O USO DE MATERIAL CONCRETO PARA O ENSINO DA PROPRIEDADE REFLEXIVA DAS CÔNICAS

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛

Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚

❖ ❯s♦ ❞❡ ▼❛t❡r✐❛❧ ❈♦♥❝r❡t♦ ♣❛r❛ ♦ ❊♥s✐♥♦ ❞❛

Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡①✐✈❛ ❞❛s ❈ô♥✐❝❛s

❋❡r♥❛♥❞♦ ❘♦❝❤❛ ❇❛r❜♦s❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❖ ❯s♦ ❞❡ ▼❛t❡r✐❛❧ ❈♦♥❝r❡t♦ ♣❛r❛ ♦ ❊♥s✐♥♦ ❞❛

Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡①✐✈❛ ❞❛s ❈ô♥✐❝❛s

❋❡r♥❛♥❞♦ ❘♦❝❤❛ ❇❛r❜♦s❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❏♦ã♦ ❇❡♥í❝✐♦ ❞❡ ▼❡❧♦ ◆❡t♦

❋■❈❍❆ ❈❆❚❆▲❖●❘➪❋■❈❆

❙❡r✈✐ç♦ ❞❡ Pr♦❝❡ss❛♠❡♥t♦ ❚é❝♥✐❝♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❇✐❜❧✐♦t❡❝❛ ❙❡t♦r✐❛❧ ❞♦ ❈❈◆

❇✷✸✽✉ ❇❛r❜♦s❛✱ ❋❡r♥❛♥❞♦ ❘♦❝❤❛

❖ ❯s♦ ❞❡ ▼❛t❡r✐❛❧ ❈♦♥❝r❡t♦ ♣❛r❛ ♦ ❊♥s✐♥♦ ❞❛ Pr♦♣r✐❡❞❛❞❡ ❘❡✲ ✢❡①✐✈❛ ❞❛s ❈ô♥✐❝❛s✴ ❋❡r♥❛♥❞♦ ❘♦❝❤❛ ❇❛r❜♦s❛✳ ✲ ❚❡r❡s✐♥❛✱ ✷✵✶✻✳

✻✻❢✳ ✐❧✳

❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧✮ ✲ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡❋❡❞❡r❛❧ ❞♦ P✐❛✉í✱

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦ã♦ ❇❡♥í❝✐♦ ❞❡ ▼❡❧♦ ◆❡t♦

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s q✉❡ ♣♦ss✐❜✐❧✐t♦✉ ♠❛✐s ❡ss❛ ❝♦♥q✉✐st❛✱ ♣♦✐s s❡♠♣r❡ ♠❡ ❞❡✉ s❛ú❞❡ ♥❡ss❡ ♣❡rí♦❞♦ ❡ ♠❡ ❣✉✐❛ ♥❛ ✈✐❞❛✳

❆♦s ♠❡✉s ♣❛✐s✱ P❡❞r♦ ❡ ▼❛r✐❛✱ ❡①❡♠♣❧♦s ❞❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❛♠♦r ❡ ❞❡❞✐❝❛çã♦ ❛♦s ✜❧❤♦s q✉❡ sã♦ ✐♥❞✐s♣❡♥sá✈❡✐s ♣❛r❛ s✉♣❡r❛r ♠✉✐t♦s ♦❜stá❝✉❧♦s✳

❆♦s ♠❡✉s ✐r♠ã♦s✱ ❈r❡❞❡♠✐r ❡ ❱❛♥❡ss❛✱ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛❥✉❞❛ ❡♠ ❞✐✈❡rs♦s ♠♦♠❡♥✲ t♦s ❞❡ss❛ ❥♦r♥❛❞❛✳

❆ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ♣❡❧♦ ❛♣♦✐♦ ❞❛❞♦ ❛ ❡ss❛ ❝♦♥q✉✐st❛✱ ♣♦r ❝♦♠♣❛rt✐❧❤❛r❡♠ ♠♦♠❡♥t♦s ❞❡ ❢❡❧✐❝✐❞❛❞❡s ❡ ❞❡ ❞✐✜❝✉❧❞❛❞❡s✳

❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ✐♥❢â♥❝✐❛✱ ❞❡ ❡♥s✐♥♦ ❜ás✐❝♦✱ ❞❡ ❣r❛❞✉❛çã♦ ❡ tr❛❜❛❧❤♦✱ ♣❡❧❛s ♦♣✐♥✐õ❡s ✈❛❧♦r♦s❛s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦s ❛♠✐❣♦s ❞❛ t✉r♠❛ ✷✵✶✹ ❞♦ P❘❖❋▼❆❚✱ ♣❡❧❛ ✉♥✐ã♦ ❡ ❛❥✉❞❛ ♥♦s ✈ár✐♦s ❞❡s❛✜♦s q✉❡ ♥♦s ❢♦r❛♠ ❝♦❧♦❝❛❞♦s✳

➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ♣❡❧❛ ✐❞❡❛❧✐③❛çã♦ ❞❡ss❡ ♠❡str❛❞♦ q✉❡ ♠❡❧❤♦r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ ♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦♦r❞❡♥❛❞♦r ❞♦ P❘❖❋▼❆❚✲❯❋P■ ♣❡❧♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❝♦♠✲ ♣❛rt✐❧❤❛❞♦s ❡ ❡♠♣❡♥❤♦ ♣❛r❛ q✉❡ ♣✉❞éss❡♠♦s s✉♣❡r❛r ❛s t❛♥t❛s ❛✈❛❧✐❛çõ❡s q✉❡ ❢♦♠♦s s✉❜✲ ♠❡t✐❞♦s✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ❏♦ã♦ ❇❡♥í❝✐♦ ❞❡ ▼❡❧♦ ◆❡t♦✱ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❡ ❞❡✲ ❞✐❝❛çã♦ ♥❛ ♦r✐❡♥t❛çã♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦✳

❆♦ ❛♠✐❣♦ Pr♦❢✳ ❋r❛♥❝✐s❝♦ ❏♦sé ❞❡ ❙♦✉s❛ ♣❡❧♦ ❞❡❞✐❝❛❞♦ tr❛❜❛❧❤♦ ❞❡ r❡✈✐sã♦ ❣r❛♠❛t✐❝❛❧✳

❙❡ ✜③❡r ✉♠❛ r❡t❛✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✐♥❞❡✜♥✐❞♦ q✉❡ ♣❛ss❛ s❡♠♣r❡ ♣♦r ✉♠ ♣♦♥t♦ ✜①♦ ✭❱ért✐❝❡✮✱ ♠♦✈❡r✲s❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ✭❜❛s❡ ❞♦ ❝♦♥❡✮ q✉❡ ♥ã♦ ❡stá ♥✉♠ ♠❡s♠♦ ♣❧❛♥♦ ❝♦♠ ♦ ♣♦♥t♦ ❞❡ ♠♦❞♦ ❛ ♣❛ss❛r s✉❝❡ss✐✈❛♠❡♥t❡ ♣♦r ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❛ r❡t❛ ♠ó✈❡❧ ❞❡s❝r❡✈❡rá ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ ❝♦♥❡ ❞✉♣❧♦✳

✭❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✮

◗✉❡♠ ❡♥t❡♥❞❡ ❆rq✉✐♠❡❞❡s ❡ ❆♣♦❧ô♥✐♦✱ ❛❞♠✐r❛rá ♠❡♥♦s ❛s r❡❛❧✐③❛çõ❡s ❞♦s ❤♦♠❡♥s ♠❛✐s ❝❡❧é❜r❡s ❞❡ é♣♦❝❛s ♣♦s✲ t❡r✐♦r❡s✳

✭▲❡✐❜♥✐③✮

❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ✜♥❛❧✐❞❛❞❡ ❛♣r❡s❡♥t❛r ♦ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ ✏❙✐♥✉❝❛ ❞❛s ❈ô♥✐❝❛s✑ ❝♦♠♦ r❡❝✉rs♦ ❞✐❞át✐❝♦ ♣❛r❛ ❢❛✈♦r❡❝❡r ♦ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ❞❡✜♥✐çõ❡s ❡ ♣r♦✲ ♣r✐❡❞❛❞❡s ❞❛s ❝ô♥✐❝❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♣r♦♣r✐❡❞❛❞❡ r❡✢❡t♦r❛✳ ❆❝r❡❞✐t❛✲s❡ q✉❡ ♦ ✉s♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ♣❛r❛ ❡♥s✐♥❛r ♠❛t❡♠át✐❝❛ é ✉♠ r❡❝✉rs♦ ❢❛❝✐❧✐t❛❞♦r ♣❛r❛ ♦ ❞❡s❡✈♦❧✈✐♠❡♥t♦ ❞✐❞át✐❝♦ ❞♦ ♣r♦❢❡ss♦r ❡ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦ ❛❧✉♥♦✳ ❆ ♣r✐♦r✐✱ ❢❛③✲s❡ ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❤✐stór✐❝♦ ❞❛s ❝ô♥✐❝❛s ❡ é ❛♣r❡s❡♥t❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s ❞❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛✳ ❊♠ s❡❣✉✐❞❛✱ ❥✉st✐✜❝❛♠✲s❡ ♦s ♥♦♠❡s ❞❛s ❝ô♥✐❝❛s ❞❛❞♦s ♣♦r ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ❡ é ❛♣r❡s❡♥t❛❞❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ss❛s ❝✉r✈❛s ♣❡❧♦s t❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥✳ ❉❛♥❞♦ ❝♦♥t✐♥✉✐❞❛❞❡✱ ❞á✲s❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ❡ ❛♥❛❧ít✐❝❛ ❞❛s ❝ô♥✐❝❛s ♥♦ ♣❧❛♥♦✱ ♦♥❞❡ s❡ ✈❡r✐✜❝❛✿ ❛ ❞❡✜♥✐çã♦ ❞♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✱ ❛ ❝♦♥str✉çã♦ ❝♦♠ ré✲ ❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ♦s ❡❧❡♠❡♥t♦s ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✳ P♦r ✜♠✱ ♣r♦♣õ❡✲s❡ ❛ ❝♦♥❢❡❝çã♦ ❞♦s ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ✏❙✐♥✉❝❛s ❞❛s ❈ô♥✐❝❛s✑ ❡ s❡ ❞✐s❝✉t❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ●❡♦♠❡tr✐❛✱ ❙❡çõ❡s ❈ô♥✐❝❛s✱ Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡①✐✈❛✱ ❙✐♥✉❝❛s ❞❛s ❈ô✲ ♥✐❝❛s✳

