❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❊st✉❞♦ ❡ ❆♣❧✐❝❛çõ❡s ❞❛s ❋✉♥çõ❡s
❍✐♣❡r❜ó❧✐❝❛s
❏♦♥❛s ❏♦sé ❈r✉③ ❞♦s ❙❛♥t♦s
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❊st✉❞♦ ❡ ❆♣❧✐❝❛çõ❡s ❞❛s ❋✉♥çõ❡s
❍✐♣❡r❜ó❧✐❝❛s
†♣♦r
❏♦♥❛s ❏♦sé ❈r✉③ ❞♦s ❙❛♥t♦s
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ r❡❞❡ ◆❛❝✐✲ ♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❥✉❧❤♦✴✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
S729e Santos, Jonas José Cruz dos.
Estudo e aplicação das funções hiperbólicas / Jonas José Cruz dos Santos.- João Pessoa, 2015.
77f. : il.
Orientador: Manassés Xavier de Souza Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Funções trigonométricas. 3. Funções hiperbólicas. 4. Catenária. 5. Velocidade da onda do mar.
❆❣r❛❞❡❝✐♠❡♥t♦s
◗✉❡r♦ ❛❣r❛❞❡❝❡r ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ s❛ú❞❡ ❡ s❛❜❡❞♦r✐❛ ♣❛r❛ r❡❛❧✐③❛r ❡st❡ ▼❡str❛❞♦✳
❆♦s ♠❡✉s ♣❛✐s✱ ▼❛ ❞❛ ❈♦♥❝❡✐çã♦ ❈r✉③ ❞♦s ❙❛♥t♦s ❡ ❆♥tô♥✐♦ ❊♠í❞✐♦ ❞♦s ❙❛♥t♦s✱
♣♦r s❡♠♣r❡ ♠❡ ❡♥s✐♥❛r ♦ ❝❛♠✐♥❤♦ ❝❡rt♦ ❛ s❡❣✉✐r✳
➚ ♠✐♥❤❛ ❡s♣♦s❛ ❙❛r❛ ❈r✉③✱ ♣♦r ❡①✐st✐r ❡ t❡r ♠❡ ❞❛❞♦ ✉♠❛ ❢❛♠í❧✐❛ tã♦ ❧✐♥❞❛ ❡ ♠❛r❛✈✐❧❤♦s❛✱ ♣♦r s❡♠♣r❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ q✉❛❧q✉❡r s✐t✉❛çã♦✱ ♣♦r ♠❡ ✐♥❝❡♥t✐✈❛r ❞✉r❛♥t❡ t♦❞❛ ♠✐♥❤❛ ❥♦r♥❛❞❛ ❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❝♦♠ ❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦s✳ ❖❜r✐❣❛❞♦ ♣❡❧♦ ❛♠♦r q✉❡ ✈♦❝ê ♠❡ ❞❛r✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ♣❡❧♦ ❡♠♣❡♥❤♦ ❞❡❞✐❝❛❞♦ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❡ à ❝♦♦r❞❡♥❛çã♦ ❞♦ P❘❖❋▼❆❚✱ ❡♠ ❡s♣❡❝✐❛❧ à t✉r♠❛ ❞❡ ✷✵✶✸✳ ➚ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛✱ Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ❡ Pr♦❢✳ ❉r❛✳ ❚❛r❝✐❛♥❛ ▼❛r✐❛ ❙❛♥t♦s ❞❛ ❙✐❧✈❛✱ ♣❡❧❛s s✉❣❡stõ❡s✱ ❛♣♦✐♦s✱ ♦r✐❡♥t❛çõ❡s ❡ ❛✈❛❧✐❛çã♦ ❞♦ tr❛❜❛❧❤♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ t✉r♠❛ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ♠♦♠❡♥t♦s ❞❡s❝♦♥tr❛í❞♦s ❡ t❛♠❜é♠ ✐♠♣♦rt❛♥t❡s ❡♥❝♦♥tr♦s ❞❡ ❡st✉❞♦s✳
❆ t♦❞♦s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ♦ ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳
➚ ❈❆P❊❙ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ✜♥❛♥❝❡✐r♦ ❝♦♠ ❛ ❜♦❧s❛ ❞❡ ♣❡sq✉✐s❛✳
❉❡❞✐❝❛tór✐❛
❆ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r ❛❝r❡❞✐t❛r ♥♦ ♠❡✉ s✉❝❡ss♦✳ ❆♦s ♠❡✉s ❧✐♥❞♦s ❡ ❛♠❛❞♦s ✜❧❤♦s ✭♠✐♥❤❛ ♣r✐♥❝❡s❛ ❆♥❛♥❞❛ ❙♦♣❤✐❛ ❡ ♠❡✉ ❛♠✐❣ã♦ ▼✐❣✉❡❧ ❈r✉③✮ q✉❡ s❡♠♣r❡ s❡rã♦ ❢♦♥t❡s ❞❡ ✐♥s♣✐r❛çã♦ ♣❛r❛ ❡✉ s❡❣✉✐r ❡♠ ❢r❡♥t❡✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ ❛♥❛❧✐s❛♥❞♦ s✉❛s s❡♠❡❧❤❛♥ç❛s ❡ ❞✐❢❡r❡♥ç❛s ❝♦♠ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✐r❝✉❧❛r❡s✳ P❛r❛ t❛♥t♦✱ ✐♥✐❝✐❛♠♦s ❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ❛ tr✐❣♦♥♦♠❡tr✐❛ ❝✐r❝✉❧❛r ❡ ❛ ❤✐♣ér✲ ❜♦❧❡✱ ❞❡s❝r❡✈❡♥❞♦ s❡✉s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s ❡ ♣r♦♣r✐❡❞❛❞❡s✳ P♦st❡r✐♦r♠❡♥t❡✱ r❡❛❧✐✲ ③❛♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ ❛♣r❡s❡♥t❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞♦ s❡♥♦✱ ❝♦ss❡♥♦ ❡ ❞❛s ❞❡♠❛✐s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❡ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❈♦♥❝❧✉í✲ ♠♦s ❝♦♠ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡st❛s ❢✉♥çõ❡s ♥♦ ❝♦t✐❞✐❛♥♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ❋✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ ❈❛t❡♥ár✐❛✱ ❱❡✲ ❧♦❝✐❞❛❞❡ ❞❛ ♦♥❞❛ ❞♦ ♠❛r✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ✐♥t❡♥❞s t♦ s❤♦✇ t❤❡ ❤②♣❡r❜♦❧✐❝ ❢✉♥❝t✐♦♥s✱ ❛♥❛❧②③✐♥❣ t❤❡✐r s✐♠✐❧❛r✐t✐❡s ❛♥❞ ❝♦♥tr❛sts ✇✐t❤ t❤❡ ❝✐r❝✉❧❛r tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳ ❚♦ t❤✐s✱ ✇❡ st❛rt s❤♦✇✐♥❣ ❛ s❤♦rt r❡✈✐❡✇ ❛❜♦✉t t❤❡ ❝✐r❝✉❧❛r tr✐❣♦♥♦♠❡tr② ❛♥❞ ❤②♣❡r❜♦❧❡✱ ❞❡s❝r✐❜✐♥❣ t❤❡✐r ♠❛✐♥ ❡❧❡♠❡♥ts ❛♥❞ ♣r♦♣❡rt✐❡s✳ ❚❤❡♥✱ ✇❡ ♠❛❞❡ ❛ st✉❞② ❛❜♦✉t ❤②♣❡r❜♦❧✐❝ ❢✉♥❝t✐♦♥s✱ ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ s✐♥❡✱ ❝♦s✐♥❡ ❛♥❞ t❤❡ ♦t❤❡r ❤②♣❡r❜♦❧✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ♠❛✐♥ ♣r♦♣❡rt✐❡s✳ ❲❡ ✜♥✐s❤❡❞ t❤✐s ✇♦r❦ ✇✐t❤ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥ ❡✈❡r②❞❛②✳
❑❡②✇♦r❞s✿ ❚r✐❣♦♥♦♠❡tr✐❝ ❋✉♥❝t✐♦♥s✱ ❍②♣❡r❜♦❧✐❝ ❋✉♥❝t✐♦♥s✱ ❈❛t❡♥❛r②✱ ❙❡❛ ❲❛✈❡ ❙♣❡❡❞✳
❙✉♠ár✐♦
✶ ❚r✐❣♦♥♦♠❡tr✐❛ ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✶
✶✳✶ ❆r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ➶♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ▼❡❞✐❞❛s ❞❡ ❆r❝♦s ❡ ➶♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❚r✐❣♦♥♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❋✉♥çõ❡s ❚r✐❣♦♥♦♠étr✐❝❛s ❈✐r❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✺✳✶ ❋✉♥çã♦ ❙❡♥♦ ❡ ❋✉♥çã♦ ❈♦ss❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✺✳✷ ❋✉♥çã♦ ❚❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✻ ❋✉♥çã♦ ❈♦t❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✼ ❋✉♥çã♦ ❙❡❝❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✽ ❋✉♥çã♦ ❈♦ss❡❝❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✾ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❘❛③õ❡s ❚r✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✼
