ARBELOS

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈
▼❡str❛❞♦ ♣r♦❢✐ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ♠❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚

❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦

❋❧❛✈✐♦ ❋❡r♥❛♥❞♦ ❞❛ ❙✐❧✈❛

❆r❜❡❧♦s

❙❛♥t♦ ❆♥❞ré ✲ ❙P
✷✵✶✹✳

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈
❈❡♥tr♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❈♦♠♣✉t❛çã♦ ❡ ❈♦❣♥✐çã♦

❆r❜❡❧♦s

❋❧❛✈✐♦ ❋❡r♥❛♥❞♦ ❞❛ ❙✐❧✈❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦
❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❙❛♥t♦ ❆♥❞ré ✲ ❙P
❆❣♦st♦ ❞❡ ✷✵✶✹✳

❆r❜❡❧♦s

❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à r❡❞❛çã♦
✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡✈✐❞❛♠❡♥t❡ ❝♦rr✐✲
❣✐❞❛ ❡ ❞❡❢❡♥❞✐❞❛ ♣♦r ❋❧❛✈✐♦ ❋❡r♥❛♥❞♦
❞❛ ❙✐❧✈❛ ❡ ❛♣r♦✈❛❞❛ ♣❡❧❛ ❝♦♠✐ssã♦
❥✉❧❣❛❞♦r❛✳
❙❛♥t♦ ❆♥❞ré✱ ✷✻ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✶✹✳

Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛
❖r✐❡♥t❛❞♦r

❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
✶✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ✭❖r✐❡♥t❛❞♦r✮ ✲ ❯❋❆❇❈
✷✳ Pr♦❢✳ ❉r✳ ❙✐♥✉❡ ❉❛②❛♥ ❇❛r❜❡r♦ ▲✉❞♦✈✐❝✐ ✲ ❯❋❆❇❈
✸✳ Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ▲②♠❜❡r♦♣♦✉❧♦s ✲ ❯❙P

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛
❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛
❯❋❆❇❈✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥✲
çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ à ♠✐♥❤❛ ❡s♣♦s❛✱ ♠❡✉s ♣❛✐s✱ ❡ ❛♠✐❣♦s❀ ❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ ♠❡
❛♣♦✐❛r❛♠ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t✉❞♦✳
❆♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✱ à ❈❆P❊❙ ♣❡❧♦
❛✉①í❧✐♦ ❝♦♥❝❡❞✐❞♦✱ à ❯❋❆❇❈ ❡ s❡✉s ♣r♦❢❡ss♦r❡s✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár❝✐♦
❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❛♦ ❛❝❡✐t❛r ♦ ♣❡❞✐❞♦ ❞❡ s❡r ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r ❛❝r❡❞✐t❛r
❡♠ ♠❡✉ ♣♦tê♥❝✐❛❧✱ ♣♦r s❡r ❝♦♠♣❡t❡♥t❡ ❡ ❡①❝❡❧❡♥t❡ ❡♠ s✉❛ ♣r♦✜ssã♦✱ ❛♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛
❞❡ ♠❡str❛❞♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ à ♠✐♥❤❛ ❡s♣♦s❛ ❘❛q✉❡❧ q✉❡ q✉❡ ♠❡ ✐❝❡♥t✐✈♦✉ ❞✉r❛♥t❡ t♦❞♦
♦ ❝✉rs♦✱ ❛❜r✐♥❞♦ ♠ã♦ ❞❡ ♣❛ss❡✐♦s ❡ ❧❛s❡r✱ ♥ã♦ ♠❡❞✐♥❞♦ ❡s❢♦rç♦ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r
❛q✉✐ ❡ ♠❡✉ ❛♠✐❣♦ ▲✉❝✐❛♥♦ q✉❡ t❛♥t♦ ♠❡ ❛❥✉❞♦✉ ❝♦♠ ♦s ❡st✉❞♦s✱ ✈✐❛❥❡♥s ❡ s✉❛s ♣❛❧❛✈r❛s
❞❡ ✐♥❝❡♥t✐✈♦✳
❆ t♦❞♦s ✈♦❝ês✱ s✐♥❝❡r❛ ❣r❛t✐❞ã♦✳

✈✐✐
❘❡s✉♠♦

■♥s♣✐r❛❞♦ ♥♦ ❛rt✐❣♦ ❞❡ ❍❛r♦❧❞ P✳ ❇♦❛s ❬✷❪✱ ♥❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♦s ❆r❜❡❧♦s ❡ s✉❛s
♣r♦♣r✐❡❞❛❞❡s✳ ❆♥❛❧✐s❛♠♦s ❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❡ ❛♣❧✐❝❛♠♦s ❡ss❛ té❝♥✐❝❛ ♥❛
❝♦♥str✉çã♦ ❞❛ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s ❡ ❞♦ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳

P❛❧❛✈r❛s✲❈❤❛✈❡

❆r❜❡❧♦s✱ ❆r❜❡❧♦s ●ê♠❡♦s✱ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s✱ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✱ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛

✈✐✐✐

❆❜str❛❝t
❇❛s❡❞ ♦♥ t❤❡ ✇♦r❦ ♦❢ ❍❛r♦❧❞ P✳ ❇♦❛s ❬✷❪✱ ✐♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ ❛r❜❡❧♦s ❛♥❞ t❤❡✐r
♣r♦♣❡rt✐❡s✳ ✇❡ ❛♥❛❧②s❡ t❤❡ ✐♥✈❡rs✐♦♥ ❛❜♦✉t ❛ ❝✐r❝❧❡ ❛♥❞ ❛♣♣❧② t❤✐s t❡❝❤♥✐q✉❡ t♦ t❤❡ ❝♦♥s✲
tr✉❝t✐♦♥ ♦❢ P❛♣♣✉s ❈❤❛✐♥ ❛♥❞ ❇❛♥❦♦✛✬s ❝✐r❝❧❡✳