❆❜str❛❝t

❚❤✐s st✉❞② ❛✐♠s t♦ ♣r❡s❡♥t t❤❡ ❝♦✉rs❡✇❛r❡ ✏P♦♦❧ ♦❢ ❈♦♥✐❝❛❧✑ ❛s ❛ t❡❛❝❤✐♥❣ r❡s♦✉r❝❡ t♦ s✉♣♣♦rt ❡❞✉❝❛t✐♦♥ ❛♥❞ ❧❡❛r♥✐♥❣ ❞❡✜♥✐t✐♦♥s ❛♥❞ ❝♦♥✐❝❛❧ ♣r♦♣❡rt✐❡s ✐♥ ❤✐❣❤ s❝❤♦♦❧✱ ❡s♣❡❝✐❛❧❧② t❤❡ r❡✢❡❝t✐✈❡ ♣r♦♣❡rt②✳ ■t ✐s ❜❡❧✐❡✈❡❞ t❤❛t t❤❡ ✉s❡ ♦❢ ❝♦♥❝r❡t❡ ♠❛t❡r✐❛❧s ❢♦r t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s ✐s ❛ ❢❛❝✐❧✐t❛t♦r r❡s♦✉r❝❡ ❢♦r t❡❛❝❤✐♥❣ ❞❡s❡✈♦❧✈✐♠❡♥t♦ t❡❛❝❤❡r ❛♥❞ st✉❞❡♥t ❧❡❛r♥✐♥❣✳ ■♥✐t✐❛❧❧②✱ ✐t ✐s ❛ ❤✐st♦r✐❝❛❧ s✉r✈❡② ♦❢ ❝♦♥✐❝❛❧ ❛♥❞ ♣r❡s❡♥ts t❤❡ ♠❡t❤♦❞ ♦❢ ❛♣♣❧✐❝❛t✐♦♥ ❛r❡❛s ♦❢ ●r❡❡❦ ❣❡♦♠❡tr②✳ ❚❤❡♥ ❥✉st✐✜❡❞ t❤❡ ♥❛♠❡s ♦❢ ❝♦♥✐❝❛❧ ❣✐✈❡♥ ❜② ❆♣♦❧❧♦♥✐✉s ♦❢ P❡r❣❛ ❛♥❞ s❤♦✇s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡s❡ ❝✉r✈❡s ❜② t❤❡♦r❡♠s ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❛♥❞ ▼♦rt♦♥ P✐❡r❝❡✳ ❈♦♥t✐♥✉✐♥❣✱ ❣✐✈❡s ②♦✉rs❡❧❢ ❛ ❣❡♦♠❡tr✐❝ ❛♥❞ ❛♥❛❧②t✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ t❤❡ ❝♦♥✐❝ ✐♥ t❤❡ ♣❧❛♥❡✱ ✇❤❡r❡ t❤❡r❡ ✐s✿ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧♦❝✉s✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ✇✐t❤ r✉❧❡r ❛♥❞ ❝♦♠♣❛ss✱ t❤❡ ❡❧❡♠❡♥ts ❛♥❞ s♦♠❡ ♣r♦♣❡rt✐❡s✳ ❋✐♥❛❧❧②✱ ♣r♦♣♦s❡ t♦ ♠❛♥✉❢❛❝t✉r❡ ♦❢ ❝♦♥❝r❡t❡ ♠❛t❡r✐❛❧s ✏❙✐♥✉❝❛s ♦❢ ❈♦♥✐❝❛❧✑ ❛♥❞ ❞✐s❝✉ss❡s t❤❡ ✉s❡ ♦❢ ❝♦♥❝r❡t❡ ♠❛t❡r✐❛❧s ❢♦r t❡❛❝❤✐♥❣ ♠❛t❤❡♠❛t✐❝s✳

❑❡②✇♦r❞s✿ ●❡♦♠❡tr②✱ ❈♦♥✐❝ ❙❡❝t✐♦♥s✱ ❘❡✢❡❝t✐✈❡ Pr♦♣❡rt②✱ ❙✐♥✉❝❛s ♦❢ ❈♦♥✐❝❛❧✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❊❧❡♠❡♥t♦s ❞♦ ❈♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷ ❙❡çõ❡s ❈ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s P❛r❛❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✺ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s ❊❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✻ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✼ ❛✮ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ tr❛❥❡tór✐❛ ❡❧í♣t✐❝❛ ❞❡ ✉♠ ♣❧❛♥❡t❛ ❛♦ r❡❞♦r ❞♦ ❙♦❧✳ ❜✮

❊♠ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦ ✐❣✉❛✐s✱ ❛s ár❡❛s ✏✈❛rr✐❞❛s✑ ♣❡❧♦ r❛✐♦ ✈❡t♦r q✉❡ ❧✐❣❛ ♦ ♣❧❛♥❡t❛ ❛♦ ❙❖▲ ✭❡♠ ✉♠ ❞♦s ❢♦❝♦s✮ t❛♠❜é♠ sã♦ ✐❣✉❛✐s✿ △_{t=t2−t1 =t4−t3} ✶✾

✷✳✶ ❙❡çõ❡s ❈ô♥✐❝❛s ❞❡✜♥✐❞❛s ♣♦r ❆♣♦❧ô♥✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❘❡t❛s t❛♥❣❡♥t❡s ❛ S q✉❡ ♣❛ss❛♠ ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✷✳✸ ❊s❢❡r❛s S1 ❡ S2 t❛♥❣❡♥t❡s ❛♦ ♣❧❛♥♦ π ❡ ❛♦ ❝♦♥❡ C ♥♦ ♠❡s♠♦ r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✷✽

✷✳✹ ❊s❢❡r❛s S1 ❡ S2 t❛♥❣❡♥t❡s ❛♦ ♣❧❛♥♦ π ❡ ❛♦ ❝♦♥❡ C ❡♠ r❛♠♦s ❞✐st✐♥t♦s ✳ ✳ ✳ ✷✾

✷✳✺ ❊s❢❡r❛ S t❛♥❣❡♥t❡ ❛♦ ♣❧❛♥♦ π1 ❡ ❛♦ ❝♦♥❡ C ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✶✺ ❊s❜♦ç♦ ❞❛ ❊❧✐♣s❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ s♦❜r❡ ♦ ❡✐①♦ OX ✳ ✳ ✳ ✳ ✳ ✹✼

✸✳✶✻ ❘❡t❛s s❡❝❛♥t❡ à ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✸✳✶✼ ❊s❜♦ç♦ ❞❛ ❍✐♣ér❜♦❧❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ s♦❜r❡ ♦ ❡✐①♦OX ✳ ✳ ✺✵

✸✳✶✽ ❊s❜♦ç♦ ❞❛ P❛rá❜♦❧❛ ❞❡ ✈ért✐❝❡ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ s♦❜r❡ ♦ ❡✐①♦ OX ✳ ✳ ✳ ✺✸

❙✉♠ár✐♦

❘❡s✉♠♦ ✺

❆❜str❛❝t ✻

■♥tr♦❞✉çã♦ ✶✶

✶ ❆ ❍✐stór✐❛ ❞❛s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ✶✸

✶✳✶ ❆ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷ ❆s ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✶ ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s P❛r❛❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✷ ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ❊❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✸ ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸ ❆ ❛♣❧✐❝❛çã♦ ❞❛s ❝ô♥✐❝❛s ❡♠ ❢❡♥ô♠❡♥♦s ❞❛ ♥❛t✉r❡③❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷ ❆s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❡ ♦s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡

▼♦rt♦♥ ✷✶

✷✳✶ ❖s ◆♦♠❡s ❞❛s ❈ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❖s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✳✶ ❆ ❘❡❧❛çã♦ ❋♦❝♦✱ ❉✐r❡çã♦ ❡ ❊①❝❡♥tr✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸ ❆s ❈ô♥✐❝❛s ♥♦ P❧❛♥♦ ✸✸

✸✳✷ ❯♠❛ ❆❜♦r❞❛❣❡♠ ❆♥❛❧ít✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷✳✶ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷✳✶✳✶ ❊❧✐♣s❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ s♦❜r❡ ♦ ❡✐①♦ OX ✳ ✹✼

✸✳✷✳✶✳✷ ❊q✉❛çã♦ ❞❛ ❘❡t❛ ❚❛♥❣❡♥t❡ à ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷✳✶✳✸ Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡t♦r❛ ❞❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷✳✷ ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷✳✷✳✶ ❍✐♣ér❜♦❧❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ s♦❜r❡ ♦ ❡✐①♦ OX ✺✵

✸✳✷✳✷✳✷ ❊q✉❛çã♦ ❞❛ ❘❡t❛ ❚❛♥❣❡♥t❡ à ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷✳✷✳✸ Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡t♦r❛ ❞❛ ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷✳✸ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✸✳✷✳✸✳✶ P❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ s♦❜r❡ ♦ ❡✐①♦

OX ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✸✳✷✳✸✳✷ ❊q✉❛çã♦ ❞❛ ❘❡t❛ ❚❛♥❣❡♥t❡ à P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✷✳✸✳✸ Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡t♦r❛ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹ ❆ ❯t✐❧✐③❛çã♦ ❞❡ ▼❛t❡r✐❛✐s ❈♦♥❝r❡t♦s ♥♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ✺✻

✺ ❙✐♥✉❝❛s ❞❛s ❈ô♥✐❝❛s ✺✾

✺✳✶ ❆ ❈♦♥❢❡❝çã♦ ❞❛s ❙✐♥✉❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷ ❙✐♥✉❝❛ ❊❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✸ ❙✐♥✉❝❛ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✹ ❙✐♥✉❝❛ P❛r❛❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✸

✶✶

■♥tr♦❞✉çã♦

❆ ♣r❡s❡♥t❡ ❞✐ss❡rt❛çã♦✱ ❞✐✈✐❞✐❞❛ ❡♠ s❡✐s ❝❛♣ít✉❧♦s✱ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛♣r❡✲ s❡♥t❛r ♦ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ ✏❙✐♥✉❝❛s ❞❛s ❈ô♥✐❝❛s✑ ❝♦♠♦ r❡❝✉rs♦ ❞✐❞át✐❝♦ ♣❛r❛ ❢❛✈♦r❡❝❡r ♦ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❝ô♥✐❝❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♣r♦♣r✐❡❞❛❞❡ r❡✢❡t♦r❛✳ ❙❡❣✉♥❞♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦❛♥✐s ✭P❈◆✬s✮✿