✷✳✶ ❆ ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✶ ❊❧❡♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✷ ❊q✉❛çã♦ ❘❡❞✉③✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✸ ❆ ❍✐♣ér❜♦❧❡ xy=k ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✶✳✹ ❍✐♣ér❜♦❧❡ ❘♦t❛❝✐♦♥❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ➶♥❣✉❧♦ ❍✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✳✶ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❡ ❊①♣♦♥❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✹ ❆❧❣✉♠❛s Pr♦♣r✐❡❞❛❞❡s ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹✳✶ ❙❡♥♦ ❡ ❈♦ss❡♥♦ ❍✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹✳✷ ❋✉♥çã♦ ■♥✈❡rs❛ ❞♦ ❙❡♥♦ ❍✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹✳✸ ❋✉♥çã♦ ■♥✈❡rs❛ ❞♦ ❈♦ss❡♥♦ ❍✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹✳✹ ❚❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹✳✺ ❋✉♥çã♦ ■♥✈❡rs❛ ❞❛ ❚❛♥❣❡♥t❡ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✹✳✻ ❈♦t❛♥❣❡♥t❡ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✹✳✼ ❋✉♥çã♦ ■♥✈❡rs❛ ❞❛ ❈♦t❛♥❣❡♥t❡ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✹✳✽ ❙❡❝❛♥t❡ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✹✳✾ ❈♦ss❡❝❛♥t❡ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✹✳✶✵ ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✺ ❈♦♠♣❛r❛♥❞♦ ❢ór♠✉❧❛s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✸ ❆♣❧✐❝❛çõ❡s ✺✹ ✸✳✶ ❈❛t❡♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✶✳✶ ❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❛ ❝❛t❡♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✶✳✷ ❆♣❧✐❝❛çõ❡s ♥♦ ❝♦t✐❞✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷ ❱❡❧♦❝✐❞❛❞❡ ❞❛ ♦♥❞❛ ❞♦ ♠❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❆r❝♦s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ➶♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ➶♥❣✉❧♦ ❈❡♥tr❛❧ AOB⌢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✹ ➶♥❣✉❧♦α✱ ❙❡t♦r ❈✐r❝✉❧❛r As ❡ ❆r❝♦ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✺ ❈✐❝❧♦ ❚r✐❣♦♥♦♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✻ ➶♥❣✉❧♦ ❝✐r❝✉❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✼ s❡♥(α) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✶✳✽ cos(α) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✾ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✵ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦ss❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✶ ❚❛♥❣❡♥t❡ ♥♦ ❈✐❝❧♦ ❚r✐❣♦♥♦♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✷ ❋✉♥çã♦ ❚❛♥❣❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✸ ❈♦t❛♥❣❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✹ ❋✉♥çã♦ ❈♦t❛♥❣❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✺ ❙❡❝❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✻ ❋✉♥çã♦ ❙❡❝❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✼ ❈♦ss❡❝❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✽ ❋✉♥çã♦ ❈♦ss❡❝❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶ ❊s❜♦ç♦ ❞❛ ❍✐♣ér❜♦❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❍✐♣ér❜♦❧❡ x2
a2 −
y2
b2 = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✸ ❍✐♣ér❜♦❧❡ y2
a2 −
x2
b2 = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✹ ❘❡tâ♥❣✉❧♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺ ❙✐♠❡tr✐❛ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✻ ❍✐♣ér❜♦❧❡ r♦t❛❝✐♦♥❛❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✼ ❙❡t♦r ❍✐♣❡r❜ó❧✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✽ ➶♥❣✉❧♦ ❍✐♣❡r❜ó❧✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✾ ❙❡t♦r ❍✐♣❡r❜ó❧✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✵ Pr❡s❡r✈❛çã♦ ❞❡ ➶♥❣✉❧♦ ❍✐♣❡r❜ó❧✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✶ ❉♦✐s ♣♦♥t♦s ♥❛ ❍✐♣ér❜♦❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✷ ❙❡t♦r ♥❛ ❍✐♣ér❜♦❧❡ r♦t❛❝✐♦♥❛❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✸ ➪r❡❛ P V SA✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✷✳✶✹ P♦♥t♦s ♥❛ ❤✐♣ér❜♦❧❡ ❡♠ r❡❧❛çã♦ ❛♦s ❡✐①♦s ❳ ❡ ❨✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✺ ❉❡✜♥✐♥❞♦ ❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✻ ❈♦♦r❞❡♥❛❞❛s ❞❡ A ♥♦ s✐st❡♠❛ XOY ❡ ♥♦ xOy✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶✳✶ ❋✉♥çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ❘❡❧❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸ ❖♣❡r❛çõ❡s ❝♦♠ ➶♥❣✉❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹ ❆r❝♦ ❉✉♣❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✺ ❆r❝♦ ▼❡t❛❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✻ ❉❡r✐✈❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶ ❘❡❧❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✷ ❋✉♥çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸ ❖♣❡r❛çõ❡s ❝♦♠ â♥❣✉❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✹ ❆r❝♦s ❞✉♣❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✺ ❆r❝♦ ▼❡t❛❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✻ ❉❡r✐✈❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
■♥tr♦❞✉çã♦
❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❡ ✉♠❛ ♣❡sq✉✐s❛ ❜✐❜❧✐♦❣rá✜❝❛ s♦❜r❡ ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
◆♦r♠❛❧♠❡♥t❡ ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s sã♦ ❡st✉❞❛❞❛s ❡♠ ❝✉rs♦s ❞❡ ❝á❧❝✉❧♦ ❞✐❢❡✲ r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧ ❡ ❣❡r❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛❞❛s s❡♠ ♥❡♥❤✉♠❛ r❡❧❛çã♦ ❝♦♠ ❛ ❤✐♣ér❜♦❧❡ ❡ s❡♠ ♥❡♥❤✉♠❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡✳ ❙ã♦ ❞❡✜♥✐❞❛s ❝♦♠ ♦ ✉s♦ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s✱ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ✐❞❡♥t✐❞❛❞❡s✱ s✉❛s ❞❡r✐✈❛❞❛s ❡ ✐♥t❡❣r❛✐s✱ ♦s ❡s❜♦ç♦s ❞♦s ❣rá✜❝♦s ❡ ❛s ❞❡✜♥✐çõ❡s ❞❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ✐♥✈❡rs❛s✳ ❈♦♠ ✐ss♦✱ ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s r❡❧❛❝✐♦♥❛♥❞♦✲❛s à ❤✐♣ér❜♦❧❡ ❡ ❛♣r❡s❡♥t❛r ♣r♦♣r✐❡❞❛❞❡s q✉❡ sã♦ ❛♥á❧♦❣❛s às ❝♦♥❤❡❝✐❞❛s ♣r♦♣r✐❡❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s✳
P❛r❛ ✜♥s ❞✐❞át✐❝♦s✱ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳ ❆ s❡❣✉✐r ❞❡t❛✲ ❧❤❛♠♦s ❝❛❞❛ ✉♠ ❞❡❧❡s✳