❑❡②✇♦r❞s
❆r❜❡❧♦s✱ ❚✇✐♥ ❛r❜❡❧♦s✱ P❛♣♣✉s ❈❤❛✐♥✱ ❇❛♥❦♦✛✬s ❝✐r❝❧❡✱ ✐♥✈❡rs✐✈❡ ❣❡♦♠❡tr②✳

❙✉♠ár✐♦
✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

✶✺

✷ ●❡♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛

✷✼

✷✳✶ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✷ ■♥✈❡rsã♦ ❞❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✸ ■♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❞❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸ ❆r❜❡❧♦s

✸✾

✸✳✶ ❆r❜❡❧♦s ❣ê♠❡♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸✳✷ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✸✳✸ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✹ ❆♣❧✐❝❛çõ❡s

✺✼

✹✳✶ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✐①



❙❯▼➪❘■❖

▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶

❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦

O

❡ r❛③ã♦

k = 2.

✶✳✷

❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦

O

❡ r❛③ã♦

k = −0, 6. ✳

✶✳✸

❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ❡①t❡r♥❛s ❞❡

Γ



Ω✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✹

❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ✐♥t❡r♥❛s ❞❡

Γ



Ω✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✺

❈ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✻

❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ r❡t❛ ❡♠ r❡t❛ ❞❡ r❛③ã♦ ❦❂✶✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✶✳✼

❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ ❝ír❝✉❧♦ ❡♠ ❝ír❝✉❧♦ ❞❡ r❛③ã♦ ❦❂✶✳✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✶✳✽

❍♦♠♦t❡t✐❛ ❞❡ r❛③ã♦ ❦❂✶✳✾

✶✾

✶✳✾

P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦

P

❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦

Γ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✶✳✶✵ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦

P

✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦

Γ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✶✳✶✸ ❈ír❝✉❧♦

Γ

♦rt♦❣♦♥❛❧ ❛♦ ❝ír❝✉❧♦

Ω✳ ✳



✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✹ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✺ ❚r✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦
✶✳✶✻ ❊❧✐♣s❡ ❞❡ ❢♦❝♦s
✶✳✶✼ ❚r✐â♥❣✉❧♦ ❆❇❈

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✶ ❙❡❣♠❡♥t♦ P❚ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦
✶✳✶✷ ❘❡t❛ ❡ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺
✶✻

✷✷
✷✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

F1



F2 ✳

✷✳✶

■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷

O

é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✷✳✸

O

♥ã♦ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

✷✳✹

❖ ♣♦♥t♦

P

é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦

Γ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

✷✳✺

❖ ♣♦♥t♦

P

é ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦

Γ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✵

①✐

✷✼

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

✶✷
✷✳✻
✷✳✼
✷✳✽
✷✳✾
✷✳✶✵
✷✳✶✶
✷✳✶✷
✷✳✶✸
✷✳✶✹
✷✳✶✺
✷✳✶✻
✷✳✶✼
✷✳✶✽

❖ ♣♦♥t♦ P ♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
■♥✈❡rsã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳ ✳ ✳ ✳
❆ r❡t❛ s é ❡①t❡r♥❛ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❆ r❡t❛ s é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❆ r❡t❛ s é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳
❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳
❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳
❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳
❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳
❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
■♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳
■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳































































































































✸✵
✸✶
✸✷
✸✷
✸✷
✸✸
✸✹
✸✹
✸✺
✸✺
✸✺
✸✻
✸✼

✸✳✶
✸✳✷
✸✳✸
✸✳✹
✸✳✺
✸✳✻
✸✳✼
✸✳✽
✸✳✾
✸✳✶✵
✸✳✶✶
✸✳✶✷
✸✳✶✸
✸✳✶✹

❆r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❆r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈♦♦r❞❡♥❛❞❛s ❞♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s
❈❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❖s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈♦♥str✉çã♦ ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ✉s❛♥❞♦ ✐♥✈❡rsã♦✳ ✳ ✳ ✳ ✳ ✳ ✳
❉✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ❛té ❛ r❡t❛ s✉♣♦rt❡✳
❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

























































































































✸✾
✹✵
✹✹
✹✹
✹✼
✹✼
✹✽
✹✽
✺✵
✺✵
✺✶
✺✷
✺✸
✺✹

✹✳✶
✹✳✷
✹✳✸
✹✳✹
✹✳✺

❱✐s✉❛❧✐③❛çã♦ ❞❛ ♣r✐♠❡✐r❛ ❝♦♥str✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳
❱✐s✉❛❧✐③❛çã♦ ❞❛ s❡❣✉♥❞❛ ❝♦♥str✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳
❱✐s✉❛❧✐③❛çã♦ ❞❡ ❝♦♠♦ ✜❝❛r✐❛ ❞❡♣♦✐s ❞❡ r❡s♦❧✈✐❞♦✳
■♥✈❡rt❡♥❞♦ r ❡♠ r❡❧❛çã♦ ❛ C ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❙♦❧✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

















