❬✳✳✳❪ ❛ ▼❛t❡♠át✐❝❛ é ❝♦♠♣♦♥❡♥t❡ ✐♠♣♦rt❛♥t❡ ♥❛ ❝♦♥str✉çã♦ ❞❛ ❝✐❞❛❞❛♥✐❛✱ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ❛ s♦❝✐❡❞❛❞❡ s❡ ✉t✐❧✐③❛✱ ❝❛❞❛ ✈❡③ ♠❛✐s✱ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❝✐❡♥✲ tí✜❝♦s ❡ r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s✱ ❞♦s q✉❛✐s ♦s ❝✐❞❛❞ã♦s ❞❡✈❡♠ s❡ ❛♣r♦♣r✐❛r✳ ❆ ♠❛t❡♠át✐❝❛ ♣r❡❝✐s❛ ❡st❛r ❛♦ ❛❧❝❛♥❝❡ ❞❡ t♦❞♦s ❡ ❛ ❞❡♠♦❝r❛t✐③❛çã♦ ❞♦ s❡✉ ❡♥s✐♥♦ ❞❡✈❡ s❡r ♠❡t❛ ♣r✐♦r✐tár✐❛ ❞♦ tr❛❜❛❧❤♦ ❞♦❝❡♥t❡✳ ❆ ❛t✐✈✐❞❛❞❡ ♠❛t❡♠át✐❝❛ ♥ã♦ é ♦❧❤❛r ♣❛r❛ ❝♦✐s❛s ♣r♦♥t❛s ❡ ❞❡✜♥✐t✐✈❛s✱ ♠❛s ❛ ❝♦♥str✉çã♦ ❡ ❛ ❛♣r♦♣r✐❛çã♦ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❡❧♦ ❛❧✉♥♦✱ q✉❡ s❡ s❡r✈✐rá ❞❡❧❡ ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❡ tr❛♥s❢♦r♠❛r s✉❛ r❡❛❧✐❞❛❞❡✳ ✭❇❘❆❙■▲✱ ✶✾✾✼✱ ♣✳ ✶✾✮✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦✱ s❡rá ❛♣r❡s❡♥t❛❞♦ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❞♦ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞❛s ❈ô♥✐❝❛s ♥❛ ❝♦♥❤❡❝✐❞❛ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛✱ s❡❣✉✐❞♦ ❞❛ ❡①♣❧❛♥❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s ❞❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛ ✉t✐❧✐③❛❞♦ ♣♦r ❆♣♦❧ô♥✐♦✳

◆♦ ❝❛♣ít✉❧♦ ✷✱ s❡rã♦ ✈❡r✐✜❝❛❞♦s ♦s s✐♥t♦♠❛s q✉❡ ❆♣♦❧ô♥✐♦ ✉s♦✉ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ❡ ❞❛r ♥♦♠❡s às s❡çõ❡s ❝ô♥✐❝❛s✿ ❡❧✐♣s❡✱ ❤✐♣ér❜♦❧❡ ❡ ♣❛rá❜♦❧❛✳ ▲♦❣♦ ❞❡♣♦✐s✱ ❞✐s❝✉t✐r✲s❡✲ã♦ ♦s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✱ ◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥✳ ❖s ❞♦✐s ♣r✐♠❡✐r♦s ❞❡♠♦♥str❛r❛♠ ❛ ❡❧✐♣s❡ ❡ ❛ ❤✐♣ér❜♦❧❡ ♣♦r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❜✐❢♦❝❛❧✱ ❡ ♦ ú❧t✐♠♦ ❞❡♠♦♥str♦✉ ❛ ♣❛rá❜♦❧❛ ♣♦r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❢♦❝♦ ❡ ❞✐r❡tr✐③✳ ❋✐♥❛❧✐③❛♥❞♦ ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ r❡❧❛çã♦ ❋♦❝♦✱ ❉✐r❡tr✐③ ❡ ❊①❝❡♥tr✐❝✐❞❛❞❡✳

❊♠ s❡❣✉✐❞❛✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❛s ❈ô♥✐❝❛s s❡rã♦ ❛❜♦r❞❛❞❛s ❞❡ ❢♦r♠❛ ❣❡♦♠étr✐❝❛ ❡ ❛♥❛❧ít✐❝❛ ♥♦ ♣❧❛♥♦✱ ❧❡✈❛♥t❛r✲s❡✲ã♦ ♦s tó♣✐❝♦s✿ ❡s❜♦ç♦ ❞❛s ❝✉r✈❛s✱ ♦s ❡❧❡♠❡♥t♦s✱ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❞✐â✲ ♠❡tr♦✱ ♣r♦♣r✐❡❞❛❞❡ ❞❛ r❡t❛ ♠❡❞✐❛tr✐③ ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ r❡✢❡t♦r❛✳ ◆❛ ❆❜♦r❞❛❣❡♠ ❆♥❛❧ít✐❝❛✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s q✉❡ ❋❡r♠❛t ✉t✐❧✐③♦✉ ♣❛r❛ ♦❜t❡r ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✳

◆♦ ❝❛♣ít✉❧♦ ✹✱ s❡rá ♠♦str❛❞♦ ❝♦♠♦ ❢♦✐ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥❢❡❝çã♦ ❞♦ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ ✏❙✐♥✉❝❛s ❞❛s ❈ô♥✐❝❛s✑ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡♥s✐♥❛r ❛ ♣r♦♣r✐❡❞❛❞❡ r❡✢❡①✐✈❛ q✉❡ ❡ss❛s ❝✉r✈❛s ♣♦ss✉❡♠✱ ❛❧é♠ ❞❡ ✐❞❡♥✜❝❛r ♦s s❡✉s ❡❧❡♠❡♥t♦s✳

✶✷

✶ ❆ ❍✐stór✐❛ ❞❛s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦

❞❡ P❡r❣❛

✶✳✶ ❆ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛

❆ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛✱ ❞❡ ✸✵✵ ❛ ✷✵✵ ❛✳ ❈✳✱ ❞❡✈❡✲s❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ três ❣r❛♥❞❡s ♠❛t❡♠át✐❝♦s ✕ ❊✉❝❧✐❞❡s✱ ❆rq✉✐♠❡❞❡s ❡ ❆♣♦❧ô♥✐♦✳ ❊st❡✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✱ ✈✐✈❡✉ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❡♥tr❡ ✷✻✷ ❡ ✶✾✵ ❛✳ ❈✳ P♦✉❝♦ s❡ s❛❜❡ s♦❜r❡ s✉❛ ✈✐❞❛✱ ❛♣❡♥❛s q✉❡ ❢♦✐ ✉♠ ❝♦♥t❡♠♣♦râ♥❡♦ ❡ r✐✈❛❧ ❞❡ ❆rq✉✐♠❡❞❡s✳ ❊ss❛ r✐✈❛❧✐❞❛❞❡ ❡r❛ ❡✈✐❞❡♥t❡ ❡♠ ✈ár✐♦s ❛ss✉♥t♦s ♠❛t❡♠át✐❝♦s ❞✐s❝✉t✐❞♦s ❞❛ é♣♦❝❛ ✭❇❖❨❊❘✱ ✷✵✵✸✮✳

▼✉✐t❛s ❞❛s ♦❜r❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡s❛♣❛r❡❝❡r❛♠ ♣❛r❝✐❛❧ ♦✉ t♦t❛❧♠❡♥t❡✱ ❞❡♥tr❡ ❡❧❛s✿ s♦❜r❡ ❉✐✈✐❞✐r ❡♠ ✉♠❛ r❛③ã♦✱ ❈♦rt❛r ✉♠❛ ár❡❛✱ ❙❡❝çã♦ ❞❡t❡r♠✐♥❛❞❛✱ ❚❛♥❣ê♥❝✐❛s ✭♦✉ ❈♦♥t❛t♦s✮✱ ■♥❝❧✐♥❛çõ❡s ❡ s♦❜r❡ ▲✉❣❛r❡s ♣❧❛♥♦s✳ ❊ss❛s ♦❜r❛s ❡ ♦✉tr❛s ❡st❛✈❛♠ ❥✉♥t❛s ❝♦♠ ❞♦✐s tr❛t❛❞♦s ❞❡ ❊✉❝❧✐❞❡s ✭❞❡ ✸✷✺ ❛ ✷✺✻ ❛✳ ❈✮ ♥✉♠❛ ❝♦❧❡çã♦ ❝❤❛♠❛❞❛ ✏❚❡s♦✉r♦ ❞❛ ❆♥á❧✐s❡✑✱ ❞❡s❝r✐t❛ ♣♦st❡r✐♦r♠❡♥t❡ ♣❡❧♦ ♠❛t❡♠át✐❝♦ P❛♣♣✉s ✭❞❡ ✸✺✵ ❛ ✷✾✵ ❛✳ ❈✮✳ ❆♣♦❧ô♥✐♦ ❢♦✐ ♦ ♣r❡❝✉rs♦r ❞♦ q✉❡ ❝❤❛♠❛♠♦s ❤♦❥❡ ❞❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛❀ ❢♦✐✱ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞❡ss❡ ♣✐♦♥❡✐r✐s♠♦✱ ❡♥tã♦ ❛♣❡❧✐❞❛❞♦ ❞❡ ❖ ●r❛♥❞❡ ●❡ô♠❡tr❛ ✭❇❖❨❊❘✱ ✷✵✵✸✮✳

❉❡♥tr❡ ❛s ♦❜r❛s ❞❡ ❆♣♦❧ô♥✐♦✱ ✉♠❛ ♣r❡s❡r✈♦✉✲s❡ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡✱ ❆s ❈ô♥✐❝❛s✱ s✉❛ ♦❜r❛ ♣r✐♠❛✳ ❉❡ss❛ ♦❜r❛✱ ❞✐s♣õ❡♠✲s❡ ❛♣❡♥❛s ❞❡ q✉❛tr♦ ❞♦ t♦t❛❧ ❞❡ ♦✐t♦ ❧✐✈r♦s q✉❡ ❛ ❝♦♠✲ ♣✉♥❤❛♠✳ ❚✐♥❤❛✲s❡ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s s❡çõ❡s ❝ô♥✐❝❛s ❝❡r❝❛ ❞❡ ✉♠ sé❝✉❧♦ ❡ ♠❡✐♦ ❛♥t❡s ❞❡ ❆♣♦❧ô♥✐♦ ❡s❝r❡✈❡r s♦❜r❡ ❡ss❛s ❝✉r✈❛s✱ t♦❞❛✈✐❛ s❡✉ tr❛❜❛❧❤♦ s✉❜st✐t✉✐✉ t❡①t♦s ❛♥t❡r✐♦r❡s ♥ã♦ ❛❝❛❜❛❞♦s ♦✉ ♣♦✉❝♦s ♠❡♥♦s r✐❣♦r♦s♦s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦✳ ❊ss❛ ♠❡s♠❛ s✉❜st✐t✉✐çã♦ ❞❡ t❡①t♦s ❛❝♦♥t❡❝❡✉ s❡♠❡❧❤❛♥t❡ ❛♦ s✉r❣✐♠❡♥t♦ ❞❡ ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s ✭❇❖❨❊❘✱ ✷✵✵✸✮✳