◆♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡✈✐sã♦ s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ tr✐❣♦♥♦♠étr✐❝❛ ❡ ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✐r❝✉❧❛r❡s✳ ❘❡❧❡♠❜r❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♣❛rt❡s ❞❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛ ✉♥✐tár✐❛✳ ❈♦♥❝❡✐t✉❛♠♦s ❡ ❛♣r❡s❡♥t❛♠♦s ❛s ❞❡♠♦♥str❛çõ❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦✲ ♥♦♠étr✐❝❛s ❡ ✜♥❛❧✐③❛♠♦s ❝♦♠ ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡st✉❞❛♠♦s ❛ ❤✐♣ér❜♦❧❡ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ➱ ❡①♣♦st❛ ✉♠❛ ❞❡✜♥✐çã♦ ❞❡ ❤✐♣ér❜♦❧❡ ♥♦ ♣❧❛♥♦ ❡ ❛♣r❡s❡♥t❛❞♦s s❡✉s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s✱ ❝♦♠♦✿ ❢♦❝♦s✱ ✈ért✐❝❡s✱ ❡✐①♦ ❢♦❝❛❧✱ ❡✐①♦ ♥ã♦ ❢♦❝❛❧ ❡ ❛ssí♥t♦t❛s✳ ❙ã♦ ❛♣r❡s❡♥t❛❞❛s ❡q✉❛çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❡ ❡s❜♦ç❛❞♦s s❡✉s r❡s♣❡❝t✐✈♦s ❣rá✜❝♦s✳ ❙ã♦ ❡①✐❜✐❞❛s t❛♠❜é♠✱ ❛s ❡q✉❛çõ❡s ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❡♠ ✉♠ s✐st❡♠❛ ✐♥✐❝✐❛❧ ❛ ✉♠ ♥♦✈♦ s✐st❡♠❛✱ q✉❡ ❢♦✐ ❢♦r♠❛❞♦ ❛tr❛✈és ❞❡ ✉♠❛ r♦t❛çã♦ ❞❡ ❡✐①♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳
❖ ❈❛♣ít✉❧♦ ✸ é ❞❡st✐♥❛❞♦ ❛ ♠♦str❛r ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s q✉❡ ♣♦❞❡♠ s❡r ❛♥❛❧✐s❛❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ➱ ❛♣r❡s❡♥t❛❞♦ ✉♠ ♣❡q✉❡♥♦ r❡s✉♠♦ ❞❛ ❤✐stór✐❛ ❞♦ ❡st✉❞♦ ❞❛ ❝❛t❡♥ár✐❛✱ ❝♦♠♦ ❢♦✐ ❞❡s❝♦❜❡rt❛ ❡q✉❛çã♦ ❞❡st❛ ❝✉r✈❛ ❡ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ♣❛r❛ tr❛ç❛r ♦ ❣rá✜❝♦✳ ❖✉tr❛ ❛♣❧✐❝❛çã♦✱ é ♦ ❝á❧❝✉❧♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛s ♦♥❞❛s ❞♦ ♠❛r✳ ❊st❡ ❝á❧❝✉❧♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ❢✉♥çã♦ t❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✳
❈❛♣ít✉❧♦ ✶
❚r✐❣♦♥♦♠❡tr✐❛ ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛
◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ r❡✈✐sã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ tr✐❣♦♥♦♠étr✐❝❛ ❡ ❞❛s ❢✉♥✲ çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✐r❝✉❧❛r❡s✳ ❊st❛ r❡✈✐sã♦ ❢♦✐ ❜❛s❡❛❞❛ ♥❛s ♦❜r❛s ❞❡ ❉♦❧❝❡✭✷✵✵✺✱ ♣✳✶✻✻✲✶✽✷✮ ❬✷❪✱ ❋❛❝❝❤✐♥✐✭✶✾✾✼✱ ♣✳ ✷✾✶✲✸✷✽✮ ❬✹❪✱ ■❡③③✐✭✶✾✾✸✱ ♣✳✸✾✲✼✶✮ ❬✻❪ ❡ ▲✐♠❛✭✷✵✵✸✱ ♣✳✷✶✸✲✷✸✷✮ ❬✽❪✳
✶✳✶ ❆r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛
❈♦♥s✐❞❡r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛✱ ♦✉ s❡❥❛✱ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❧♦❝❛❧✐③❛❞♦ ♥❛ ♦r✐❣❡♠ ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❡ r❛✐♦ ❞❡ ♠❡❞✐❞❛ ✶✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ ❆ ❡ ❇ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s q✉❡ ❞✐✈✐❞❡♠ ❡♠ ❞✉❛s ♣❛rt❡s ❛ ❝✐r✲ ❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛✳ ❈❛❞❛ ✉♠❛ ❞❡ss❛s ♣❛rt❡s ❝❤❛♠❛✲s❡ ❛r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❖s ♣♦♥t♦s ❆ ❡ ❇ sã♦ ❛s ❡①tr❡♠✐❞❛❞❡s ❞♦s ❛r❝♦s q✉❡ ❞❡t❡r♠✐♥❛♠✳
❋✐❣✉r❛ ✶✳✶✿ ❆r❝♦s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳
✶✳✷✳ ➶◆●❯▲❖
❖❜s❡r✈❛çã♦ ✶✳✷✳ ◗✉❛♥❞♦ ❛s ❡①tr❡♠✐❞❛❞❡s ❆ ❡ ❇ ❝♦✐♥❝✐❞❡♠✱ t❡♠♦s ✉♠ ❛r❝♦ ❞❡ ✉♠❛ ✈♦❧t❛ ♦✉ ✉♠ ❛r❝♦ ♥✉❧♦✳
❖❜s❡r✈❛çã♦ ✶✳✸✳ ◗✉❛♥❞♦ ❛s ❡①tr❡♠✐❞❛❞❡s ❆ ❡ ❇ ❞♦ ❛r❝♦ sã♦ t❛♠❜é♠ ❛s ❡①tr❡♠✐✲ ❞❛❞❡s ❞❡ ✉♠ ❞✐â♠❡tr♦✱ ♦ ❛r❝♦ AB⌢ é ❝❤❛♠❛❞♦ ❛r❝♦ ❞❡ ♠❡✐❛✲✈♦❧t❛✳
✶✳✷ ➶♥❣✉❧♦
❈♦♥s✐❞❡r❡ ❞✉❛s s❡♠✐rr❡t❛s−→OA❡−−→OB ❞❡ ♠❡s♠❛ ♦r✐❣❡♠O✱ ♥ã♦ ❝♦✐♥❝✐❞❡♥t❡s ❡ ♥ã♦
♦♣♦st❛s✳
❋✐❣✉r❛ ✶✳✷✿ ➶♥❣✉❧♦✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❆s ❞✉❛s r❡❣✐õ❡s✱ ■ ❡ ■■✱ ❞❡t❡r♠✐♥❛❞❛s ♥♦ ♣❧❛♥♦✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✶✳✷✱ sã♦ ❝❤❛♠❛❞❛s ❞❡ â♥❣✉❧♦s✳ ❖ ♣♦♥t♦ ❖ é ♦ ✈ért✐❝❡ ❞♦s â♥❣✉❧♦s ❡ ❛s s❡♠✐rr❡t❛s −→
OA ❡ −−→OB sã♦ ♦s ❧❛❞♦s ❞♦s â♥❣✉❧♦s✳
❖❜s❡r✈❛çã♦ ✶✳✺✳ ❈❛s♦ ❛s s❡♠✐rr❡t❛s−→OA ❡ −−→OB ❝♦✐♥❝✐❞❛♠✱ ❞✐③❡♠♦s✱ ♣♦r ❡①t❡♥sã♦✱
q✉❡ ❡❧❛s ❞❡t❡r♠✐♥❛♠ ✉♠ â♥❣✉❧♦ ♥✉❧♦ ✭q✉❡ é ❛ ♣ró♣r✐❛ s❡♠✐rr❡t❛✮ ♦✉ ✉♠ â♥❣✉❧♦ ❞❡ ✉♠❛ ✈♦❧t❛ ✭q✉❡ é ♦ ♣ró♣r✐♦ ♣❧❛♥♦✮✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ❛s s❡♠✐rr❡t❛s −→OA ❡ −−→OB
sã♦ ♦♣♦st❛s✱ ❞✐③❡♠♦s✱ t❛♠❜é♠ ♣♦r ❡①t❡♥sã♦✱ q✉❡ ❞❡t❡r♠✐♥❛♠ ✉♠ â♥❣✉❧♦ r❛s♦ ✭q✉❡ é ♦ ♣ró♣r✐♦ s❡♠✐♣❧❛♥♦✮✳
❉❡✜♥✐çã♦ ✶✳✻✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❖ ❡ r❛✐♦ ❘ ❡✱ ♥❡❧❛✱ ❞♦✐s ♣♦♥t♦s✱ ❆ ❡ ❇✳ ❖ â♥❣✉❧♦ ❞❡ ❧❛❞♦s −→OA ❡ −−→OB✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✶✳✸✱ é ❝❤❛♠❛❞♦ ❞❡
â♥❣✉❧♦ ❝❡♥tr❛❧ ✭s❡✉ ✈ért✐❝❡ é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✮✳
❖❜s❡r✈❛çã♦ ✶✳✼✳ ❆ ❝❛❞❛ ❛r❝♦AB⌢ ❝♦rr❡s♣♦♥❞❡ ✉♠ â♥❣✉❧♦ ❝❡♥tr❛❧AOB⌢ ❡ ✈✐❝❡✲✈❡rs❛✳
❙❡ ❛❞♦t❛r♠♦s ✉♠❛ ♠❡s♠❛ ✉♥✐❞❛❞❡ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ♠❡❞✐❞❛s ❞❡ ❛r❝♦s ❡ â♥❣✉❧♦s✱ ❛ ♠❡❞✐❞❛ ❞♦ ❛r❝♦ s❡rá ✐❣✉❛❧ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡✳
✶✳✸✳ ▼❊❉■❉❆❙ ❉❊ ❆❘❈❖❙ ❊ ➶◆●❯▲❖❙
❋✐❣✉r❛ ✶✳✸✿ ➶♥❣✉❧♦ ❈❡♥tr❛❧AOB⌢ ✳
✶✳✸ ▼❡❞✐❞❛s ❞❡ ❆r❝♦s ❡ ➶♥❣✉❧♦s
P❛r❛ ♠❡❞✐r ✉♠❛ ❣r❛♥❞❡③❛✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♦✉tr❛ ❣r❛♥❞❡③❛ ❞❡ ♠❡s♠♦ ♣❛❞rã♦ ♣❛r❛ ❛ q✉❛❧ ❝♦♥✈❡♥❝✐♦♥❛♠♦s ♠❡❞✐❞❛ ✉♥✐tár✐❛ ❡✱ ❞❡♣♦✐s ❞✐ss♦✱ ♣r♦❝✉r❛✲ s❡ ❛ r❛③ã♦ ❡♥tr❡ ❡ss❛s ❞✉❛s ❣r❛♥❞❡③❛s ❞❡ ♠❡s♠♦ ♣❛❞rã♦✳
❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ❝♦♥s✐❞❡r❛r❡♠♦s ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ R ❡ ❝❡♥tr♦ O✳
❉❡✜♥✐çã♦ ✶✳✽✳ ❆ ♠❡❞✐❞❛ ❞❡ ✉♠ ❛r❝♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛AB⌢ é ♦ ♥ú♠❡r♦ r❡❛❧ α✱ ♥ã♦
♥❡❣❛t✐✈♦✱ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ r❛③ã♦ ❡♥tr❡ ♦ ❛r❝♦ AB⌢ ❛ s❡r ♠❡❞✐❞♦ ❡ ✉♠ ❛r❝♦ ✉♥✐tár✐♦
✉ ❞❛ ♠❡s♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛
α=
⌢
AB u .
❉❡✜♥✐çã♦ ✶✳✾✳ ❉✐✈✐❞✐♥❞♦ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ ✸✻✵ ❛r❝♦s ❝♦♥❣r✉❡♥t❡s✱ ❝❛❞❛ ✉♠ ❞❡s✲ s❡s ❛r❝♦s s❡rá ❝❤❛♠❛❞♦ ❛r❝♦ ❞❡ ✉♠ ❣r❛✉ ✭✶♦✮✳ ❖ â♥❣✉❧♦ ❝❡♥tr❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡
s❡rá ❞❡ ✶♦ t❛♠❜é♠✳
❉❡✜♥✐çã♦ ✶✳✶✵✳ ❖ ❛r❝♦ ♠❡❞❡ ✉♠ r❛❞✐❛♥♦ ✭✶ r❛❞✮ s❡ ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦ s ❢♦r
✐❣✉❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ R ❞♦ r❛✐♦✳ ❖ â♥❣✉❧♦ ❝❡♥tr❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡ s❡rá✱ t❛♠❜é♠✱ ❞❡
✉♠ r❛❞✐❛♥♦ ✭✶ r❛❞✮✳
❊♥tã♦ ♣❛r❛ s❛❜❡r ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ❛r❝♦ ✭♦✉ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡✮ ❡♠ r❛❞✐❛♥♦s✱ ❜❛st❛ ❝❛❧❝✉❧❛r q✉❛♥t❛s ✈❡③❡s ♦ r❛✐♦ ❞❡ ♠❡❞✐❞❛R ✧❝❛❜❡✧ ♥❡ss❡ ❛r❝♦ ❞❡
❝♦♠♣r✐♠❡♥t♦s✳
P♦rt❛♥t♦✱
α= s
R.