✺✽
✻✵
✻✶
✻✷
✻✷

























































■♥tr♦❞✉çã♦
❖s ❛r❜❡❧♦s ❢♦r❛♠ ❡st✉❞❛❞♦s ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ❆rq✉✐♠❡❞❡s ❡♠ s❡✉ ▲✐✈r♦ ❞❡ ▲❡♠❛s✱
P❛♣♣✉s t❛♠❜é♠ ❡st✉❞♦✉ ♦s ❛r❜❡❧♦s ♥♦ ❧✐✈r♦ ■❱ ❞❡ s✉❛ ❝♦❧❡çã♦✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ✹ ❝❛♣ít✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✶ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s
r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s ❝♦♠♦✿ ❤♦♠♦t❡t✐❛✱ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦✱ r❡t❛s ❡ ❝ír❝✉❧♦s ♦rt♦✲
❣♦♥❛✐s✱ ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦✱
❡❧✐♣s❡ ❡ ❛ ❢ór♠✉❧❛ ❞❡ ❍❡r♦♥✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ❛❜♦r❞❛r❡♠♦s ❛ ❣❡♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛✳ ❙❡rá ❞❡✜♥✐❞♦ ♦ q✉❡ é ✐♥✈❡rsã♦ ❡ ♦
q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ ✐♥✈❡rt❡♠♦s ✉♠❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❡ ✉♠ ❝ír❝✉❧♦ ❡♠ r❡❧❛çã♦
❛ ♦✉tr♦✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ s❡rá ❢❡✐t♦ ♦ ❡st✉❞♦ s♦❜r❡ ❛r❜❡❧♦s✱ ❛ss✐♠ ❝♦♠♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐✲
❡❞❛❞❡s ❡ ❛✐♥❞❛ ✉s❛r❡♠♦s ❛ ✐♥✈❡rsã♦ ♣❛r❛ ❢❛❧❛r ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ❡ ❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳
◆♦ ❝❛♣ít✉❧♦ ✹✱ s❡rã♦ ❛♣❧✐❝❛❞♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s ♣❛r❛ ♣r♦✲
❞✉③✐r ❡①❡♠♣❧♦s ❞❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ♣♦❞❡♠ s❡r ✉t✐✲
❧✐③❛❞♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❛s ❛t✐✈✐❞❛❞❡s✱ s✉❣❡r✐♠♦s ♦ ✉s♦ ❞❡ ❛❧❣✉♠
s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✱ ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ q✉❡ ❢♦✐ ✉t✐❧✐③❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ♠❛s
❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ♣♦❞❡♠ s❡r r❡❛❧✐③❛❞❛s ✉s❛♥❞♦ ❛♣❡♥❛s ré❣✉❛ ❡ ❝♦♠♣❛ss♦✳ ❊st❛ é ✉♠❛
❜♦❛ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦s ❛❧✉♥♦s ❡ ❤❛❜✐❧✐❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❞❡
♣r♦❜❧❡♠❛s q✉❡ ✉s❛♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❡ ❛❧❣é❜r✐❝♦✳

✶✸

✶✹

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

❈❛♣ít✉❧♦ ✶
❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s
♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳
◆♦ ❝❛♣ít✉❧♦ ✷ tr❛t❛r❡♠♦s ❞❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✳ P❛r❛ ✐st♦✱ ♣r❡❝✐s❛r❡♠♦s✱
♣♦r ❡①❡♠♣❧♦✱ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❤♦♠♦t❡t✐❛ ❡ ❞❡ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠
❝ír❝✉❧♦✱ ❛❧é♠ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝ír❝✉❧♦s ♦rt♦❣♦♥❛✐s✱ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r✳
P♦r ❝ír❝✉❧♦ q✉❡r❡♠♦s ❞✐③❡r ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ♥♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ ✉♠
❞❛❞♦ ♣♦♥t♦✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ F ✉♠❛ r❡❣✐ã♦ ❞♦ ♣❧❛♥♦✱ O ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❡ k ✉♠ ♥ú♠❡r♦ r❡❛❧

♥ã♦✲♥✉❧♦✳ ❆ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k é ❛ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ ❛ss♦❝✐❛ ❛
−→
❝❛❞❛ ♣♦♥t♦ P ❞❡ F ♦ ♣♦♥t♦ P ′ s♦❜r❡ ❛ s❡♠✐rr❡t❛ OP ✱ ❞❡ ♦r✐❣❡♠ O✱ t❛❧ q✉❡ OP ′ = k · OP ✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ❙❡ k > 0✱ ❛ ❤♦♠♦t❡t✐❛ ❞❡ r❛③ã♦ k é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❤♦♠♦t❡t✐❛ ❞✐r❡t❛✱ ❡

s❡ k < 0✱ ❝♦♠♦ ❤♦♠♦t❡t✐❛ ✐♥✈❡rs❛✱ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ❛ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ❛ ❤♦♠♦t❡t✐❛
❞✐r❡t❛ ❞❡ r❛③ã♦ −k > 0 ❝♦♠ ❛ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣♦♥t♦ O✳

❋✐❣✉r❛ ✶✳✶✿ ❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k = 2.

✶✺

✶✻

❈❆P❮❚❯▲❖ ✶✳

❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❋✐❣✉r❛ ✶✳✷✿ ❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k = −0, 6.

❉♦✐s ❝ír❝✉❧♦s sã♦ s❡♠♣r❡ ❤♦♠♦tét✐❝♦s✶ ✳ ◆❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ❡❧❡s
❛❞♠✐t❡♠ ❞✉❛s ❤♦♠♦t❡t✐❛s✱ ✉♠❛ ❞✐r❡t❛ ❡ ✉♠❛ ✐♥✈❡rs❛✳ ◆♦ ❝❛s♦ ❞❡ ❝ír❝✉❧♦s ❞✐s❥✉♥t♦s✱ ♦s
❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ sã♦ ❛s ✐♥t❡rs❡❝çõ❡s ❞❛s
t❛♥❣❡♥t❡s ❝♦♠✉♥s ✐♥t❡r♥❛s ✷ ✭✐♥✈❡rs❛✮ ❡ ❞❛s t❛♥❣❡♥t❡s ❝♦♠✉♥s ❡①t❡r♥❛s ✸ ✭❞✐r❡t❛✮✳ ❊st❡s
r❡s✉❧t❛❞♦s ❡stã♦ ♣r♦✈❛❞♦s ♥❛s ❞✉❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s ❡ ✐❧✉str❛❞♦s ♥❛s ✜❣✉r❛s ✭✶✳✸✮✱
✭✶✳✹✮ ❡ ✭✶✳✺✮✳

