrpc qÖ cl cj m`hcrm* _n_pcacpÅl amkm jÖlc_q n_p_jcj_q cl rmb_q j_q tgqr_q* _ kclmq osc qc _jglccl slm bcrpÅq bcj mrpm, Cqr_ gldmpk_agâl nscbc qcp årgj as_lbm sq+
rcb cqrÉ bgqrpg`swclbm sl bg`shm* qm`pc rmbm qg qc rgclc sl_ qs+ ncpdgagc amknjch_ glajgl_b_ s m`jgas_ aml `mpbcq n_p_jcjmq, Cl j_ dgesp_ 3,22 qc kscqrp_ sl chcknjm bc jÖlc_q n_p_jcj_q cl j_q tgqr_q bc sl bg`shm,
Jmlegrsb tcpb_bcp_
3,20 Npmwcaagmlcq bc sl `mpbc glajgl_bm,
Qcqe_b_
3,21 Npmwcaagmlcq bc sl `mpbc m`jgasm,
p+j† Z
&_' Nj_lmq n_p_jcjmq glrcpqca_bmq nmp mrpm nj_lm
3,22 Jálc_q n_p_jcj_q,
&`' Jálc_q /* 0 w 1* 2 n_p_jcj_q clrpc qá w n_p_jcj_q _j nj_lm fmpgxmlr_j
&a' Jálc_q /* 0 w 1* 2 n_p_jcj_q clrpc qá w n_p_jcj_q _j nj_lm dpmlr_j
n_p_jcj_q clrpc qá w m`jgas_q _ rmbmq jmq nj_lmq
8/19/2019 Teoria Isometria y Proyecciones
32/34
/6 . A ? N ≠ R S J M 3 N P M W C A A G ± L M P R M E P ™ D G A ?
™lesjm _r_k_ãm ,tcpb_bcpm
&_' ™lesjm cl sl nj_lm lmpk_j
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