❉❛í✱
s=α·R.
❉❡✜♥✐çã♦ ✶✳✶✶✳ ❈❤❛♠❛♠♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ✐♥❞✐❝❛❞♦ ♣♦r C✱ ♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ 2πR✳ ❆❧❣❡❜r✐❝❛♠❡♥t❡
C = 2πR. ✭✶✳✶✮
✶✳✸✳ ▼❊❉■❉❆❙ ❉❊ ❆❘❈❖❙ ❊ ➶◆●❯▲❖❙
❖❜s❡r✈❛çã♦ ✶✳✶✷✳ ◗✉❛♥❞♦ ♦ ❛r❝♦sé ♦ ❛r❝♦ ❞❡ ✉♠❛ ✈♦❧t❛✱ ❡♥tã♦sé ♦ ❝♦♠♣r✐♠❡♥t♦ C ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❊ ♣♦r ❞❡✜♥✐çã♦✱ t❡♠✲s❡
α= s
R =
2πR R .
❊♥tã♦✱
α= 2πr❛❞.
▲♦❣♦✱ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ✉♠ ❛r❝♦ ❞❡2πr❛❞ ❡ ♦ â♥❣✉❧♦ ❝❡♥tr❛❧ ❞❡ ✉♠❛ ✈♦❧t❛ ♠❡❞❡
2πr❛❞.
❆ s❡❣✉✐r s❡rá ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ α ❡ ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ❞❡t❡r♠✐♥❛❞♦ ♣♦r ❡st❡ â♥❣✉❧♦✳
Pr♦♣♦s✐çã♦ ✶✳✶✸✳ ❈♦♥s✐❞❡r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ R✳ ❉❡♥♦t❛♥❞♦ α ♦ â♥❣✉❧♦
❝❡♥tr❛❧ ❡♠ r❛❞✐❛♥♦s ❡As❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ❞❡t❡r♠✐♥❛❞♦ ♣♦r ❡st❡ â♥❣✉❧♦ ❝❡♥tr❛❧✱
♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳
❋✐❣✉r❛ ✶✳✹✿ ➶♥❣✉❧♦ α✱ ❙❡t♦r ❈✐r❝✉❧❛r As ❡ ❆r❝♦ α✳
❊♥tã♦✱ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ α é
α= 2·As
R2 .
❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ ♦ â♥❣✉❧♦ ❝❡♥tr❛❧ ❞❡ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ R ✐❣✉❛❧ ❛ 2πr❛❞ ❡
s✉❛ ár❡❛ ❞❡t❡r♠✐♥❛❞❛ ♣♦rπR2✳ ❚❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦rçã♦ ❡♥tr❡ â♥❣✉❧♦ ❝❡♥tr❛❧ ❡
ár❡❛
2πrad→πR2 αrad →As.
❘❡❛❧✐③❛♥❞♦ ❛s ❞❡✈✐❞❛s ♦♣❡r❛çõ❡s ♦❜t❡♠♦s
α= 2·As
R2 .
✶✳✹✳ ❈■❘❈❯◆❋❊❘✃◆❈■❆ ❚❘■●❖◆❖▼➱❚❘■❈❆
❖❜s❡r✈❛çã♦ ✶✳✶✹✳ ◆❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛ t❡♠♦s
α= 2·As.
❈♦♥❝❧✉✐✲s❡ ❡♥tã♦ q✉❡ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ éα r❛❞✐❛♥♦s q✉❛♥❞♦ ❛ ár❡❛ ❞♦ s❡✉
s❡t♦r ❝✐r❝✉❧❛r é α
2 ✉♥✐❞❛❞❡s ❞❡ ár❡❛✳
❖❜s❡r✈❛çã♦ ✶✳✶✺✳ ❙❡ ♦ â♥❣✉❧♦ ❝❡♥tr❛❧ α✱ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛✱ ❡st✐✈❡r ❡①✲
♣r❡ss♦ ❡♠ ❣r❛✉s✱ t❡♠♦s
α = 360·As
πR2 ,
❈♦♠♦ R= 1✱
α = 360·As
π .
❖❜s❡r✈❛çã♦ ✶✳✶✻✳ P♦❞❡✲s❡ r❡❧❛❝✐♦♥❛r ❛ ♠❡❞✐❞❛ r❛❞✐❛♥♦ ❝♦♠ ❛ ♠❡❞✐❞❛ ❣r❛✉✳ ❉❡ ❢❛t♦✱ ✉♠❛ ✈♦❧t❛ ❝♦♠♣❧❡t❛ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦ss✉✐ ✸✻✵♦ ♦✉✱ ❡♠ r❛❞✐❛♥♦s✱ 2πr❛❞✱
❡♥tã♦ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡
180♦ =πr❛❞, ♦✉ s❡❥❛, 90♦= π 2 r❛❞.
❯♠ â♥❣✉❧♦ ♣♦❞❡ t❡r ♦ ✈❛❧♦r r❡❛❧ q✉❡ ❞❡s❡❥❛r✳ ◆♦ ❡♥t❛♥t♦✱ ❛ s❡♠✐rr❡t❛ q✉❡ ❞á ♦ â♥❣✉❧♦ ✭❝♦♠ ♦✉tr❛ s❡♠✐rr❡t❛ ✜①❛✱ ❞❡ r❡❢❡rê♥❝✐❛✮ ❝♦♠♣❧❡t❛ ✉♠❛ ✈♦❧t❛ ❛♣ós 360♦✱
❞✉❛s ✈♦❧t❛s ❛♣ós 720♦✱ ❡t❝✳✱ ♦✉ ✉♠❛ ✈♦❧t❛ ♥♦ s❡♥t✐❞♦ ❝♦♥trár✐♦✱ ❡ ♥❡ss❡ ❝❛s♦ ❞✐③✲s❡
q✉❡ ❞❡s❝r❡✈❡✉ ✉♠ â♥❣✉❧♦ ❞❡ −360♦✳ ❊♥tã♦ ✉♠ â♥❣✉❧♦ φ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛✿
φ =α+k·360◦,
❡♠ q✉❡ k é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❡ α é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ φ ♣♦r 360 ❡ t❛♠❜é♠ ♦
â♥❣✉❧♦ ❞❡s❡❥❛❞♦✳
◆♦ ❡♥t❛♥t♦✱ é ♥❡❝❡ssár✐♦ ❞❡✜♥✐r ✉♥✐✈♦❝❛♠❡♥t❡ ❛ ❛♣❧✐❝❛çã♦ q✉❡ ❞á ♦ â♥❣✉❧♦ ❞❡✲ ✜♥✐❞♦ ♣♦r ❞✉❛s r❡t❛s q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠✳ P♦rt❛♥t♦✱ ❡ ♣❛r❛ ❡ss❡ ❡❢❡✐t♦✱ ♠❡❞❡♠✲s❡ ♦s â♥❣✉❧♦s ♥✉♠ ❞♦♠í♥✐♦ q✉❡ ✈❛✐ ❞❡ 0◦ ❛ 360◦ ✭♦✉✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡✱ ❞❡ 0 ❛ 2π
r❛❞✐❛♥♦s✮✳
✶✳✹ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❚r✐❣♦♥♦♠étr✐❝❛
❆♥t❡s ❞❡ ❞❡✜♥✐r ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ tr✐❣♦♥♦♠étr✐❝❛✱ s❡❣✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛ ♦r✐❡♥t❛❞❛✳
❉❡✜♥✐çã♦ ✶✳✶✼✳ ❯♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❖ ❡ r❛✐♦ ❘ é ❞✐t❛ ♦r✐❡♥t❛❞❛ q✉❛♥❞♦ ♣♦ss✉✐ ♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦ ❞♦ ♣❡r❝✉rs♦ ❝♦♠♦ ♣♦s✐t✐✈♦✳
❉❡✜♥✐çã♦ ✶✳✶✽✳ ❈✐r❝✉♥❢❡rê♥❝✐❛ tr✐❣♦♥♦♠étr✐❝❛ ✭♦✉ ❝✐❝❧♦ tr✐❣♦♥♦♠étr✐❝♦✮ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦r✐❡♥t❛❞❛ ❞❡ r❛✐♦ ✉♥✐tár✐♦ ♣❛r❛ ❛ q✉❛❧ s❡ ❛❞♦t❛ ❝♦♠♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ♦ ❛♥t✐✲❤♦rár✐♦ ❡ s❡ ❡s❝♦❧❤❡ ✉♠ ♣♦♥t♦ A = (1,0) ❝♦♠♦ ♦r✐❣❡♠ ❞♦s
❛r❝♦s✳
✶✳✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙ ❈■❘❈❯▲❆❘❊❙
❋✐❣✉r❛ ✶✳✺✿ ❈✐❝❧♦ ❚r✐❣♦♥♦♠étr✐❝♦✳
❖❜s❡r✈❛çã♦ ✶✳✶✾✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ✜❣✉r❛ ❛❝✐♠❛ ♦ ❝✐❝❧♦ tr✐❣♦♥♦♠étr✐❝♦ ❡stá ❞✐✲ ✈✐❞✐❞♦ ❡♠ q✉❛tr♦ q✉❛❞r❛♥t❡s✱ ♥✉♠❡r❛❞♦s ❞❡ ✶ ❛ ✹✱ ❛ ♣❛rt✐r ❞♦ s❡❣♠❡♥t♦ OA✱ ♥♦
s❡♥t✐❞♦ ♣♦s✐t✐✈♦✳
✶✳✺ ❋✉♥çõ❡s ❚r✐❣♦♥♦♠étr✐❝❛s ❈✐r❝✉❧❛r❡s