❖❜s❡r✈❛çã♦ ✶✳✷✳

❙❡❥❛♠ Γ ❡ Ω ❞♦✐s ❝ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳ ❆s t❛♥❣❡♥t❡s ❝♦♠✉♥s ❡①t❡r♥❛s
❝r✉③❛♠✲s❡ ❡♠ O q✉❡ é ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛✳

Pr♦♣♦s✐çã♦ ✶✳✶✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ {O} = t1 ∩ t2 ✱ ✜❣✉r❛ ✭✶✳✸✮✱ ♦s ♣♦♥t♦s A ❡ B ♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s
Γ ❡ Ω r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ X ❡ Y ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞❡ t1 ❝♦♠ ♦s ❝ír❝✉❧♦s Γ ❡ Ω ❝♦♠♦
✐♥❞✐❝❛❞♦ ♥❛ ✜❣✉r❛ ✭✶✳✺✮✳ P❡❧♦ ❝❛s♦ ❞❡ s❡♠❡❧❤❛♥ç❛ AA t❡♠♦s q✉❡ △OAX ∼ △OBY ✳
▲♦❣♦✱
BY
OB
=
= k ⇒ BY = k · AX.
OA
AX

❈♦♠♦ O é ú♥✐❝♦✱ ❡♥tã♦ O é ♦ ❝❡♥tr♦ ❞❛ ❤♦♠♦t❡t✐❛✳
Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡❥❛♠ Γ ❡ Ω ❝ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳
❆s t❛♥❣❡♥t❡s ❝♦♠✉♥s ✐♥t❡r♥❛s
❝r✉③❛♠✲s❡ ❡♠ O1 q✉❡ é ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛✳

✶ ❉♦✐s

✷ P❛r❛

❝ír❝✉❧♦s sã♦ ❤♦♠♦tét✐❝♦s s❡ ❡①✐st✐r ✉♠❛ ❤♦♠♦t❡t✐❛ q✉❡ ❛♣❧✐❝❛ ✉♠ s♦❜r❡ ♦✉tr♦✳
❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s t❛♥❣❡♥t❡s✱ ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❛♦s ❞♦✐s ❝ír❝✉❧♦s ❡stã♦ ❡♠ s❡♠✐♣❧❛♥♦s

♦♣♦st♦s ❡♠ r❡❧❛çã♦ à r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s✳

✸ P❛r❛

❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s t❛♥❣❡♥t❡s✱ ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❛♦s ❞♦✐s ❝ír❝✉❧♦s ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦

❡♠ r❡❧❛çã♦ à r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s✳

✶✼

❋✐❣✉r❛ ✶✳✸✿ ❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ❡①t❡r♥❛s ❞❡

Γ



{O1 } = t1 ∩ t2 ✱ ♦s ♣♦♥t♦s A ❡ B ♦s ❝❡♥tr♦s ❞♦s
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ P ❡ Q ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞❡ t1 ❝♦♠ ♦s ❝ír❝✉❧♦s Γ
❞❡ s❡♠❡❧❤❛♥ç❛ AA t❡♠♦s q✉❡ △O1 AP ∼ △O1 BQ✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠

Ω✳

❝ír❝✉❧♦s


Ω✳

▲♦❣♦✱

BQ
O1 B
=
= k ⇒ BQ = k · AP.
O1 A
AP

❋✐❣✉r❛ ✶✳✹✿ ❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ✐♥t❡r♥❛s ❞❡

❋✐❣✉r❛ ✶✳✺✿ ❈ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳

Γ



Ω✳

Γ





P❡❧♦ ❝❛s♦

✶✽

❈❆P❮❚❯▲❖ ✶✳

❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❍♦♠♦t❡t✐❛s ❧❡✈❛♠ r❡t❛s ❡♠ r❡t❛s✳

r ✉♠❛ r❡t❛ ❡ Ho,k ❛ ❤♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k ✳ ❚♦♠❡ B ❡ C
r✳


❙❡❥❛♠ B = Ho,k (B) ❡ C
= Ho,k (C)✳ P❡❧♦ ❝❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛✱ t❡♠♦s q✉❡ ♦s
′ ′
tr✐â♥❣✉❧♦s △OBC ❡ △OB C sã♦ s❡♠❡❧❤❛♥t❡s✳
←→
❙❡ P ❢♦r ✉♠ ♦✉tr♦ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡ r = BC ✱ ❡♥tã♦✱ ♣❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦✱ ❝♦♥❝❧✉í♠♦s
←−


′ ′

q✉❡ P = Ho,k (P ) ♣❡rt❡♥❝❡ à r❡t❛ B C ✳ P♦rt❛♥t♦ r = Ho,k (r) é ✉♠❛ r❡t❛✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠
♣♦♥t♦s ❞❡

❋✐❣✉r❛ ✶✳✻✿ ❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ r❡t❛ ❡♠ r❡t❛ ❞❡ r❛③ã♦ ❦❂✶✳✺