❆♥t❡s ❞❡ss❡ ♠❛t❡♠át✐❝♦✱ ❛s ❝ô♥✐❝❛s ❡r❛♠ ♦❜t✐❞❛s ❝♦♠♦ s❡çõ❡s ❞❡ três t✐♣♦s ❜❡♠ ❞✐✲ ❢❡r❡♥t❡s ❞❡ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ♣♦r ✉♠ ♣❧❛♥♦✱ ❝♦♥❢♦r♠❡ ♦ â♥❣✉❧♦ ❞♦ ♣❧❛♥♦ ❝♦♠ ♦ ❡✐①♦ ❞♦

❆ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛ ✶✹

❝♦♥❡✳ ❆♣♦❧ô♥✐♦ ❣❡♥❡r❛❧✐③♦✉ q✉❡ ♥ã♦ s❡r ♥❡❝❡ssár✐♦ t♦♠❛r s❡çõ❡s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❡❀ q✉❡ ❞❡ ✉♠ ú♥✐❝♦ ❝♦♥❡ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s t♦❞❛s ❛s s❡çõ❡s ❝ô♥✐❝❛s ❡ q✉❡ ♦ ❝♦♥❡ ♥ã♦ ♣r❡❝✐s❛ s❡r r❡t♦✳ ❆✐♥❞❛✱ s✉❜st✐t✉✐✉ ♦ ❝♦♥❡ ❞❡ ✉♠❛ só ❢♦❧❤❛ ♣♦r ✉♠ ❞✉♣❧♦ ❡ ❞❡✉ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❡ ❝✐r❝✉❧❛r ✉s❛❞❛ ❛té ❤♦❥❡✿

❙❡ ✜③❡r ✉♠❛ r❡t❛✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✐♥❞❡✜♥✐❞♦ q✉❡ ♣❛ss❛ s❡♠♣r❡ ♣♦r ✉♠ ♣♦♥t♦ ✜①♦ ✭❱ért✐❝❡✮✱ ♠♦✈❡r✲s❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ✭❜❛s❡ ❞♦ ❝♦♥❡✮ q✉❡ ♥ã♦ ❡stá ♥✉♠ ♠❡s♠♦ ♣❧❛♥♦ ❝♦♠ ♦ ♣♦♥t♦ ❞❡ ♠♦❞♦ ❛ ♣❛ss❛r s✉❝❡ss✐✈❛♠❡♥t❡ ♣♦r ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❛ r❡t❛ ♠ó✈❡❧ ❞❡s❝r❡✈❡rá ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ ❝♦♥❡ ❞✉♣❧♦✳ ✭❇❖❨❊❘✱ ✷✵✵✸✮

❊ss❛s ❣❡♥❡r❛❧✐③❛çõ❡s ✜③❡r❛♠ ❞❛ ❤✐♣ér❜♦❧❡ ❛ ❝✉r✈❛ ❞❡ ❞♦✐s r❛♠♦s ❝♦♠♦ é ❝♦♥❤❡❝✐❞❛ ❛t✉❛❧♠❡♥t❡✳

◆♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦ ❧❡✐❛✿ r = ←→AB ❛ r❡t❛ r q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s A ❡ B❀ ✉♠

♣♦♥t♦A✱ s♦❜r❡ ✉♠❛ r❡t❛ r✱ ❛ ❞✐✈✐❞❡ ❡♠ ❞♦✐s ♣❡❞❛ç♦s ❝❤❛♠❛❞♦s s❡♠✐rr❡t❛s✱ s❡ B ❡ C sã♦

♣♦♥t♦s s♦❜r❡ ❝❛❞❛ ✉♠ ❞❡ t❛✐s ♣❡❞❛ç♦s✱ ♣♦❞❡✲s❡ ❞❡♥♦t❛r ❛s s❡♠✐rr❡t❛s ❞❡ ♦r✐❣❡♠ A ♣♦r

−→

AB ❡ −→AC❀AB ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ é ❛ ♣♦rçã♦ ❞❛ r❡t❛ r q✉❡ ❛ ❝♦♥té♠✱ s✐t✉❛❞❛ ❞❡A ❛ B❀AB ♦✉ ❞(A, B)❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦sA ❡B ♦✉ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AB❀

❞(P, r) ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ P à r❡t❛r❀ABC ♦ tr✐â♥❣✉❧♦ ❞❡ ✈ért✐❝❡s A✱ B ❡ C❀∠_AOB ♦

â♥❣✉❧♦ ❞❡ ❧❛❞♦s−→OA ❡ −−→OB❀ AOBb ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦∠_AOB❀ _{A1A2· · ·An} ♦ ♣♦❧í❣♦♥♦ ❞❡

✈ért✐❝❡sA1✱ A2✱ · · ·✱ An✳

❉❡ss❛ ❢♦r♠❛✱ ❞❛❞♦ ✉♠ ❝♦♥❡ ❞❡ ❞✉♣❧❛ ❢♦❧❤❛ ❝♦♠ ✈ért✐❝❡ V ❝♦rt❛❞♦ ♣♦r ✉♠ ♣❧❛♥♦ q✉❡

❝♦♥té♠ ♦ ❡✐①♦✳ ◆♦ r❛♠♦ ✐♥❢❡r✐♦r✱ ❞❡t❡r♠✐♥❛ ♦ ❞✐â♠❡tr♦AB ❡ ♦ tr✐â♥❣✉❧♦V AB✱ ❝❤❛♠❛❞♦

tr✐â♥❣✉❧♦ ❛①✐❛❧✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✶✳✶✿

❋✐❣✉r❛ ✶✳✶✿ ❊❧❡♠❡♥t♦s ❞♦ ❈♦♥❡

❆ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛ ✶✺

❆s ❝ô♥✐❝❛s ♣❛ss❛♠ ❛ s❡r ✐♥t❡r♣r❡t❛❞❛s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❝♦♠♦ ❛s ✐♥t❡rs❡çõ❡s ❞❡st❡ ❝♦♥❡ ♣♦r ♣❧❛♥♦s q✉❡ ❝♦rt❛♠ ♦ s❡❣♠❡♥t♦ AB ♦✉ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ AB s♦❜r❡ ❛ r❡t❛ r ✭♣❡r✲

♣❡♥❞✐❝✉❧❛r ❛AB✮✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✶✳✷✿

❋✐❣✉r❛ ✶✳✷✿ ❙❡çõ❡s ❈ô♥✐❝❛s

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r ←→

P X é ❛ ✐♥t❡rs❡çã♦ ❞♦ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ ♦ tr✐â♥❣✉❧♦ ❛①✐❛❧ ❝♦♠ ♦ ♣❧❛♥♦ q✉❡ ❝♦rt❛ ♦

s❡❣♠❡♥t♦AB ♦✉ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ AB s♦❜r❡ ❛ r❡t❛r✳ ▲♦❣♦✱ ❛ ❝ô♥✐❝❛ s❡rá ✉♠❛✿

• ❊❧✐♣s❡✱ s❡ ←→P X ✐♥t❡rs❡❝t❛ ♦s ❞♦✐s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧❀

❆s ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ✶✻

• ❍✐♣ér❜♦❧❡✱ s❡ ←→P X ✐♥t❡rs❡❝t❛ ✉♠ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧ ❡ ♦ ♣r♦❧♦❣❛♠❡♥t♦ ❞♦

♦✉tr♦ ✭▲❖P❊❙✱ ✷✵✶✶✮✳

❯♠ r❡❝✉rs♦ ✐♠♣♦rt❛♥t❡ ✉s❛❞♦ ♣♦r ❆♣♦❧ô♥✐♦✱ ❡♠♣r❡❣❛❞♦ ♣♦r ❛♥t❡❝❡ss♦r❡s✱ é ❛ ❝❛r❛❝✲ t❡r✐③❛çã♦ ❞❛s ❝ô♥✐❝❛s ♣♦r ♠❡✐♦ ❞❡ ✉♠ s✐♥t♦♠❛ q✉❡ é ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❣r❛♥❞❡③❛s q✉❡ ❝❛r❛❝t❡r✐③❛ ♦s ♣♦♥t♦s q✉❡ ❡stã♦ s♦❜r❡ ❛ ❝ô♥✐❝❛ ✭❘❖◗❯❊❀ ❈❆❘❱❆▲❍❖✱ ✷✵✶✷✮✳ ❊♠ s✉♠❛✱ ♠❛r❝❛✲s❡ ✉♠ ♣♦♥t♦ K s♦❜r❡ ❛ s❡çã♦ ❡ ❢❛③✲s❡ ♣❛ss❛r ♣♦r ❡st❡ ✉♠ ♣❧❛♥♦ ♣❛r❛❧❡❧♦ à ❜❛s❡

✭▲❖P❊❙✱ ✷✵✶✶✮✳

❆s r❡❢❡r✐❞❛s ♥♦♠❡♥❝❧❛t✉r❛s ✭❡❧✐♣s❡✱ ♣❛rá❜♦❧❛✱ ❤✐♣ér❜♦❧❡✮ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣♦r ❡❧❡ q✉❛♥❞♦ ♦❜s❡r✈♦✉ ❛s s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s ♣♦r ❛♣❧✐❝❛çõ❡s ❞❡ ár❡❛s ❞❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛ ✭❇❖❨❊❘✱ ✷✵✵✸✮✳ ❙❡❣✉♥❞♦ ❘♦q✉❡ ❡ ❈❛r✈❛❧❤♦ ✭✷✵✶✷✱ ♣✳✶✻✶✮✿ ✏❛ ❛♣❧✐❝❛çã♦ ❡①❛t❛ é ❝❤❛♠❛❞❛ ❞❡ ❵♣❛rá❜♦❧❛✬✱ ❛♣❧✐❝❛çã♦ ❝♦♠ ❢❛❧t❛ é ✉♠❛ ❵❡❧✐♣s❡✬ ❡ ❛♣❧✐❝❛çã♦ ❝♦♠ ❡①❝❡ss♦ é ✉♠❛ ❵❤✐♣ér❜♦❧❡✬✧✳