◆❡st❛ s❡çã♦ ❡st✉❞❛r❡♠♦s ❛s ❢✉♥çõ❡s s❡♥♦✱ ❝♦ss❡♥♦✱ t❛♥❣❡♥t❡✱ ❝♦t❛♥❣❡♥t❡✱ s❡❝❛♥t❡ ❡ ❝♦ss❡❝❛♥t❡ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛✳ ❊st❛s ❢✉♥çõ❡s t❡♠ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ q✉❡ é ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡✱ ♦ q✉❡ ♣❡r♠✐t❡ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ♥❛ ♠❛t❡♠át✐❝❛ ❡ ❡♠ ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♥❛ ❋ís✐❝❛✳
❘❡❧❡♠❜r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣❡r✐ó❞✐❝❛✳
❉❡✜♥✐çã♦ ✶✳✷✵✳ ❯♠❛ ❢✉♥çã♦ f : R −→ R ❝❤❛♠❛✲s❡ ♣❡r✐ó❞✐❝❛ q✉❛♥❞♦ ❡①✐st❡ ✉♠
♥ú♠❡r♦ ♥ã♦ ♥✉❧♦ T ∈R t❛❧ q✉❡ f(x+T) =f(x) ♣❛r❛ t♦❞♦ x∈R✳
❖❜s❡r✈❛çã♦ ✶✳✷✶✳ ❙❡ ✉♠❛ ❢✉♥çã♦ é ♣❡r✐ó❞✐❝❛✱ ❡♥tã♦ f(x+kT) = f(x) ♣❛r❛ t♦❞♦
x ∈ R ❡ t♦❞♦ k ∈ Z✳ ❖ ♠❡♥♦r ♥ú♠❡r♦ T > 0 t❛❧ q✉❡ f(x+T) = f(x) ♣❛r❛ t♦❞♦
x∈R ❝❤❛♠❛✲s❡ ♦ ♣❡rí♦❞♦ ❞❛ ❢✉♥çã♦ f✳
✶✳✺✳✶ ❋✉♥çã♦ ❙❡♥♦ ❡ ❋✉♥çã♦ ❈♦ss❡♥♦
❈♦♥s✐❞❡r❡ ♦ s❡t♦r ❝✐r❝✉❧❛r AOP⌢ ♣❛r❛ ❞❡✜♥✐r ❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦✱ r❡s♣❡❝t✐✲
✈❛♠❡♥t❡✱ ♠❛r❝❛✲s❡ ❛ ♣r♦❥❡çã♦ ❞♦ ♣♦♥t♦P ♥♦ ❡✐①♦Oy ❡ ♥♦ ❡✐①♦Ox✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛
✶✳✻✳
✶✳✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙ ❈■❘❈❯▲❆❘❊❙
❋✐❣✉r❛ ✶✳✻✿ ➶♥❣✉❧♦ ❝✐r❝✉❧❛r✳
P♦rt❛♥t♦✱
s❡♥(α) = OH ❡ cos(α) = OM.
❖ s❡♥♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ é ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ ❝♦♥t✐❞♦ ♥♦ ❡✐①♦Oy✱ ♦♥❞❡ ✉♠❛ ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s é ♦ ♣♦♥t♦ (0,0) ❡ ❛ ♦✉tr❛ é ❛ ♣r♦❥❡çã♦ ❞♦
♣♦♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❞❡t❡r♠✐♥❛ ♦ â♥❣✉❧♦ ❝❡♥tr❛❧✱ s♦❜r❡ ♦ ♣ró♣r✐♦ ❡✐①♦ Oy✱
❋✐❣✉r❛ ✶✳✼✳
❋✐❣✉r❛ ✶✳✼✿ s❡♥(α)
◆♦ ❝❛s♦ ❞♦ ❝♦ss❡♥♦ ❞❡ ❝❡rt♦ â♥❣✉❧♦ ❝❡♥tr❛❧✱ ❡❧❡ é ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ ❝♦♥t✐❞♦ ♥♦ ❡✐①♦Ox ✱ ♦♥❞❡ ✉♠❛ ❡①tr❡♠✐❞❛❞❡ t❛♠❜é♠ é ♦ ♣♦♥t♦(0,0)❡ ❛ ♦✉tr❛ é ❛ ♣r♦❥❡çã♦
❞♦ ♠❡s♠♦ ♣♦♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ s♦❜r❡ ♦ ♣ró♣r✐♦ ❡✐①♦Ox✱ ❋✐❣✉r❛ ✶✳✽✳
✶✳✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙ ❈■❘❈❯▲❆❘❊❙
❋✐❣✉r❛ ✶✳✽✿ cos(α)
❏á ❛ ❋✐❣✉r❛ ✶✳✾ ✐❧✉str❛ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡♥♦✳ ❆tr❛✈és ❞❡st❡ ❣rá✜❝♦ ✜❝❛ ❢á❝✐❧ ✈❡r q✉❡ ❛ ❢✉♥çã♦ s❡♥♦ é ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦2π✱ ♦ ❞♦♠í♥✐♦ é t♦❞♦ ♦ ❝♦♥❥✉♥t♦
❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ ❛ ✐♠❛❣❡♠ é ♦ ✐♥t❡r✈❛❧♦[−1,1]✳ ❊st❡ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡♥♦
♣♦❞❡ s❡r ❝❤❛♠❛❞♦ ❞❡ s❡♥♦✐❞❡✳
❋✐❣✉r❛ ✶✳✾✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡♥♦✳
❆ ❋✐❣✉r❛ ✶✳✶✵ r❡♣r❡s❡♥t❛ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦ss❡♥♦✳ ❚❛♠❜é♠ ❛♣ós ✉♠❛ ❛♥á❧✐s❡✱ ♦❜s❡r✈❛✲s❡ q✉❡ ❛ ❢✉♥çã♦ ❝♦ss❡♥♦ ♣♦ss✉✐ ♦ ♠❡s♠♦ ♣❡rí♦❞♦ ✭2π✮✱ ❞♦♠í♥✐♦ ✭R✮ ❡
✐♠❛❣❡♠ ✭[−1,1]✮ ❞❛ ❢✉♥çã♦ s❡♥♦✳ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦ss❡♥♦ t❛♠❜é♠ ♣♦❞❡ r❡❝❡❜❡r
✉♠ ♥♦♠❡ ❡s♣❡❝í✜❝♦✱ ❝♦ss❡♥♦✐❞❡✳
✶✳✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙ ❈■❘❈❯▲❆❘❊❙
❋✐❣✉r❛ ✶✳✶✵✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦ss❡♥♦✳
✶✳✺✳✷ ❋✉♥çã♦ ❚❛♥❣❡♥t❡
❉❡✜♥✐❞❛s ❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦✱ ♣♦❞❡✲s❡ ❞❡✜♥✐r ♦✉tr❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐✲ ❝❛s✳
❉❡✜♥✐çã♦ ✶✳✷✷✳ ❉❡✜♥❡✲s❡ ❛ ❢✉♥çã♦ t❛♥❣❡♥t❡✱ ♣♦r
tan(α) = sen(α) cos(α),
s❡♠♣r❡ q✉❡ cos(α)6= 0✳
❆ ❢✉♥çã♦ t❛♥❣❡♥t❡ é ❞❡✜♥✐❞❛ ❝♦♠♦ q✉♦❝✐❡♥t❡ ❡♥tr❡ ❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦✱ ❛ss✐♠ s❡✉ ❞♦♠í♥✐♦ ✜❝❛ r❡str✐t♦ ❛♦s ♥ú♠❡r♦s r❡❛✐s ♣❛r❛ ♦s q✉❛✐s ♦ ❞❡♥♦♠✐♥❛❞♦r é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❈♦♠♦ cos(α) = 0 ♣❛r❛ t♦❞♦ α = π
2 +kπ✱ ❝♦♠ k ✐♥t❡✐r♦✱ ✐ss♦
✐♠♣❧✐❝❛ q✉❡ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ t❛♥❣❡♥t❡ é
D={α∈R|α6= π
2 +kπ,comk∈Z}.
❯t✐❧✐③❛♥❞♦ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ❡♥tr❡ ♦s tr✐â♥❣✉❧♦s OAT ❡OM P✱ ♣♦❞❡♠♦s
❞❡✜♥✐r✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❛ t❛♥❣❡♥t❡ ❞❡ α ❝♦♠♦ s❡♥❞♦ ❛ ♠❡❞✐❞❛ ❛❧❣é❜r✐❝❛ ❞♦
s❡❣♠❡♥t♦AT✱ ♣♦✐sAT = sen(cos(αα))✳ ❈♦♠A= (1,0) ❡T ❛ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛←→OP ❝♦♠ ❛
r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦Ox ♥♦ ♣♦♥t♦A✭❡✐①♦ ❞❛s t❛♥❣❡♥t❡s✮✳ ❖♥❞❡ O é ❛ ♦r✐❣❡♠
❞♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ❡P é ❛ ♦✉tr❛ ❡①tr❡♠✐❞❛❞❡ ❞♦ ❛r❝♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ â♥❣✉❧♦
❝❡♥tr❛❧ α✱ ❡M ❛ ♣r♦❥❡çã♦ ❞♦ ♣♦♥t♦P s♦❜r❡ ♦ ❡✐①♦ Ox✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✶✳✶✶✳
❊♥tã♦ t❡♠♦s✱
tan(α) = sen(α)
cos(α) =AT .