Pr♦♣♦s✐çã♦ ✶✳✹✳ ❍♦♠♦t❡t✐❛s ❧❡✈❛♠ ❝ír❝✉❧♦s ❡♠ ❝ír❝✉❧♦s✳

❉❡♠♦♥str❛çã♦✳ ❉❡♥♦t❡ ♣♦r

Ho,k
A

q✉❛❧q✉❡r ❞♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦

k · OP

O ❡ r❛③ã♦ k ✳ ❙❡❥❛♠ P ✉♠ ♣♦♥t♦
P ′ = Ho,k (P ) ❡ A′ = Ho,k (A)✳ ❊♥tã♦ OP ′ =

❛ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦
❡ r❛✐♦

r



OA′ = k · OA✳
′ ′
P❡❧♦ ❝❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛✱ ♦s tr✐â♥❣✉❧♦s △OAP ❡ △OA P sã♦ s❡♠❡❧❤❛♥t❡s✱ ❞❡ ♠♦❞♦
′ ′


q✉❡ P A = k · r ✳ ❆ss✐♠✱ P ♣❡rt❡♥❝❡ ❛ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ A ❡ r❛✐♦ kr ✳


❋✐❣✉r❛ ✶✳✼✿ ❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ ❝ír❝✉❧♦ ❡♠ ❝ír❝✉❧♦ ❞❡ r❛③ã♦ ❦❂✶✳✾

✶✾

Pr♦♣♦s✐çã♦ ✶✳✺✳ ❍♦♠♦t❡t✐❛s ♣r❡s❡r✈❛♠ t❛♥❣ê♥❝✐❛ ❡♥tr❡ r❡t❛s ❡ ❝ír❝✉❧♦s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ Ho,k ❛ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k ✱ r ✉♠❛ r❡t❛✱ Γ ✉♠ ❝ír❝✉❧♦
❞❡ ❝❡♥tr♦ A ❡ r❛✐♦ R✱ ❡ P ✉♠ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❡♥tr❡ Γ ❡ r✳
❙❡ B ❡ C ♣❡rt❡♥❝❡ ❛ r ❡♥tã♦ r′ = Ho,k (r) é ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r B ′ = Ho,k (B) ❡
C ′ = Ho,k (C)✳ ❆❧é♠ ❞✐st♦✱ Γ′ = Hk,o (Γ) é ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ A′ ❡ r❛✐♦ kR✳
P♦r s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ❝♦♥❝❧✉í♠♦s q✉❡ △OAP ∼ △OA′ P ′ ❡ △OP B ∼ △OP ′ B ′ ✳
π
❈♦♥s❡q✉❡♥t❡♠❡♥t❡ ∠B ′ P ′ A′ = ∠BP A = ❞♦♥❞❡ s❡❣✉❡ q✉❡ P ′ é ✉♠ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛
2
❡♥tr❡ Γ′ ❡ r′ ✳

❋✐❣✉r❛ ✶✳✽✿ ❍♦♠♦t❡t✐❛ ❞❡ r❛③ã♦ ❦❂✶✳✾

❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❙❡ ✉♠ ♣♦♥t♦ P ❡stá ❛ ✉♠❛

❞✐stâ♥❝✐❛ d ❞❡ O✱ ❞❡✜♥✐♠♦s ❛ ♣♦tê♥❝✐❛ ❞♦ ♣♦♥t♦ P ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ Γ ❝♦♠♦
P ot(P ) = d2 − r2 .

P❡❧❛ ❞❡✜♥✐çã♦ ❛♣r❡s❡♥t❛❞❛✱ s❡ P é ❡①t❡r✐♦r ❛ Γ✱ s✉❛ ♣♦tê♥❝✐❛ é ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦❀
s❡ P ♣❡rt❡♥❝❡ ❛ Γ✱ s✉❛ ♣♦tê♥❝✐❛ é ③❡r♦✱ s❡ P é ✐♥t❡r✐♦r ❛ Γ✱ s✉❛ ♣♦tê♥❝✐❛ é ♥❡❣❛t✐✈❛✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛♠ ❞❛❞♦s ✉♠ ❝ír❝✉❧♦ Γ✱ ❞❡ r❛✐♦ r ❡ ❝❡♥tr♦ O✱ ❡ ✉♠ ♣♦♥t♦ P ✳ ❙❡ ❛ r❡t❛

t q✉❡ ♣❛ss❛ ♣♦r P ❡ ✐♥t❡rs❡❝t❛ ♦ ❝ír❝✉❧♦ Γ ♥♦s ♣♦♥t♦s A ❡ B ✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ P A.P B é
✉♠❛ ❝♦♥st❛♥t❡ ✭✐st♦ é ✐♥❞❡♣❡♥❞❡ ❞❡ r✮✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ♦ ❝ír❝✉❧♦ Γ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❙❡❥❛ P ✉♠ ♣♦♥t♦ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛
Γ ❝♦♠ P O = d✳ ❙❡❥❛♠ t ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s P ✱ A ❡ B ❡ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦
s❡❣♠❡♥t♦ AB ✳ ❙❡❥❛ ❛❣♦r❛ M A = M B = x✳ P❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ LLL✱ t❡♠♦s q✉❡ ♦s
tr✐â♥❣✉❧♦s △AOM ❡ △BOM sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❈♦♥s❡q✉❡t❡♠❡♥t❡ ✱ ♦s â♥❣✉❧♦s ∠AM O ❡
∠BM O sã♦ r❡t♦s✱ ♣♦✐s sã♦ ❝♦♥❣r✉❡♥t❡s ❡ s✉♣❧❡♠❡♥t❛r❡s✳ ❖✉ s❡❥❛✱ AM ⊥ OM ✳ P♦❞❡♠♦s
❞✐③❡r q✉❡✿

✷✵

❈❆P❮❚❯▲❖ ✶✳

✶✳ ❙❡

P

é ❡①t❡r✐♦r ❛♦ ❝ír❝✉❧♦

❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

Γ✱

P A · P B = (P M − x)(P M + x) = P M 2 − x2 = P O2 − OM 2 − x2 =
= P O2 − (OM 2 + x2 ) = d2 − r2 = P ot(P ).