✶✳✷ ❆s ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s

❖ ♠ét♦❞♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ✉♠❛ ♣♦❧✐❣♦♥❛❧ ❛ ✉♠ s❡❣♠❡♥t♦ ❞❛❞♦✱ ✐st♦ é✱ ❝♦♥s✐st❡ ❡♠ ❝♦♥str✉✐r ✉♠❛ ♦✉tr❛ ♣♦❧✐❣♦♥❛❧ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ s❡❥❛ ✉♠ ❞❡ s❡✉s ❧❛❞♦s✳ ❊♠ s✉♠❛✱ ❡①✐❣❡✲s❡ q✉❡ ❛ ♣♦❧✐❣♦♥❛❧ ❝♦♥str✉í❞❛ t❡♥❤❛ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s✳

P♦r ❡①❡♠♣❧♦✱ ❞❛❞♦ ABCDEF ✉♠ ♣♦❧í❣♦♥♦ ❡ XY ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✳ ❆♣❧✐❝❛r ✉♠

♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ ár❡❛ ✐❣✉❛❧ à ❞❡ABCDEF ❛♦ s❡❣♠❡♥t♦ XY✱ ❝♦♥s✐st❡ ❡♠ ❝♦♥str✉✐r ✉♠

♣❛r❛❧❡❧♦❣r❛♠♦ XY ZW ❡♠ q✉❡ XY é ✉♠ ❞♦s ❧❛❞♦s✱ ❡ ❝✉❥❛ ár❡❛ s❡❥❛ ✐❣✉❛❧ à ár❡❛ ❞❡ ABCDEF✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✶✳✸✿

❋✐❣✉r❛ ✶✳✸✿ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s

A

B C D E

F

X Y

W Z

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

❆s ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ✶✼

❛♣❧✐❝❛❞♦ é ✉♠ r❡tâ♥❣✉❧♦✳ ◆❛ ●ré❝✐❛✱ ✉s❛✈❛✲s❡ três t✐♣♦s ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s✿ ❛♣❧✐❝❛çã♦ ♣❛r❛❜ó❧✐❝❛❀ ❛♣❧✐❝❛çã♦ ❡❧í♣t✐❝❛ ❡ ❛♣❧✐❝❛çã♦ ❤✐♣❡r❜ó❧✐❝❛ ✭❘❖◗❯❊❀ ❈❆❘❱❆▲❍❖✱ ✷✵✶✷✮✳

✶✳✷✳✶ ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s P❛r❛❜ó❧✐❝❛

❆ ❛♣❧✐❝❛çã♦ ♣❛r❛❜ó❧✐❝❛ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦XY ZW ❝♦♠ ár❡❛ ✐❣✉❛❧

à ❞❡ ✉♠❛ ✜❣✉r❛ S ❛ ✉♠ s❡❣♠❡♥t♦ XY✱ ❝♦♠ ✉♠ â♥❣✉❧♦ ∠ABC ❡s♣❡❝✐✜❝❛❞♦✱ ❝♦♥❢♦r♠❡

✜❣✉r❛ ✶✳✹✿

❋✐❣✉r❛ ✶✳✹✿ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s P❛r❛❜ó❧✐❝❛

S A

B C

X Y

W Z

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

✶✳✷✳✷ ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ❊❧í♣t✐❝❛

❯♠❛ ❛♣❧✐❝❛çã♦ ❡❧í♣t✐❝❛ ♦✉ ❝♦♠ ❢❛❧t❛ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ XY′

Z′

W

❝♦♠ â♥❣✉❧♦s ❞❛❞♦s ✭â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦P✮ ❡ ár❡❛ ✐❣✉❛❧ à ❞❡ ✉♠ ♣♦❧í❣♦♥♦

❞❛❞♦ S ❛ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ XY✳ ❉❡ ♠❛♥❡✐r❛ q✉❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ Y′

Y ZZ′

✱ q✉❡ ❢❛❧t❛ ♣❛r❛ ❝♦♠♣❧❡t❛r ❛ ✜❣✉r❛ ❡ ❝♦❜r✐r t♦❞♦ ♦ s❡❣♠❡♥t♦ XY✱ s❡❥❛ ✉♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦

s❡♠❡❧❤❛♥t❡ ❛♦ P✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ Y′

Y ZZ′

é ♦ q✉❡ ❢❛❧t❛ ♣❛r❛ q✉❡

❆s ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ✶✽

❋✐❣✉r❛ ✶✳✺✿ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s ❊❧í♣t✐❝❛

S

P

Z′ _Z

Y Y′ X

W

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

✶✳✷✳✸ ❆♣❧✐❝❛çõ❡s ❞❡ ➪r❡❛s ❍✐♣❡r❜ó❧✐❝❛

❯♠❛ ❛♣❧✐❝❛çã♦ ❤✐♣❡r❜ó❧✐❝❛ ♦✉ ❝♦♠ ❡①❝❡ss♦ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦

XY′

Z′

W ❝♦♠ â♥❣✉❧♦s ❞❛❞♦s ✭â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦P✮ ❡ ár❡❛ ✐❣✉❛❧ à ❞❡ ✉♠

♣♦❧í❣♦♥♦ ❞❛❞♦S ❛ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ XY✳ ❉❡ ♠❛♥❡✐r❛ q✉❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ Y Y′

Z′

Z✱

q✉❡ ❡①❝❡❞❡ ♣❛r❛ ❝♦♠♣❧❡t❛r ❛ ✜❣✉r❛✱ s❡❥❛ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ s❡♠❡❧❤❛♥t❡ ❛♦P✳ ❊♠ ♦✉tr❛s

♣❛❧❛✈r❛s✱ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ Y Y′

Z′

Z é ♦ ❡①❝❡ss♦ ♣❛r❛ q✉❡ XY ZW ❡st❡❥❛ ❛♣❧✐❝❛❞♦ ❛ XY✱

❝♦♥❢♦r♠❡ ✜❣✉r❛ ✶✳✻✿

❋✐❣✉r❛ ✶✳✻✿ ❆♣❧✐❝❛çã♦ ❞❡ ➪r❡❛s ❍✐♣❡r❜ó❧✐❝❛

S

P

Z Z′

Y′ Y X

W

❆ ❛♣❧✐❝❛çã♦ ❞❛s ❝ô♥✐❝❛s ❡♠ ❢❡♥ô♠❡♥♦s ❞❛ ♥❛t✉r❡③❛ ✶✾

✶✳✸ ❆ ❛♣❧✐❝❛çã♦ ❞❛s ❝ô♥✐❝❛s ❡♠ ❢❡♥ô♠❡♥♦s ❞❛ ♥❛t✉r❡③❛

❆s ❞❡✜♥✐çõ❡s ❞❡ r❡t❛ t❛♥❣❡♥t❡ ❡ ♥♦r♠❛❧ ❛ ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❝ô♥✐❝❛ ❥á ❡r❛♠ ❝♦♥❤❡❝✐❞❛s ❡ ❛♣r❡s❡♥t❛❞❛s ♣♦r ❆♣♦❧ô♥✐♦ ♥♦ tr❛❜❛❧❤♦ r❡❧❛t✐✈♦ ❛ ❘❡t❛s ▼á①✐♠❛s ❡ ▼í♥✐♠❛s✱ ♣r❡s❡♥t❡ ♥♦ ❧✐✈r♦ ❱✳ ❊♠❜♦r❛ ❝♦♥❤❡❝✐❞❛s✱ ♥ã♦ t✐♥❤❛ ♣❡rs♣❡❝t✐✈❛s ❡♠ ❛♣❧✐❝á✲❧❛s✳ ❍♦❥❡✱ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ss❛ ❞❡✜♥✐çã♦ é ❡①t❡♥s❛ ✭❇❖❨❊❘✱ ✷✵✵✸✮✳ ❋♦❝♦ ❡ ❡①❝❡♥tr✐❝✐❞❛❞❡ ❡r❛♠ ♥♦♠❡s q✉❡ t❛♠❜é♠ ♥ã♦ ❢❛③✐❛♠ ♣❛rt❡ ❞❛ ♦❜r❛❀ ♣♦ré♠✱ ♦ ♠❛t❡♠át✐❝♦ ❢❛③✐❛ r❡❢❡rê♥❝✐❛s ✐♥❞✐r❡t❛✲ ♠❡♥t❡ ✭s❡♠ ✉s❛r ❛s ♥♦♠❡♥❝❧❛t✉r❛s ❛t✉❛✐s✮ ❡♠ ❞✐✈❡rs♦s t❡♦r❡♠❛s✳ ❈♦♥❢♦r♠❡ ❇♦②❡r ✭✷✵✵✸✮✿ é ♣♦ssí✈❡❧ q✉❡ ❛❧❣✉♠❛s ♦✉ t♦❞❛s ❡ss❛s ♦♠✐ssõ❡s s♦♠❛❞❛s r❡s✉❧t❡♠ ❞♦ ❢❛t♦ ❞❡ t❡r❡♠ s✐❞♦ tr❛t❛❞❛s ❡♠ ❛❧❣✉♠ ❧✉❣❛r✱ ❡♠ ♦❜r❛s ♣❡r❞✐❞❛s ❞❡ ❆♣♦❧ô♥✐♦ ♦✉ ♦✉tr♦s ❛✉t♦r❡s✳

P♦r q✉❛s❡ ✷✵✵✵ ❛♥♦s✱ ❛s ❝ô♥✐❝❛s ❡r❛♠ t✐❞❛s ❝♦♠♦ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❜str❛t♦ s❡♠ ❛♣❧✐❝❛çõ❡s ♥♦s ❢❡♥ô♠❡♥♦s ♥♦ ♠✉♥❞♦ r❡❛❧✱ ❛té q✉❡✱ ❝♦♠ ♦s ❡st✉❞♦s ❛♣❧✐❝❛❞♦s à ❛str♦♥♦♠✐❛ ❞❡ ❏♦❤❛♥♥❡s ❑❡♣❧❡r ✭✶✺✼✶✲✶✻✸✵✮✱ ❡♠ ✶✻✵✾✱ ❞❡s❝♦❜r✐✉✲s❡ ❛s ór❜✐t❛s ❡❧í♣t✐❝❛s ❞❡s❝r✐t❛s ♣❡❧♦s ♣❧❛♥❡t❛s✳ ❑❡♣❧❡r ❝♦♥tr✐❜✉✐ ❝♦♠ ♥♦✈❛s ❛❜♦r❞❛❣❡♥s s♦❜r❡ ❛s ❝ô♥✐❝❛s ✭♦ t❡r♠♦ ❢♦❝♦ ❢♦✐ ❡♠♣r❡❣❛❞♦ ♣♦r ❡❧❡ ♥❡ss❡ ♣❡rí♦❞♦✮ ❡ ❡ss❡s r❡s✉❧t❛❞♦s ❛❜r✐r❛♠ ❝❛♠✐♥❤♦s ♣❛r❛ s✉❛s ❞✉❛s ❧❡✐s ❞❛ ❛str♦♥♦♠✐❛ ❝♦♥❤❡❝✐❞❛s ❛té ❤♦❥❡✿ ✶ ✲ ♦s ♣❧❛♥❡t❛s ❞❡s❝r❡✈❡♠ ór❜✐t❛s ❡❧í♣t✐❝❛s ❡♠ t♦r♥♦ ❞♦ ❙♦❧✱ ❝♦♠ ♦ ❙♦❧ ♦❝✉♣❛♥❞♦ ✉♠ ❞♦s ❢♦❝♦s✱ ❡ ✷ ✲ ♦ r❛✐♦ ✈❡t♦r q✉❡ ✉♥❡ ✉♠ ♣❧❛♥❡t❛ ❛♦ ❙♦❧ ✈❛rr❡ ár❡❛s ✐❣✉❛✐s ❡♠ t❡♠♣♦s ✐❣✉❛✐s ✭❇❖❨❊❘✱ ✷✵✵✸✮✳