✶✳✻✳ ❋❯◆➬➹❖ ❈❖❚❆◆●❊◆❚❊
❋✐❣✉r❛ ✶✳✶✶✿ ❚❛♥❣❡♥t❡ ♥♦ ❈✐❝❧♦ ❚r✐❣♦♥♦♠étr✐❝♦✳
❯s❛♥❞♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ t❛♥❣❡♥t❡ ♣♦ss✉✐ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❋✐❣✉r❛ ✶✳✶✷✿ ❋✉♥çã♦ ❚❛♥❣❡♥t❡✳
❖❜s❡r✈❛çã♦ ✶✳✷✸✳ ◆♦t❡ q✉❡ ❡♥q✉❛♥t♦ ♦ ❞♦♠í♥✐♦ é ♦ ❝♦♥❥✉♥t♦ D = {α ∈ R|α 6=
π
2 +kπ,comk ∈Z}✱ ❛ ✐♠❛❣❡♠ é t♦❞♦ ♦ ❝♦♥❥✉♥t♦R✳
✶✳✻ ❋✉♥çã♦ ❈♦t❛♥❣❡♥t❡
❆❣♦r❛ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡✳
❉❡✜♥✐çã♦ ✶✳✷✹✳ ❆ ❢✉♥çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❝♦t❛♥❣❡♥t❡ é ❞❡✜♥✐❞❛ ♣♦r
cot(α) = cos(α) sen(α),
s❡♠♣r❡ q✉❡ sen(α)6= 0✳
✶✳✻✳ ❋❯◆➬➹❖ ❈❖❚❆◆●❊◆❚❊
❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ❢✉♥çã♦ t❛♥❣❡♥t❡✱ ♦ ❞♦♠í♥✐♦ ❞❛ ❝♦t❛♥❣❡♥t❡ ❡st❛rá ❞❡✜♥✐❞♦ ♣❛r❛ sen(α) 6= 0✳ ❈♦♠♦ sen(α) = 0 ♣❛r❛ t♦❞♦ α = kπ✱ ❝♦♠ k ✐♥t❡✐r♦✱ ❡♥tã♦ ♦
❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡ é
D={α ∈R|α6=kπ,comk∈Z}.
P♦❞❡♠♦s ✉t✐❧✐③❛r ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ❡♥tr❡ ♦s tr✐â♥❣✉❧♦s OHT ❡ OM P
♣❛r❛ ♠♦str❛r q✉❡ HT = cos(sen(αα))✳ ❈♦♠ O s❡♥❞♦ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ❡ P
é ❛ ♦✉tr❛ ❡①tr❡♠✐❞❛❞❡ ❞♦ ❛r❝♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ â♥❣✉❧♦ ❝❡♥tr❛❧ α✱ H = (0,1)✱ T
❛ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛ ←→OP ❝♦♠ ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ Oy ♥♦ ♣♦♥t♦ H ✭❡✐①♦
❞❛s ❝♦t❛♥❣❡♥t❡s✮ ❡ M ❛ ♣r♦❥❡çã♦ ❞♦ ♣♦♥t♦ P s♦❜r❡ ♦ ❡✐①♦ Ox✳ ❱❡❥❛ ❋✐❣✉r❛ ✶✳✶✸✳
❊♥tã♦ ❞❡✜♥✐✲s❡✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❝♦t❛♥❣❡♥t❡ ❞❡ α ❝♦♠♦ s❡♥❞♦ ❛ ♠❡❞✐❞❛ ❛❧❣é❜r✐❝❛
❞♦ s❡❣♠❡♥t♦ HT✳
❋✐❣✉r❛ ✶✳✶✸✿ ❈♦t❛♥❣❡♥t❡✳
❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡ ❡stá r❡♣r❡s❡♥t❛❞♦ ❛❜❛✐①♦✳
❋✐❣✉r❛ ✶✳✶✹✿ ❋✉♥çã♦ ❈♦t❛♥❣❡♥t❡✳
❖❜s❡r✈❛çã♦ ✶✳✷✺✳ ❆ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡ é ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
✶✳✼✳ ❋❯◆➬➹❖ ❙❊❈❆◆❚❊
✶✳✼ ❋✉♥çã♦ ❙❡❝❛♥t❡
❉❡✜♥❡✲s❡ ❛❣♦r❛ ❛ ❢✉♥çã♦ s❡❝❛♥t❡✳
❉❡✜♥✐çã♦ ✶✳✷✻✳ ❆ ❢✉♥çã♦ s❡❝❛♥t❡ é ❞❡✜♥✐❞❛ ♣♦r
sec(α) = 1 cos(α),
❝♦♠ cos(α)6= 0✳
❈♦♥❢♦r♠❡ ❞❡✜♥✐çã♦✱ ❛ ❢✉♥çã♦ s❡❝❛♥t❡ só ❡st❛rá ❞❡✜♥✐❞❛ q✉❛♥❞♦ cos(α) 6= 0✳
❊♥tã♦ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ s❡❝❛♥t❡ é
D={α∈R|α6= π
2 +kπ,comk∈Z}.
❆♣❧✐❝❛♥❞♦ ♥♦s tr✐â♥❣✉❧♦s OSP ❡ OM P ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ♣♦❞❡♠♦s
♣❡r❝❡❜❡r q✉❡ ♦ s❡❣♠❡♥t♦ OS= 1
cos(α)✳ ❈♦♠O = (0,0)✱P é ❛ ♦✉tr❛ ❡①tr❡♠✐❞❛❞❡ ❞♦
â♥❣✉❧♦ ❝❡♥tr❛❧α✱M ❛ ♣r♦❥❡çã♦ ❞♦ ♣♦♥t♦P s♦❜r❡ ♦ ❡✐①♦Ox❡S ❛ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛
t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥♦ ♣♦♥t♦ P ❡ ♦ ❡✐①♦ Ox✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✶✳✶✺✳
❊♥tã♦✿
sec(α) = OS = 1 cos(α).
❋✐❣✉r❛ ✶✳✶✺✿ ❙❡❝❛♥t❡✳
❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡❝❛♥t❡ ❡stá r❡♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✶✻✳
✶✳✽✳ ❋❯◆➬➹❖ ❈❖❙❙❊❈❆◆❚❊
❋✐❣✉r❛ ✶✳✶✻✿ ❋✉♥çã♦ ❙❡❝❛♥t❡✳
❖❜s❡r✈❛çã♦ ✶✳✷✼✳ ❆ ❢✉♥çã♦ s❡❝❛♥t❡ ♣♦ss✉✐ ❝♦♠♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠
Im(sec(α)) = (−∞,−1]∪[1,+∞).
✶✳✽ ❋✉♥çã♦ ❈♦ss❡❝❛♥t❡
❉❡✜♥✐r❡♠♦s ❛❣♦r❛ ❛ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡✳
❉❡✜♥✐çã♦ ✶✳✷✽✳ ❆ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡ é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
csc(α) = 1 sen(α),
❝♦♠ sen(α)6= 0✳
❱✐❞❡ ❞❡✜♥✐çã♦✱ ❛ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡ s♦♠❡♥t❡ ❡st❛rá ❞❡✜♥✐❞❛ q✉❛♥❞♦ sen(α)6= 0✳
▲♦❣♦ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡ ❞❡α é ♦ ❝♦♥❥✉♥t♦
D={α ∈R|α6=kπ,comk∈Z}.
◆❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ tr✐❣♦♥♦♠étr✐❝❛✱ ✉t✐❧✐③❛♥❞♦ ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ❡♥tr❡
OM P ❡ ORP t❡♠✲s❡ q✉❡ OR = 1
sen(α)✳ ❈♦♠ O = (0,0) ❡ R ❛ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛
t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥♦ ♣♦♥t♦P ❝♦♠ ♦ ❡✐①♦Oy✳ ❖♥❞❡ P é ♦✉tr❛ ❡①tr❡♠✐❞❛❞❡
❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧α ❡M é ❛ ♣r♦❥❡çã♦ ❞♦ ♣♦♥t♦P s♦❜r❡ ♦ ❡✐①♦Ox✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛
✶✳✶✼✳
❉❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ à ❢✉♥çã♦ s❡❝❛♥t❡✱ ♣♦❞❡✲s❡ ❡s❜♦ç❛r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦s✲ s❡❝❛♥t❡✱ ❋✐❣✉r❛ ✶✳✶✽✳
❖❜s❡r✈❛çã♦ ✶✳✷✾✳ ◆♦t❡ q✉❡ ❛ ✐♠❛❣❡♠ é ♦ ❝♦♥❥✉♥t♦
Im(csc(α)) = (−∞,−1]∪[1,+∞).