❋✐❣✉r❛ ✶✳✾✿ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦

✷✳ ❙❡

P

é ✐♥t❡r✐♦r ❛

P

❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦

Γ✳

Γ✱

−P A · P B = −(x − P M )(x + P M ) = P M 2 − x2 = P O2 − OM 2 − x2 =
= P O2 − (OM 2 + x2 ) = d2 − r2 = P ot(P ).

❋✐❣✉r❛ ✶✳✶✵✿ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦

P

✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦

Γ✳

✷✶

❖❜s❡r✈❛çã♦ ✶✳✸✳

❙❡

P

♣❡rt❡♥❝❡ ❛

Γ✱

❡♥tã♦

P

s❡rá

A

♦✉

B✱

❞❡ ♠♦❞♦ q✉❡

pot(P ).

❖❜s❡r✈❛çã♦ ✶✳✹✳
♥♦ ♣♦♥t♦

T

❡♥tã♦

❙❡ P é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω
P T 2 = P A · P B ✳ ❉❡ ❢❛t♦✱

❈♦♥s✐❞❡r❡ ✉♠❛ r❡t❛

t

t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦



♣♦✐s✱
❝♦r❞❛

❡ ✉♠❛ ❞❛s r❡t❛s é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦

T ❡ ✉♠❛ r❡t❛ s q✉❡ ❝r✉③❛ ♦
△AT P ❡ △P T B s❡♠❡❧❤❛♥t❡s✱



AT ✮✳

♦✉

P T 2 = P A · P B.

❋✐❣✉r❛ ✶✳✶✶✿ ❙❡❣♠❡♥t♦ P❚ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦

❉❡✜♥✐çã♦ ✶✳✸✳
Γ

✐♥s❝r✐t♦ ❛ ✉♠❛ ♠❡s♠❛

▲♦❣♦✱ s❡✉s ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ♣r♦♣♦r❝✐♦♥❛✐s✱ ✐st♦ é✱

PA
PT
=
PT
PB



C

♥✉♠ ♣♦♥t♦

Ω ♥♦s ♣♦♥t♦s A ❡ B ✳ ❚❡♠♦s ❡♥tã♦ ❞♦✐s tr✐â♥❣✉❧♦s
∠T P A = ∠T P B ❡ ∠AT P ∼
= ∠T BP ✭â♥❣✉❧♦s ❞❡ s❡❣♠❡♥t♦

❝ír❝✉❧♦

PA·PB = 0 =

❙❡❥❛♠ ✉♠ ❝ír❝✉❧♦

s❡ ❡ s♦♠❡♥t❡ s❡

r

Γ

❞❡ ❝❡♥tr♦

♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡

O

❡ ✉♠❛ r❡t❛

r✳

Ω✳

❉✐③❡♠♦s q✉❡

Γ✳

❋✐❣✉r❛ ✶✳✶✷✿ ❘❡t❛ ❡ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧✳

r

é ♦rt♦❣♦♥❛❧

✷✷

❈❆P❮❚❯▲❖ ✶✳

❋✐❣✉r❛ ✶✳✶✸✿ ❈ír❝✉❧♦

❉❡✜♥✐çã♦ ✶✳✹✳
♣♦♥t♦

I

❉♦✐s ❝ír❝✉❧♦s

Γ





Γ

❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

♦rt♦❣♦♥❛❧ ❛♦ ❝ír❝✉❧♦



sã♦ ❞✐t♦s ♦rt♦❣♦♥❛✐s s❡ ❡❧❡s s❡ ✐♥t❡rs❡❝t❛♠ ♥✉♠

❢♦r♠❛♥❞♦ â♥❣✉❧♦s r❡t♦s✱ ✐st♦ é✱ s❡ s❡✉s r❛✐♦s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♥♦ ♣♦♥t♦

I

❞❡

✐♥t❡rs❡❝çã♦✳

◆♦ ❝❛♣ít✉❧♦ ✸ ✉s❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛ s❡❣✉✐r✳

Pr♦♣♦s✐çã♦ ✶✳✻✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ s❡✱ ❡ só s❡✱ ❞✉❛s ❞✐❛❣♦♥❛✐s
s❡ ✐♥t❡rs❡❝t❛♠ ♥♦s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s ♠é❞✐♦s✳

M ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝✲

çã♦ ❞❡ s✉❛s ❞✐❛❣♦♥❛✐s✳ ❉❡ AB k CD ✱ s❡❣✉❡ q✉❡ ∠BAM = ∠DCM ❡ ∠ABM ∼
= ∠CDM ✳
❈♦♠♦ ❥á s❛❜❡♠♦s q✉❡ AB = CD ✱ s❡❣✉❡ q✉❡ ♦s tr✐â♥❣✉❧♦s △ABM ❡ △CDM sã♦ ❝♦♥❣r✉✲
❡♥t❡s ♣❡❧♦ ❝❛s♦ ALA✳ ▲♦❣♦✱ AM = CM ❡ BM = DM ✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ ABCD ✉♠ q✉❛❞r✐❧át❡r♦ t❛❧ q✉❡ s✉❛s ❞✐❛❣♦♥❛✐s AC ❡ BD s❡ ✐♥t❡rs❡❝✲
t❛♠ ❡♠ M ✱ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❛♠❜❛s✳ ❊♥tã♦ M A = M C ✱ BM = DM ❡ ∠AM B ∼
= ∠CM D
✭â♥❣✉❧♦s ♦♣st♦s ♣❡❧♦ ✈ért✐❝❡✮✱ ❞❡ ♠♦❞♦ q✉❡ ♦s tr✐â♥❣✉❧♦s △ABM ❡ △CDM sã♦ ❝♦♥❣r✉✲
❡♥t❡s✱ ♣♦r LAL✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ △BCM ❡ △DAM t❛♠❜é♠ sã♦ ❝♦♥❣r✉❡♥t❡s ♣♦r LAL✳
❚❛✐s ❝♦♥❣r✉ê♥❝✐❛s ♥♦s ❞ã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ AB = CD ❡ BC = AD ✱ ♦ q✉❡ ❥á s❛❜❡♠♦s
s❡r ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❢❛t♦ ❞❡ ABCD s❡r ♣❛r❛❧❡❧♦❣r❛♠♦✳