❋✐❣✉r❛ ✶✳✼✿ ❛✮ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ tr❛❥❡tór✐❛ ❡❧í♣t✐❝❛ ❞❡ ✉♠ ♣❧❛♥❡t❛ ❛♦ r❡❞♦r ❞♦ ❙♦❧✳ ❜✮ ❊♠ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦ ✐❣✉❛✐s✱ ❛s ár❡❛s ✏✈❛rr✐❞❛s✑ ♣❡❧♦ r❛✐♦ ✈❡t♦r q✉❡ ❧✐❣❛ ♦ ♣❧❛♥❡t❛ ❛♦ ❙❖▲ ✭❡♠ ✉♠ ❞♦s ❢♦❝♦s✮ t❛♠❜é♠ sã♦ ✐❣✉❛✐s✿ △_{t =t2−t1 =t4−t3}

❋♦♥t❡✿ ❙❆◆❚✬❆◆◆❆ ❡t✳ ❛❧✳✱ ✷✵✶✵✱ ♣✳ ✷✺✾

❆ ❛♣❧✐❝❛çã♦ ❞❛s ❝ô♥✐❝❛s ❡♠ ❢❡♥ô♠❡♥♦s ❞❛ ♥❛t✉r❡③❛ ✷✵

q✉❡ t♦❞♦s ♦s ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✱ ❞✐s❝✉t✐❞♦s ♣♦r ❆♣♦❧ô♥✐♦✱ ♣♦❞❡r✐❛♠ s❡r ♦❜t✐❞♦s ♥❛ ❢♦r♠❛ ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s ✭❉❊▲●❆❉❖❀ ❋❘❊◆❙❊▲❀ ❈❘■❙❙❆❋❋✱ ✷✵✶✸✮✳

❖ ♠❛t❡♠át✐❝♦ ❛♣❧✐❝♦✉ ❛ ♠❡s♠❛ ✐❞❡✐❛ ♥❛ ♦❜r❛ ❈ô♥✐❝❛s ❡✱ ❛tr❛✈és ❞❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❛s s❡çõ❡s ❝ô♥✐❝❛s✱ ♦❜t❡✈❡ s✉❛s ❡q✉❛çõ❡s✳ ❈♦♠ ♦s ❛✈❛♥ç♦s ❞♦s ❡st✉❞♦s ♥❡ss❛ ♥♦✈❛ ♣❡r♣❡❝t✐✈❛✱ ❋❡r♠❛t ❡st❛❜❡❧❡❝❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠ ♦s ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✱ ❞✐✈✐❞✐♥❞♦ ❛ ❢❛♠í❧✐❛ ❞❡ ❡q✉❛çõ❡s ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s ❛té ♦ ✷♦ _{❣r❛✉ ❡♠ ✼ s✉❜❢❛♠í❧✐❛s✿ r❡t❛✱ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛✱}

♣❛r ❞❡ r❡t❛s ❝♦♥❝♦rr❡♥t❡s✱ ♣❛rá❜♦❧❛✱ ❝ír❝✉❧♦✱ ❡❧✐♣s❡ ❡ ❤✐♣ér❜♦❧❡ ❛①✐❛❧✳ P♦❞❡♥❞♦ ❡ss❛s s✉❜❢❛♠í❧✐❛s s❡r❡♠ ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s q✉❡✱ ❡s❝r✐t❛ ♥❛ ❧✐♥❣✉❛❣❡♠ ❛t✉❛❧✱ é ❝♦♥❢♦r♠❡ ❡q✉❛çã♦ ✶✳✶✿

Ax2+Bxy+Cy2 +Dx+Ey+F = 0 ✭✶✳✶✮

❱❛r✐❛♥❞♦ ♦s ✈❛❧♦r❡s ❞♦s ❝♦❡✜❝✐❡♥t❡s ✭❉❊▲●❆❉❖❀ ❋❘❊◆❙❊▲❀ ❈❘■❙❙❆❋❋✱ ✷✵✶✸✮✳

❆♣ós q✉❛s❡ s❡ss❡♥t❛ ❛♥♦s ❞❛s ❞❡s❝♦❜❡rt❛s ❞❡ ❑❡♣❧❡r✱ ■s❛❛❝ ◆❡✇t♦♥ ✭✶✻✹✷✲✶✼✶✻✮ ♣r♦✈♦✉ q✉❡ ❛s ór❜✐t❛s ❡❧í♣t✐❝❛s ❞❡s❝r✐t❛s ♣❡❧♦s ♣❧❛♥❡t❛s sã♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ s✉❛ ▲❡✐ ❞❡ ❆tr❛çã♦ ❯♥✐✈❡rs❛❧ ❡✱ ❡♠ ✶✻✽✼✱ ♣✉❜❧✐❝♦✉ ❛ ❚❡♦r✐❛ ❞❛ ●r❛✈✐t❛çã♦ ❯♥✐✈❡rs❛❧✱ ❡①♣❧✐❝❛♥❞♦ ♦ s✐st❡♠❛ s♦❧❛r ✭❇❖❨❊❘✱ ✷✵✵✸✮✳

❊♠ ✶✽✷✷✱ ♦ ♠❛t❡♠át✐❝♦ ❜❡❧❣❛ ●❡r♠✐♥❛❧ P✐❡rr❡ ❉❛♥❞❡❧✐♥ ✭✶✼✾✹✲✶✽✹✼✮ ❝♦♠ ❛ ❛❥✉❞❛ ❞❡ s❡✉ ❛♠✐❣♦ ▲❛♠❜❡rt ❆❞♦❧♣❤ ❏❛❝q✉❡s ◗✉❡t❡❧❡t ✭✶✼✾✻✲✶✽✼✹✮✱ ❞❡♠♦♥str❛ ♦ ❚❡♦r❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ✉t✐❧✐③❛♥❞♦ ❡s❢❡r❛s ✐♥s❝r✐t❛s ♥♦ ❝♦♥❡✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❊s❢❡r❛s ❞❡ ❉❛♥❞❡❧✐♥✳ ❊ss❡ t❡♦r❡♠❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t✱ ❛✜r♠❛ q✉❡✿ ❉❛❞♦ ✉♠ ♣❧❛♥♦ q✉❡ s❡❝❝✐♦♥❛ ✉♠ ❝♦♥❡✱ ❡①✐st❡ ✉♠❛ ♦✉ ❞✉❛s ❡s❢❡r❛s q✉❡ sã♦ t❛♥❣❡♥t❡s ❛♦ ♣❧❛♥♦ ❡ ❛♦ ❝♦♥❡✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ t❡♠✲s❡ ✉♠❛ ♣❛rá❜♦❧❛ ❡✱ ♥♦ s❡❣✉♥❞♦✱ ✉♠❛ ❡❧✐♣s❡ ♦✉ ❤✐♣ér❜♦❧❡ ✭❙❖❯❙❆✱ ✷✵✶✹✮✳

✷ ❆s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❡ ♦s

❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡

P✐❡r❝❡ ▼♦rt♦♥

✷✳✶ ❖s ◆♦♠❡s ❞❛s ❈ô♥✐❝❛s

❖ ♠ét♦❞♦ ♣❡❧♦ q✉❛❧ ❆♣♦❧ô♥✐♦ ❛tr✐❜✉✐✉ ♦s ♥♦♠❡s às ❝ô♥✐❝❛s é ❜❛s❡❛❞♦ ♥♦ ♠ét♦❞♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ ár❡❛s q✉❡ ❢♦✐ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❛❞♦t❛❞❛ ♣❡❧♦ ❣r❛♥❞❡ ❣❡ô♠❡tr❛ ✭▲❖P❊❙✱ ✷✵✶✶✮✱ ❝♦♥❢♦r♠❡ ❛s s❡çõ❡s ❝ô♥✐❝❛s ❞❛ ✜❣✉r❛ ✷✳✶✿

❋✐❣✉r❛ ✷✳✶✿ ❙❡çõ❡s ❈ô♥✐❝❛s ❞❡✜♥✐❞❛s ♣♦r ❆♣♦❧ô♥✐♦

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

❖s ◆♦♠❡s ❞❛s ❈ô♥✐❝❛s ✷✷

❉❛s ❝♦♥str✉çõ❡s✱ tê♠✲s❡✿ ✭❛✮ AB é ❜❛s❡ ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧❀

✭❜✮ ♦ s❡❣♠❡♥t♦ CD ♣❡rt❡♥❝❡♥t❡ ❛♦ ♣❧❛♥♦ ❞❛ ❜❛s❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ AB ♦✉ ♦ ♣r♦❧♦♥❣❛✲

♠❡♥t♦ ❞❡AB❀

✭❝✮ ♦s s❡❣♠❡♥t♦s XY ✭♥❛ ❡❧✐♣s❡✮ ❡ P X ✭♥❛ ♣❛rá❜♦❧❛ ❡ ❤✐♣ér❜♦❧❡✮ sã♦ ❛ ✐♥t❡rs❡çã♦ ❞♦

tr✐â♥❣✉❧♦ ❛①✐❛❧ ❡ ❞♦ ♣❧❛♥♦ ❞❡ s❡çã♦❀

✭❞✮ ♦ s❡❣♠❡♥t♦V F✱ ♥❛ ❡❧✐♣s❡ ❡ ❤✐♣ér❜♦❧❡✱ é ♣❛r❛❧❡❧♦ ❛P X ❡ ❡♥❝♦♥tr❛ AB ♦✉ ♦ ♣r♦❧♦♥❣❛✲