✶✳✾✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆❙ ❘❆❩Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙
❋✐❣✉r❛ ✶✳✶✼✿ ❈♦ss❡❝❛♥t❡✳
❋✐❣✉r❛ ✶✳✶✽✿ ❋✉♥çã♦ ❈♦ss❡❝❛♥t❡✳
✶✳✾ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❘❛③õ❡s ❚r✐❣♦♥♦♠étr✐❝❛s
❆ ♣❛rt✐r ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ■■ sã♦ ❡st✉❞❛❞❛s ❛s r❛③õ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s✱ q✉❡ ♣♦r s✉❛ ✈❡③ ❛✉①✐❧✐❛♠ s✐❣♥✐✜❝❛t✐✲ ✈❛♠❡♥t❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s tr✐❣♦♥♦♠étr✐❝♦s✳ P♦rt❛♥t♦✱ ♥❡st❛ s❡çã♦ ✐r❡♠♦s ❢❛③❡r ✉♠ ❜r❡✈❡ r❡s✉♠♦ ❞❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s r❛③õ❡s tr✐❣♦♥♦♠étr✐❝❛s✳ ❯♠❛ ♣r♦✈❛ ❞❡t❛❧❤❛❞❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✾❪ ❡ ❬✶✷❪✳
❆ ♣r✐♠❡✐r❛ t❛❜❡❧❛ r❡❢❡r❡✲s❡ ❛ ❞❡✜♥✐çã♦ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❞❡r✐✈❛❞❛s ❞❛s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦✳
✶✳✾✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆❙ ❘❆❩Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙
❚❛❜❡❧❛ ✶✳✶✿ ❋✉♥çõ❡s✳
tan(x) = sen(cos(xx)) cot(x) = sen(cos(xx)) csc(x) = 1
sen(x) sec(x) = cos(1x)
❊st❛ s❡❣✉♥❞❛ t❛❜❡❧❛ ❛♣r❡s❡♥t❛ ❛s ♣r✐♥❝✐♣❛✐s r❡❧❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s tr✐❣♦♥♦♠étr✐✲ ❝❛s✳
❚❛❜❡❧❛ ✶✳✷✿ ❘❡❧❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s✳
cos2(x) + sen2(x) = 1 1 + tan2(x) = sec2(x) cot2(x) + 1 = csc2(x)
❊st❛ ♣ró①✐♠❛ t❛❜❡❧❛ ✐❧✉str❛ ♦ s❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ❞❛ s♦♠❛ ❡ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞♦✐s â♥❣✉❧♦s✳
❚❛❜❡❧❛ ✶✳✸✿ ❖♣❡r❛çõ❡s ❝♦♠ ➶♥❣✉❧♦s✳
sen(β+γ) = sen(β) cos(γ)−sen(γ) cos(β) cos(β+γ) = cos(β) cos(γ)−sen(β)sen(γ)
tan(β+γ) = 1tan(−tan(β)+tan(β) tan(γγ))
sen(β−γ) = sen(β) cos(γ) + sen(γ) cos(β) cos(β−γ) = cos(β) cos(γ) + sen(β)sen(γ)
tan(β−γ) = 1+tan(tan(β)β−) tan(tan(γγ)).
❆ q✉❛rt❛ t❛❜❡❧❛ ♥♦s ♠♦str❛ ♦ s❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ❞❡ ❛r❝♦s ❞✉♣❧♦s✳
✶✳✾✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆❙ ❘❆❩Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙
❚❛❜❡❧❛ ✶✳✹✿ ❆r❝♦ ❉✉♣❧♦✳
sen(2γ) = 2sen(γ) cos(γ) cos(2γ) = cos2(γ)−sen2(γ)
tan(2γ) = 12 tan(−tan2γ) (γ)
❊st❛ t❛❜❡❧❛ ❛♣r❡s❡♥t❛ ♦ s❡♥♦✱ ❝♦ss❡♥♦ ❡ t❛♥❣❡♥t❡ ❞♦ ❛r❝♦ ♠❡t❛❞❡✳
❚❛❜❡❧❛ ✶✳✺✿ ❆r❝♦ ▼❡t❛❞❡✳
sen θ
2
=±
q
1−cos(θ) 2
cos θ
2
=±
q
1+cos(θ) 2 tan θ2
=±q11+cos(−cos(θθ))
P❛r❛ ❡st❛ ú❧t✐♠❛ t❛❜❡❧❛ ❛♣r❡s❡♥t❛♠♦s ❛s ❞❡r✐✈❛❞❛s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✳
❚❛❜❡❧❛ ✶✳✻✿ ❉❡r✐✈❛❞❛s✳
d
dx(sen(x)) = cos(x) d
dx(cos(x)) =−sen(x) d
dx(tan(x)) = sec
2(x)
d
dx(cot(x)) =−csc
2(x)
d
dx(sec(x)) = sec(x) tan(x) d
dx(csc(x)) =−csc(x) cot(x)
❈❛♣ít✉❧♦ ✷
❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s
◆❡st❡ ❝❛♣ít✉❧♦ r❡✈✐s❛r❡♠♦s ❛ ❤✐♣ér❜♦❧❡✱ ❢♦❝❛♥❞♦ ♥❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ ❝✐t❛♥❞♦ ❛s s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ s✉❛s ✐♥✈❡rs❛s✱ ❛s ❞❡r✐✈❛❞❛s✱ ❛❧é♠ ❞✐ss♦✱ ♥♦ ✜♥❛❧ ❢❛r❡✲ ♠♦s ❛❧❣✉♠❛s ❝♦♠♣❛r❛çõ❡s ❡♥tr❡ ❛s ❢✉♥çõ❡s ❝✐r❝✉❧❛r❡s ❡ ❤✐♣❡r❜ó❧✐❝❛s✳ ❊st❡ ❝❛♣ít✉❧♦ ❢♦✐ ❜❛s❡❛❞♦ ♥❛s ♦❜r❛s ❞❡ ❆❧❤❛❞❛s✭✷✵✶✸✮ ❬✶❪✱ ❲✐♥t❡r❧❡✭✷✵✵✵✱ ♣✳✶✾✸✲✷✵✹✮ ❬✶✼❪✳
✷✳✶ ❆ ❍✐♣ér❜♦❧❡
❙❡❥❛θ ✉♠ ♣❧❛♥♦ q✉❛❧q✉❡r✱ ❡♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❤✐♣ér❜♦❧❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳
❉❡✜♥✐çã♦ ✷✳✶✳ ❈❤❛♠❛♠♦s ❞❡ ❍✐♣ér❜♦❧❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ θ ❝✉❥❛
❛ ❞✐❢❡r❡♥ç❛ ❞❛s ❞✐stâ♥❝✐❛s✱ ❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s ❞❡ θ é ❝♦♥st❛♥t❡✳
❆♥❛❧✐t✐❝❛♠❡♥t❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❤✐♣ér❜♦❧❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❈♦♥s✐❞❡r❛♥❞♦ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s F1 ❡ F2✱ ♣❡rt❡♥❝❡♥t❡s ❛ θ✱ 2c ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡ss❡s ♣♦♥t♦s ❡
✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦a❞❡ ♠♦❞♦ q✉❡2a <2c✱ ♦♥❞❡2aé ❛ ❝♦♥st❛♥t❡ ❞❛ ❉❡✜♥✐çã♦
✷✳✶✱ t❡♠♦s
h={P ∈θ/|d(P, F1)−d(P, F2)|= 2a},
❡♠ q✉❡ d(P, F1) ❡ d(P, F2) ❞❡♥♦t❛♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s P ❡ F1✱ P ❡ F2✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳
P❛r❛ ♣♦ss✐❜✐❧✐t❛r ✉♠ tr❛ç❛❞♦ ♠❡❧❤♦r ❞❛ ❤✐♣ér❜♦❧❡✱ t❡❝❡r♠♦s ❝♦♥s✐❞❡r❛çõ❡s ❛ r❡s✲ ♣❡✐t♦ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✱ ❢❛r❡♠♦s ❛ ❝♦♥str✉çã♦ ❞❛ ❋✐❣✉r❛ ✷✳✶ ❛ s❡❣✉✐r ❡①♣❧❛♥❛❞❛✳
❈♦♥s✐❞❡r❡ ♥♦ ♣❧❛♥♦ θ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r F1 ❡ F2 ❝♦♠ d(F1, F2) = 2c✳ ❉❡♥♦✲
t❛♥❞♦ C ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ F1F2✱ tr❛❝❡♠♦s ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦
C ❡ r❛✐♦ c✳
❉❛❞♦ ✉♠ ✈❛❧♦r ❛r❜✐trár✐♦ a < c✱ s♦❜r❡ F1F2 ♠❛rq✉❡ ❛ ♣❛rt✐r ❞❡ C ♦s ♣♦♥t♦s
A1 ❡ A2 t❛✐s q✉❡ d(C, A1) =d(C, A2) = a✳ P♦r ❡st❡s ♣♦♥t♦s tr❛❝❡♠♦s ❞✉❛s ❝♦r❞❛s
♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦ ❞✐â♠❡tr♦ F1F2✳ ❆s ❡①tr❡♠✐❞❛❞❡ ❞❡st❛s ❝♦r❞❛s sã♦ ♦s ✈ért✐❝❡s ❞❡
✉♠ r❡tâ♥❣✉❧♦ ▼◆P◗✳ ❚r❛❝❡♠♦s ❛s r❡t❛s r ❡ s q✉❡ ❝♦♥tê♠ ❛s ❞✐❛❣♦♥❛✐s ❞♦ r❡❢❡r✐❞♦ r❡tâ♥❣✉❧♦ ❡✱ ♣♦r ✜♠✱ ❛ ❤✐♣ér❜♦❧❡ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✷✳✶ ✭❝✉r✈❛sγ ❡ δ✮✳
✷✳✶✳ ❆ ❍■P➱❘❇❖▲❊
❋✐❣✉r❛ ✷✳✶✿ ❊s❜♦ç♦ ❞❛ ❍✐♣ér❜♦❧❡✳
❈♦♠ ❜❛s❡ ♥❡st❛ ✜❣✉r❛ t❡♠♦s ♦s ❡❧❡♠❡♥t♦s ❞❛ ❤✐♣ér❜♦❧❡✳
✷✳✶✳✶ ❊❧❡♠❡♥t♦s
• ❋♦❝♦s✿ sã♦ ♦s ♣♦♥t♦s F1 ❡F2❀
• ❉✐stâ♥❝✐❛ ❢♦❝❛❧✿ é ❛ ❞✐stâ♥❝✐❛ 2c❡♥tr❡ ♦s ❢♦❝♦s❀
• ❈❡♥tr♦✿ é ♦ ♣♦♥t♦ ♠é❞✐♦ ❈ ❞♦ s❡❣♠❡♥t♦ F1F2❀
• ❱ért✐❝❡s✿ sã♦ ♦s ♣♦♥t♦s A1 ❡ A2❀
• ❊✐①♦ r❡❛❧✿ é ♦ s❡❣♠❡♥t♦ A1A2 ❞❡ ❝♦♠♣r✐♠❡♥t♦ 2a❀
• ❊✐①♦ ✐♠❛❣✐♥ár✐♦✿ é ♦ s❡❣♠❡♥t♦B1B2 ❞❡ ❝♦♠♣r✐♠❡♥t♦2b✱ ❝♦♠B1B2 ⊥A1A2
❡♠ ❈ ❀
• ❆ssí♥t♦t❛s✿ sã♦ ❛s r❡t❛s r ❡ s❀
• ❘❛♠♦s ❞❛ ❤✐♣ér❜♦❧❡✿ γ ❡ δ✳
❖❜s❡r✈❛çã♦ ✷✳✷✳ ◆♦t❡ q✉❡ ♦s ♣♦♥t♦s A1 ❡ A2 sã♦ ♣♦♥t♦s ❞❛ ❤✐♣ér❜♦❧❡ ♣♦rq✉❡ s❛t✐s✲
❢❛③❡♠ ❛ ❞❡✜♥✐çã♦ ✷✳✶✳ ◆❛ ✈❡r❞❛❞❡✱ ♣❛r❛A1✱ t❡♠✲s❡
d(A1, F1) = c−a ❡ d(A1, F2) = a+c
❡
|d(A1, F1)−d(A1, F2)|=| −2a|= 2a.