❉❡♠♦♥str❛çã♦✳

Pr✐♠❡✐r❛♠❡♥t❡✱ s❡❥❛

ABCD

✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡

❋✐❣✉r❛ ✶✳✶✹✿ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳

✷✸

Pr♦♣♦s✐çã♦ ✶✳✼✳

❙❡ ✉♠ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥✉♠ s❡♠✐❝ír❝✉❧♦

Γ

❞❡ ❝❡♥tr♦

O

❡ r❛✐♦

r

t❡♠

✉♠ ❧❛❞♦ ❝✉❥❛ ♠❡❞✐❞❛ é ✐❣✉❛❧ ❛♦ s❡✉ ❞✐â♠❡tr♦✱ ❡♥tã♦ ❡❧❡ é ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡ ❡ss❡
❞✐â♠❡tr♦ é ❤✐♣♦t❡♥✉s❛ ❞♦ tr✐â♥❣✉❧♦✳

❙❡❥❛♠ D✱ A ❡ B ♣♦♥t♦s ❞❡ Γ t❛✐s q✉❡ A✱ O ❡ B s❡❥❛♠ ❝♦❧✐♥❡❛r❡s✳ t❡♠♦s
q✉❡ ♦ tr✐â♥❣✉❧♦ △BOD é ✐sós❝❡❧❡s✱ ❞❡ ♠♦❞♦ q✉❡ ∠ODB ∼
= ∠OBD✱ ❞❡ ♠❡❞✐❞❛ α✳
❆♥á❧♦❣❛♠❡♥t❡✱ ∠OAD ∼
= ∠ODA✱ ❞❡ ♠❡❞✐❞❛ β ✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

❉❡♠♦♥str❛çã♦✳

2α + 2β = π.
π
2

▲♦❣♦✱ α + β = ✳ ❆ss✐♠✱ ∠ADB ♠❡❞❡

π
❡ ♦ tr✐â♥❣✉❧♦ △ADB é r❡tâ♥❣✉❧♦ ❡♠ D✳
2

❋✐❣✉r❛ ✶✳✶✺✿ ❚r✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦
❆ ❞❡✜♥✐çã♦ ❞❡ ❡❧✐♣s❡ s❡rá ♥❡❝❡ssár✐❛ ♣❛r❛ ❢❛❧❛r♠♦s ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s q✉❡ s❡rá
❛❜♦r❞❛❞❛ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✳

❉❡✜♥✐çã♦ ✶✳✺✳

❋✐①❛❞♦ ❞♦✐s ♣♦♥t♦s

F1



F2

❞♦ ♣❧❛♥♦ ✉♠❛ ❡❧✐♣s❡

ξ

❞❡ ❢♦❝♦s

é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ ❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❛

F1

2a > 0✱
d(F1 , F2 ) = 2c✱

❖✉ s❡❥❛✱

❝♦♥st❛♥t❡

♠❛✐♦r ❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s

2c ≥ 0✳

ξ = {P |d(P, F1 ) + d(P, F2 ) = 2a}

❋✐❣✉r❛ ✶✳✶✻✿ ❊❧✐♣s❡ ❞❡ ❢♦❝♦s F1 ❡ F2 ✳



F2

F1



F2

é ✐❣✉❛❧ ✉♠❛

0≤c 0 ❡ ♠♦str❛♠♦s ❝♦♠♦ ✐♥✈❡rt❡r r❡t❛s ❡ ❝ír❝✉❧♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❞❛❞♦
❝ír❝✉❧♦✳ ❆ ✐♥✈❡rsã♦ é ✉♠❛ té❝♥✐❝❛ ✉s❛❞❛ ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦ ♥♦ ❡st✉❞♦ ❞♦s ❛r❜❡❧♦s✳
P♦r ❡①❡♠♣❧♦✱ ♥❛ ❝♦♥str✉çã♦ ❞❛s ❝❛❞❡✐❛s ❞❡ P❛♣♣✉s ❡ ❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳
✷✳✶

■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r > 0 ✜①❛❞♦s✱ ♥♦ ♣❧❛♥♦✳ ❆ ✐♥✈❡rsã♦

I ❡♠ r❡❧❛çã♦ ❛ Γ é ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ P ✱ ❞✐st✐♥t♦ ❞❡ O✱
−→
♥♦ ♣❧❛♥♦ ✉♠ ú♥✐❝♦ ♣♦♥t♦ P ′ ♣❡rt❡♥❝❡♥t❡ à s❡♠✐rr❡t❛ OP t❛❧ q✉❡ OP · OP ′ = r2 ✳

❋✐❣✉r❛ ✷✳✶✿ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✳
❆ s❡❣✉✐r✱ ❞❡t❡r♠✐♥❛♠♦s ✉♠❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛ ♣❛r❛ ❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ Γ✳
❈♦♠❡ç❛♠♦s s✉♣♦♥❞♦ q✉❡ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳
✷✼