♠❡♥t♦ ❞❡AB❀

✭❡✮ K é ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ ❝✉r✈❛ ❡ O é ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡K ❛♦ tr✐â♥✲

❣✉❧♦ ❛①✐❛❧✳ ❆ss✐♠✱KO é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ XY ❡ P X❀

✭❢✮ ♦ s❡❣♠❡♥t♦ A′

B′

é ♦ ❞✐â♠❡tr♦ ❞❛ ❝ír❝✉❧♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ✉♠ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ ♦ s❡❣♠❡♥t♦KO ❡ é ♣❛r❛❧❡❧♦ ❛ AB ❡♠ K✳ ❙❡❣✉❡ q✉❡✱ KO é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ A′

B′

❡♠ O❀

✭❣✮ ♦s ♣♦♥t♦sX ❡ Y sã♦ ❛s ✐♥t❡rs❡çõ❡s ❞❛s ❝✉r✈❛s ❝♦♠ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧❀

✭❤✮ XL é ✉♠ s❡❣♠❡♥t♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ P X✱ ♦♥❞❡ XL✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❛ s❡r ❞❡✜♥✐❞♦✱

❡stá ❝♦♥t✐❞♦ ♥♦ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❝✉r✈❛ q✉❡ ♣❛ss❛ ♣♦rX❀

✭✐✮ ♦ r❡tâ♥❣✉❧♦ XOSL❡stá s✐t✉❛❞♦ ♥✉♠ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❞❡ s❡çã♦❀

✭❥✮ ♦ ♣♦♥t♦ R é ❛ ✐♥t❡rs❡çã♦ ❞♦s s❡❣♠❡♥t♦s OS ❡ Y L ♦✉ ❞♦s ♣r♦❧♦♥❣❛♠❡♥t♦s ❞❡ss❡s ❞♦✐s

s❡❣♠❡♥t♦s✳

❖ s❡❣♠❡♥t♦ XL✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏▲❛❝t✉s ❊r❡t✉♠✑✱ s❡rá r❡s♣♦♥sá✈❡❧ ♣♦r ❝❛r❛❝t❡r✐③❛r

❛ ❝✉r✈❛✳ ➱ ✉♠ ♣❛râ♠❡tr♦ q✉❡ ❆♣♦❧ô♥✐♦ ❞❡✜♥✐✉ ❡♠ ❢✉♥çã♦ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧ ❡ ❞♦s s❡❣♠❡♥t♦sV X ❡ V F✱ ❞❡ ♠❛♥❡✐r❛ q✉❡✿

✶✳ q✉❛♥❞♦ ❛ s❡çã♦ ❢♦r ❡❧✐♣s❡ ♦✉ ✉♠❛ ❤✐♣ér❜♦❧❡✿

XL= AF ×F B

V F2 ×

XY _⇔ XL XY =

AF _×F B

V F2 ✭✷✳✶✮

✷✳ q✉❛♥❞♦ ❢♦r ✉♠❛ ♣❛rá❜♦❧❛✿

XL= AB

2

V A_×V B ×V X ⇔ XL V X =

AB2

V A_×V B ✭✷✳✷✮

❆♣♦❧ô♥✐♦ ❡s❝r❡✈❡✉ ❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛ ❞❛s três ❝✉r✈❛s r❡❧❛❝✐♦♥❛♥❞♦ ♦ q✉❡ s❡ ♣♦❞❡ ✐♠❛❣✐♥❛r ❝♦♠♦ ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠ ♣♦♥t♦ K q✉❛❧q✉❡r ❞❛ s❡çã♦ ❝♦♠ ♦ ♣❛râ♠❡tr♦ XL✱

✐st♦ é✱ ❞❡t❡r♠✐♥♦✉ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❝❛❞❛ ❝ô♥✐❝❛ ♣♦r ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦s s❡❣♠❡♥t♦s

KO✱XO ❡ XL ❡✱ ❞❛í✱ ❛s ✐♥t❡r♣r❡t❛çõ❡s ❣❡♦♠étr✐❝❛s ❞❡r❛♠ ♦r✐❣❡♠ ❛♦s ♥♦♠❡s ❞❛s s❡çõ❡s

❝ô♥✐❝❛s ✭▲❖P❊❙✱ ✷✵✶✶✮✳

❙❛❜❡✲s❡ q✉❡ A′

B′

❖s ◆♦♠❡s ❞❛s ❈ô♥✐❝❛s ✷✸

é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ s❡❣♠❡♥t♦ A′

B′

✱ ♣♦✐s KO é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ tr✐â♥❣✉❧♦ ❛①✐❛❧✳ P❡❧❛

♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ s❡❣✉❡✲s❡ q✉❡✿

KO2 =A′_O

×OB′ ✭✷✳✸✮

❊ss❛ r❡❧❛çã♦ é ✈á❧✐❞❛ ♣❛r❛ ❛s três ❝✉r✈❛s ❡ ❛ ❛♥á❧✐s❡ s❡rá ❢❡✐t❛ s❡♣❛r❛❞❛♠❡♥t❡✳ ❚❡♠✲s❡ ✉♠ ❞♦s ❝❛s♦s✿

✐✮ ♦ s❡❣♠❡♥t♦P X é ♣❛r❛❧❡❧♦ ❛♦ ❛♦ ❧❛❞♦V A✭◆♦t❛çã♦✿ P X _kV A✮ ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧❀ ♦✉

✐✐✮ ♦ s❡❣♠❡♥t♦ P X ♥ã♦ é ♣❛r❛❧❡❧♦ ❛♦ ❧❛❞♦ V A ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧❀

◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❝♦♠♦ P X _kV A ❡ A′

B′

kAB✱ ♦s tr✐â♥❣✉❧♦s V AB✱ XOB′

❡ V A′

B′

sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦ ❝❛s♦ AA ✭➶♥❣✉❧♦✲➶♥❣✉❧♦✮✳ ❙❡❣✉❡ q✉❡✿

OB′

XO = AB V A ❡

A′_O

V X = AB

V B ✭✷✳✹✮

❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ✷✳✸ ♣♦rXO_×V X✱ ♦❜té♠✲s❡✿ KO2

XO_×V X = A′_O

×OB′

XO_×V X ✭✷✳✺✮

❙✉❜st✐t✉✐♥❞♦ ❛s ❡q✉❛çõ❡s ✷✳✷ ❡ ✷✳✹ ❡♠ ✷✳✺✱ r❡s✉❧t❛ ❡♠✿

KO2

XO_×V X = A′_O

×OB′

XO_×V X = AB

V B ×V X× AB

V A ×XO XO_×V X

= AB

2

V B_×V A = XL V X

⇔KO2 = XL_×XO ✭✷✳✻✮

❆ r❡❧❛çã♦ ✷✳✻ é ♦ s✐♥t♦♠❛ ❞❛ ♣❛rá❜♦❧❛ r❡s✉❧t❛♥t❡ ❞❛ ✐♥t❡rs❡❝çã♦ ❞♦ ❝♦♥❡ ♣❡❧♦ ♣❧❛♥♦ ❞❡ s❡çã♦ ♣❛r❛❧❡❧♦ ❛ ✉♠ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧✳ ❆❧é♠ ❞✐ss♦✱ ❛ ♣❛rá❜♦❧❛ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦sK q✉❡ s❛t✐s❢❛③❡♠ ✷✳✻✳

❆ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛ é q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦KO

é ✐❣✉❛❧ à ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s XL ❡ OX✳ ❆ ❝✉r✈❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s C✱ X ❡ D é ❞❡♥♦♠✐♥❛❞❛ ♣❛rá❜♦❧❛✱ q✉❡ ✈❡♠ ❞♦ ❣r❡❣♦ ♣❛r❛❜♦❧✐✱ ✐st♦ é✱ ❝♦rr❡s♣♦♥❞❡ à ❛♣❧✐❝❛çã♦ ❞❡

❖s ◆♦♠❡s ❞❛s ❈ô♥✐❝❛s ✷✹

◗✉❛♥❞♦ P X ♥ã♦ é ♣❛r❛❧❡❧♦ ❛♦ ❧❛❞♦ V A ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧✱ ♣♦❞❡✲s❡ t❡r P X ✐♥t❡rs❡❝✲

t❛♥❞♦ ♦ s❡❣♠❡♥t♦V B ♦✉ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡V B✳ ❙❛❜❡✲s❡ q✉❡✿ A′

B′

kAB ❡V F _kP X❀

❞❛í VAFb =XAb′_O ❡ _{VF A}b

=XOAb ′

✳ ▲♦❣♦✱ ♦s tr✐â♥❣✉❧♦s V AF ❡ XA′

O sã♦ s❡♠❡❧❤❛♥t❡s

♣❡❧♦ ❝❛s♦AA✳ ❉❛í✱

A′_O

XO = AF

V F ✭✷✳✼✮

OB′

k F B ❡ Y O _k V F❀ ❞❛í YOBb ′

= VF Bb ❡ YBc′_{O = VBF}b ✳ ▲♦❣♦✱ ♦s tr✐â♥❣✉❧♦s

Y OB′

❡V BF sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦ ❝❛s♦ AA✳ ❉❛í✱ OB′

Y O = F B

V F ✭✷✳✽✮

❆✐♥❞❛✱ XL_kOR ❡XOSL é r❡tâ♥❣✉❧♦❀ ❞❛í✱ YXLb =YORb ❡YLXb =YROb ✳ ▲♦❣♦✱ ♦s

tr✐â♥❣✉❧♦s Y XL❡ Y OR sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦ ❝❛s♦ AA✳ ❉❛í✱ XL

XY = OR OY =

OR

Y O ✭✷✳✾✮

➱ ❢á❝✐❧ ♥♦t❛r q✉❡ ♦s tr✐â♥❣✉❧♦sY XL ❡ LSR sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦ ❝❛s♦ AA✳ ❉❛í✱ RS SL = LX XY = XL XY ✭✷✳✶✵✮

❉❛ ❡q✉❛çã♦ ✷✳✶✵✱ ❝♦♠♦ XO ❡ SL tê♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ♦❜té♠✲s❡

RS = XL

XY ×XO ✭✷✳✶✶✮

❆ ❡①♣r❡ssã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❡❧✐♣s❡ ❡ ❛ ❤✐♣ér❜♦❧❡ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❞✐✈✐❞✐♥❞♦ ✷✳✸ ♣♦r