❖❜s❡r✈❛çã♦ ✷✳✸✳ ❖ r❡tâ♥❣✉❧♦M N P Qt❡♠ ❞✐♠❡♥sõ❡s 2a ❡2b✱ ♦♥❞❡at❡♠ ❛ ♠❡❞✐❞❛
❞♦ s❡♠✐✲❡✐①♦ r❡❛❧ ❡ b ❛ ♠❡❞✐❞❛ ❞♦ s❡♠✐✲❡✐①♦ ✐♠❛❣✐♥ár✐♦✳ ❉♦ tr✐â♥❣✉❧♦ CA2B1✱
♦❜t❡♠♦s ❛ r❡❧❛çã♦
c2 =a2+b2. ✭✷✳✶✮
✷✳✶✳ ❆ ❍■P➱❘❇❖▲❊
❆s ❛ssí♥t♦t❛s sã♦ r❡t❛s ❞❛s q✉❛✐s ❛ ❤✐♣ér❜♦❧❡ s❡ ❛♣r♦①✐♠❛ ❝❛❞❛ ✈❡③ ♠❛✐s à ♠❡❞✐❞❛ q✉❡ ♦s ♣♦♥t♦s s❡ ❛❢❛st❛♠ ❞♦s ✈ért✐❝❡s✳ ❆ t❡♥❞ê♥❝✐❛ ❞❛ ❤✐♣ér❜♦❧❡ é t❛♥❣❡♥❝✐❛r s✉❛s ❛ssí♥t♦t❛s ♥♦ ✐♥✜♥✐t♦✳ ❊st❛ ♣❛rt✐❝✉❧❛r✐❞❛❞❡ ❝♦♥st✐t✉✐ ✉♠ ❡①❝❡❧❡♥t❡ ❣✉✐❛ ♣❛r❛ tr❛ç❛r ♦ ❡s❜♦ç♦ ❞♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❤✐♣ér❜♦❧❡✳
❈♦♠ ♦ q✉❡ ❢♦✐ ✈✐st♦ ♥❛ ❝♦♥str✉çã♦ ❞❛ ❤✐♣ér❜♦❧❡✱ ❡st❛ ✜❝❛ ❞❡t❡r♠✐♥❛❞❛ q✉❛♥❞♦ s❡ ❝♦♥❤❡❝❡ ♦ ❝❡♥tr♦ ❈ ❡ ♦s ✈❛❧♦r❡s ❛ ❡ ❜ ✭♦✉ ❛ ❡ ❝ ♦✉ ❜ ❡ ❝✮✳ ❉❡ ❢❛t♦✱ ❛ ♣❛rt✐r ❞❡st❡s ❡❧❡♠❡♥t♦s ❝♦♥stró✐✲s❡ ♦ r❡tâ♥❣✉❧♦ ▼◆P◗ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛s ❛ssí♥t♦t❛sr ❡ s✱
❡ ❞❛í✱ ♦s ❞♦✐s r❛♠♦s ❞❛ ❤✐♣ér❜♦❧❡✳
❖ â♥❣✉❧♦ β ❛ss✐♥❛❧❛❞♦ ♥❛ ❋✐❣✉r❛ ✷✳✶ é ❝❤❛♠❛❞♦ ❛❜❡rt✉r❛ ❞❛ ❤✐♣ér❜♦❧❡✳
❈❤❛♠❛✲s❡ ❡①❝❡♥tr✐❝✐❞❛❞❡ ❞❛ ❤✐♣ér❜♦❧❡ ♦ ♥ú♠❡r♦
e= c
a >1.
❖❜s❡r✈❛çã♦ ✷✳✹✳ ◗✉❛♥t♦ ♠❛✐♦r ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡✱ ♠❛✐♦r s❡rá ❛ ❛❜❡rt✉r❛✱ ♦✉ s❡❥❛✱ ♠❛✐s ✧❛❜❡rt♦s✧ ❡st❛rã♦ ♦s r❛♠♦s ❞❛ ❤✐♣ér❜♦❧❡✳ ◗✉❛♥❞♦a=b✱ ♦ r❡tâ♥❣✉❧♦ ▼◆P◗ s❡
tr❛♥s❢♦r♠❛ ♥✉♠ q✉❛❞r❛❞♦ ❡ ❛s ❛ssí♥t♦t❛s s❡rã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳ ❆ ❤✐♣ér❜♦❧❡ ♥❡st❡ ❝❛s♦ é ❞❡♥♦♠✐♥❛❞❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛✳
✷✳✶✳✷ ❊q✉❛çã♦ ❘❡❞✉③✐❞❛
❆♣ós t❡r♠♦s ❞❡✜♥✐❞♦ ❛ ❤✐♣ér❜♦❧❡ ❡ ❝✐t❛❞♦ ♦s s❡✉s ❡❧❡♠❡♥t♦s✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❣♦r❛ s✉❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛ q✉❛♥❞♦ ♦s ❢♦❝♦s sã♦ ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✳
❈♦♥s✐❞❡r❡♠♦s ❛ ❤✐♣ér❜♦❧❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ ❡✐①♦ r❡❛❧ s♦❜r❡ ♦ ❡✐①♦Ox✳ ❙❡♥❞♦ P(x, y) ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ ❤✐♣ér❜♦❧❡ ✭❋✐❣✉r❛ ✷✳✷✮ ❞❡ ❢♦❝♦sF1(−c,0)❡ F2(c,0)✳
❋✐❣✉r❛ ✷✳✷✿ ❍✐♣ér❜♦❧❡ x2
a2 −
y2
b2 = 1✳
P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✶✱ t❡♠♦s
|d(P, F1)−d(P, F2)|= 2a,
♦✉✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s
|p(x+c)2+ (y−0)2−p(x−c)2+ (y−0)2|= 2a.
✷✳✶✳ ❆ ❍■P➱❘❇❖▲❊
❉❛í✱
p
(x+c)2+ (y−0)2−p(x−c)2+ (y−0)2 = 2a.
❙♦♠❛♥❞♦ p
(x−c)2+ (y−0)2 ❛♦s ❞♦✐s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠✲s❡
p
(x+c)2+ (y−0)2 =p
(x−c)2+ (y−0)2+ 2a.
❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱
(x+c)2+y2 = (x−c)2+y2+ 4ap(x−c)2+y2+ 4a2.
■s♦❧❛♥❞♦ ♦ t❡r♠♦ ❝♦♠ r❛❞✐❝❛❧ ♦❜tê♠✲s❡✱
4ap(x−c)2+y2 = (x+c)2+y2
−(x−c)2−y2−4a2 = 4cx−4a2.
▲♦❣♦✱
ap(x−c)2+y2 =cx−a2.
❊❧❡✈❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛♦ q✉❛❞r❛❞♦✱
a2(x−c)2+a2y2 =c2x2−2a2cx+a4.
❊♥tã♦✱
a2x2−2a2cx+a2c2+a2y2 =c2x2−2a2cx+a4.
❈❛♥❝❡❧❛♥❞♦ ♦s t❡r♠♦s ✐❣✉❛✐s✱
a2x2+a2c2+a2y2 =c2x2+a4.
❆ss✐♠✱
c2x2 −a2x2−a2y2 =a2c2−a4.
❈♦❧♦❝❛♥❞♦ ♦ ❢❛t♦r ❝♦♠✉♠ ❡♠ ❡✈✐❞ê♥❝✐❛ t❡♠ ❝♦♠♦ r❡s✉❧t❛❞♦✱
(c2−a2)x2−a2y2 =a2(c2−a2).
❖❜s❡r✈❡ q✉❡ ❡♠ ✭✷✳✶✮ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r b2 =c2 −a2✱ ❛ss✐♠✱
b2x2−a2y2 =a2b2.
❈♦♠ ✐ss♦ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❞❛ ❤✐♣ér❜♦❧❡ é
x2
a2 −
y2
b2 = 1. ✭✷✳✷✮
◆♦ ❝❛s♦ ❞❡ ❢♦❝♦s s♦❜r❡ ♦ ❡✐①♦ Oy✱ ♦❜s❡r✈❛♥❞♦ ❛ ❋✐❣✉r❛ ✷✳✸ ❡ ❝♦♠ ♣r♦❝❡❞✐♠❡♥t♦
❛♥á❧♦❣♦✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛
y2
a2 −
x2
b2 = 1. ✭✷✳✸✮