✷✽

❈❆P❮❚❯▲❖ ✷✳

●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆

✶◦ ❝❛s♦✿ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ✐st♦ é✱ O = (0, 0)✳

❋✐❣✉r❛ ✷✳✷✿ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳
❙❡❥❛♠ P = (x, y) 6= (0, 0) ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦ ❡ P ′ = (x′ , y ′ ) ❛ ✐♠❛❣❡♠ ❞❡ P
−→
♣❡❧❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ Γ✱ ✐st♦ é✱ P ′ = I(P )✳ ❈♦♠♦ P ′ ∈ OP ✱ ❡♥tã♦

(x′ , y ′ ) = λ(x, y), ❝♦♠ λ ∈ R+
∗.
p
p
P♦r ❞❡✜♥✐çã♦✱ OP · OP ′ = r2 ✱ ❞♦♥❞❡ t❡♠♦s q✉❡ x2 + y 2 · (x′ )2 + (y ′ )2 = r2 ✳
r2
r2
=
✳ P♦rt❛♥t♦✱
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ λ2 · (x2 + y 2 )2 = r4 ✳ ▲♦❣♦✱ λ = 2
x + y2
(OP )2
I(x, y) = P ′ =

−→
r2
· OP .
2
(OP )

❈❛s♦ ❣❡r❛❧✿ O = (x0 , y0 ) é ❞✐st✐♥t♦ ❞❡ (0, 0)✿
◆♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ♦r✐❣✐♥❛❧✱ s✉♣♦♥❤❛ q✉❡ P t❡♥❤❛ ❝♦♦r❞❡♥❛❞❛s
P = (x1 , y1 )✳ ◆♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s q✉❡ t❡♠ O = (x0 , y0 ) ❝♦♠♦ ♦r✐❣❡♠✱ ❛s ❝♦♦r❞❡✲
♥❛❞❛s ❞❡ P sã♦ P = (x1 − x0 , y1 − y0 )✳ ❊♠ r❡❧❛çã♦ ❛ ❡st❡ s✐st❡♠❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦

−→
r2
· OP ✳ P❛r❛ ♦❜t❡r ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ P ′ ♥♦
2
(OP )
s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♦r✐❣✐♥❛❧✱ ♣r❡❝✐s❛♠♦s tr❛♥s❧❛❞❛r ♦s ❡✐①♦s ❞❡ ♠♦❞♦ q✉❡ (0, 0) s❡❥❛
♥♦✈❛♠❡♥t❡ ❛ ♦r✐❣❡♠✳ ■st♦ é ❢❡✐t♦ s♦♠❛♥❞♦✲s❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ O às ❝♦♦r❞❡♥❛❞❛s ♥♦

♣♦♥t♦ P ′ = I(P ) sã♦ ❞❛❞❛s ♣♦r P ′ =

♦✉tr♦ s✐st❡♠❛✳ ❖✉ s❡❥❛✱

(x′ , y ′ ) =

−→
r2
·
OP + O.
(OP )2

✷✳✶✳

✷✾

■◆❱❊❘❙➹❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼ ❈❮❘❈❯▲❖

❋✐❣✉r❛ ✷✳✸✿ O ♥ã♦ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳
❊♠ ❝♦♦r❞❡♥❛❞❛s t❡♠♦s✱
I(x, y) = (x′ , y ′ ) = (x0 , y0 ) +

r2
· (x1 − x0 , y1 − y0 ).
(x1 − x0 )2 + (y1 − y0 )2

❉❡st❛ ❢♦r♠❛✱ ❛ ❡①♣r❡ssã♦ ❣❡r❛❧ ♣❛r❛ ❛ ✐♥✈❡rsã♦ I : R2 \ {O} −→ R2 \ {O} ❡♠ r❡❧❛çã♦
❛ ✉♠ ❝ír❝✉❧♦ Γ ❞❡ ❝❡♥tr♦ O é
r2 −→
· OP .
I(P ) = O +
OP 2

✭✷✳✶✮

❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ✐♠❡❞✐❛t❛s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥✈❡rsã♦✿
−−→

✶✳ OP ′ =

r2 −→
· OP ✱ ❞❡ ♠♦❞♦ q✉❡ OP ′ · OP = r2 ✳
OP 2

✷✳ ❙❡ P ❢♦r ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r ❞❡ Γ✱ ✐st♦ é✱ OP > r ❡♥tã♦ r2 = OP · OP ′ > r · OP ′ ❀
❧♦❣♦✱ OP ′ < r✱ ✐st♦ é✱ P ′ ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ Γ✳

❋✐❣✉r❛ ✷✳✹✿ ❖ ♣♦♥t♦ P é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳

✸✵

❈❆P❮❚❯▲❖ ✷✳

❋✐❣✉r❛ ✷✳✺✿ ❖ ♣♦♥t♦

✸✳ ❙❡

P

❧♦❣♦✱

✹✳ ❙❡

P

❢♦r ✉♠ ♣♦♥t♦ ❞♦ ✐♥t❡r✐♦r ❞❡

OP > r✱


✐st♦ é✱

P



❢♦r ✉♠ ♣♦♥t♦ ❞❡ ❞❡

♦✉ s❡❥❛✱

P ′ ∈ Γ✳

Γ✱

I

Γ✳

é ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦

OP < r
❞❡ Γ✳

✐st♦ é✱

❡stá ♥♦ ❡①t❡r✐♦r

❡♥tã♦

r2 = OP · OP < r · OP ❀

Γ✱ ✐st♦ é✱ OP = r ❡♥tã♦ r2 = r ·OP ′ ✱ ❞❡ ♠♦❞♦ q✉❡✱OP ′ = r✱

❋✐❣✉r❛ ✷✳✻✿ ❖ ♣♦♥t♦

✺✳

P

●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆

P

♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦

Γ✳

é ✉♠❛ ✐♥✈♦❧✉çã♦✱ ✐st♦

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