XO_×Y O✱ ♦❜té♠✲s❡✿

KO2 XO_×Y O =

A′_O

×OB′

XO_×Y O ✭✷✳✶✷✮

❙✉❜st✐t✉✐♥❞♦ ✷✳✼ ❡ ✷✳✽ ❡♠ ✷✳✶✷✱ ♦❜té♠✲s❡✿

KO2

XO_×Y O = A′_O

×OB′

XO_×Y O = AF

V F ×XO× F B

V F ×Y O XO_×Y O

⇔ KO

2

XO_×Y O =

AF _×BF

F B2 ✭✷✳✶✸✮

❖s ◆♦♠❡s ❞❛s ❈ô♥✐❝❛s ✷✺

KO2

XO_×Y O = XL XY =

OR Y O

⇔KO2 = OR_×XO ✭✷✳✶✹✮

❆ss✐♠✱ KO2 =OR_×XO é ✉♠❛ ❡①♣r❡ssã♦ ❣❡r❛❧ ♣❛r❛ ❛s ❞✉❛s ❝✉r✈❛s✱ ♦ q✉❡ ✐rá ❞✐❢❡✲

r❡♥❝✐❛r é OR✳ ❊s❝r❡✈❡♥❞♦ OR ❡♠ ❢✉♥çã♦ ❞❛ ❝♦♥st❛♥t❡ XL ❡ ❛♥❛❧✐s❛♥❞♦ ❛s ❞✉❛s ❝✉r✈❛s

♥❛ ✜❣✉r❛ ✷✳✶✱ tê♠✲s❡✿

✭❛✮ P❛r❛ P X ✐♥t❡rs❡❝t❛♥❞♦ V B✱ ♦❜s❡r✈❛✲s❡ q✉❡ OR=XL₋RS✳ ❊♥tã♦✱

KO2 =XO XL₋RS ✭✷✳✶✺✮

❙✉❜st✐t✉✐♥❞♦ ✷✳✶✶ ❡♠ ✷✳✶✺✱ ♦❜té♠✲s❡✿

KO2 =XO

XL₋ XL

XY ×XO

✭✷✳✶✻✮ ❆ r❡❧❛çã♦ ✷✳✶✻ é ♦ s✐♥t♦♠❛ ❞❛ ❡❧✐♣s❡ r❡s✉❧t❛♥t❡ ❞❛ ✐♥t❡rs❡❝çã♦ ❞♦ ❝♦♥❡ ♣❡❧♦ ♣❧❛♥♦ ❞❡ s❡çã♦ q✉❡ ✐♥t❡rs❡❝t❛ ♦s ❞♦✐s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❛①✐❛❧✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❡❧✐♣s❡ é ♦ ❧✉❣❛r ❣❡♦✲ ♠étr✐❝♦ ❞♦s ♣♦♥t♦sK q✉❡ s❛t✐s❢❛③❡♠ ✷✳✶✻✳

❆ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ❡q✉❛çã♦ ✷✳✶✻ é q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ KO é

✐❣✉❛❧ à ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s XO ❡ XL ♠❡♥♦s ♦ ✈❛❧♦r XL

XY ×XO✳ ❆ ❝✉r✈❛✱ ♥❡ss❡

❝❛s♦✱ é ❞❡♥♦♠✐♥❛❞❛ ❡❧✐♣s❡ q✉❡ ✈❡♠ ❞♦ ❣r❡❣♦ ❡❧❧✐♣✐s✱ ✐st♦ é✱ ❝♦rr❡s♣♦♥❞❡ à ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛ ❝♦♠ ❢❛❧t❛✳

✭❜✮ P❛r❛P X ✐♥t❡rs❡❝t❛♥❞♦ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡V B✱ ♦❜s❡r✈❛✲s❡ q✉❡OR =XL+RS✳

❊♥tã♦✱

KO2 =XO XL+RS ✭✷✳✶✼✮

❙✉❜st✐t✉✐♥❞♦ ✷✳✶✶ ❡♠ ✷✳✶✼✱ ♦❜té♠✲s❡✿

KO2 =XO

XL+ XL

XY ×XO

❖s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥ ✷✻

❆ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ❡q✉❛çã♦ ✷✳✶✽ é q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ KO é

✐❣✉❛❧ à ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s XO ❡ XL ♠❛✐s ♦ ✈❛❧♦r XL

XY ×XO✳ ❆ ❝✉r✈❛✱ ♥❡ss❡

❝❛s♦✱ é ❞❡♥♦♠✐♥❛❞❛ ❤✐♣ér❜♦❧❡ q✉❡ ✈❡♠ ❞♦ ❣r❡❣♦ ②♣❡r❜♦❧✐✱ ✐st♦ é✱ ❝♦rr❡s♣♦♥❞❡ à ❛♣❧✐❝❛çã♦ ❞❡ ár❡❛ ❝♦♠ ❡①❝❡ss♦✳

❯s❛♥❞♦ ❛ ❧✐♥❣✉❛❣❡♠ ❛❧❣é❜r✐❝❛ ❛t✉❛❧✱ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ s✐st❡♠❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❢❛③❡♥❞♦ KO = y✱ XY = d✱ XL = p ✭é ✉♠❛ ❝♦♥st❛♥t❡✮ ❡ XO = x❀ ♦s

s✐♥t♦♠❛s ♣♦❞❡r✐❛♠ s❡r tr❛❞✉③✐❞♦s ❡♠✿

y2 = px ✭✷✳✶✾✮

y2 = xp₋ p d ×x

✭✷✳✷✵✮

y2 = xp+p

d ×x

✭✷✳✷✶✮ q✉❡ sã♦ ❛s ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❞❛ ♣❛rá❜♦❧❛✱ ❡❧✐♣s❡ ❡ ❤✐♣ér❜♦❧❡ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

✷✳✷ ❖s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥

❆s ❞❡✜♥✐çõ❡s ❞❡ ❝ô♥✐❝❛s✱ q✉❡ s❡ ❝♦♥❤❡❝❡ ❛t✉❛❧♠❡♥t❡✱ ❢♦r❛♠ ❞❡♠♦♥str❛❞❛s ♣❡❧♦s ♠❛✲ t❡♠át✐❝♦s ❉❛♥❞❡❧✐♥✱ ◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥✳ ❖s ❞♦✐s ♣r✐♠❡✐r♦s ❞❡♠♦♥str❛r❛♠ ❛ ❡❧✐♣s❡ ❡ ❛ ❤✐♣ér❜♦❧❡ ♣♦r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❜✐❢♦❝❛❧✱ ❡ ♦ ú❧t✐♠♦ ❞❡♠♦♥str❛ ❛ ♣❛rá❜♦❧❛ ♣♦r ♣♦r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❢♦❝♦ ❡ ❞✐r❡tr✐③✱ ❝♦♥❢♦r♠❡ s❡❣✉❡✲s❡✿

P❛r❛ t❛✐s✱ ♦s ♠❛t❡♠át✐❝♦s ✜③❡r❛♠ ✉s♦ ❞♦ t❡♦r❡♠❛✿

▲❡♠❛ ✷✳✶✳ ❙❡❥❛♠P ✉♠ ♣♦♥t♦ ❡①t❡✐♦r à ❡s❢❡r❛ _S ❡ ❞✉❛s r❡t❛s ❞✐st✐♥t❛s t❛♥❣❡♥t❡s ❛_S ♥♦s

♣♦♥t♦s A ❡ B q✉❡ ♣❛ss❛♠ ♣♦r P✱ ❡♥tã♦ P A ❡ P B✱ sã♦ ✐❣✉❛✐s✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ O ❡ r = OA = OB ♦ ❝❡♥tr♦ ❡ r❛✐♦ ❞❛ ❡s❢❡r❛ S✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥ ✷✼

❋✐❣✉r❛ ✷✳✷✿ ❘❡t❛s t❛♥❣❡♥t❡s ❛S q✉❡ ♣❛ss❛♠ ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r P

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

◆♦t❡ q✉❡ OB = OA ❡ OP é ❧❛❞♦ ❝♦♠✉♠ ❛♦s tr✐â♥❣✉❧♦s AOP ❡ BOP✳ ❙✉♣♦♥❤❛ P′

✉♠ ♣♦♥t♦ s♦❜r❡ ♦ s❡❣♠❡♥t♦ P B t❛❧ q✉❡ P′_{B = P A}✱ ❞❛í ♣❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ _LAL✱

t❡♠✲s❡ q✉❡ ♦s tr✐â♥❣✉❧♦s AOP ❡ BOP′

sã♦ ❝♦♥❣r✉❡♥t❡s ⇒ OP′ _{= OP}✱ ✐st♦ é✱ _P′ é ♦

♣ró♣r✐♦P✳ P♦rt❛♥t♦✱ ❖s tr✐â♥❣✉❧♦sAOP ❡BOP sã♦ ❝♦♥❣r✉❡♥t❡s✳

❚❡♦r❡♠❛ ✷✳✶ ✭❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ✲ ❊❧✐♣s❡✮✳ ❙❡❥❛♠C ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ❞❡ ✉♠❛ ❢♦❧❤❛✱ ✉♠ ♣❧❛♥♦ π q✉❡ ♦ ✐♥t❡rs❡❝t❛ ❡ ❞✉❛s ❡s❢❡r❛s✱ S1 ❡ S2✱ t❛♥❣❡♥t❡s ❛ C ✭✐♥t❡r♥❛♠❡♥t❡✮ ❡

❛ π✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✷✳✸✳ ❙❡ ♦s ♣♦♥t♦s F1 ❡ F2✱ ❞✐st✐♥t♦s✱ sã♦ ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦

❞❛s ❡s❢❡r❛s S1 ❡ S2 ❝♦♠ ♦ ♣❧❛♥♦ π r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♥tã♦ ❞❛❞♦ ✉♠ ♣♦♥t♦ P ∈ C ∩π ✱

t❡♠✲s❡ q✉❡ P F1+P F2 ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ P✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♥❡st❡ ❝❛s♦✱ t♦❞♦ ♣♦♥t♦ ♥❛

✐♥t❡rs❡❝ã♦ ❞♦ ❝♦♥❡ ❝♦♠ ♦ ♣❧❛♥♦ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ❛ s♦♠❛ ❞❛s s✉❛s ❞✐stâ♥❝✐❛s ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s✱ F1 ❡ F2✱ é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ss❡ ♣♦♥t♦ ❡s❝♦❧❤✐❞♦✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛F1 ❡ F2 ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❡s❢❡r❛sS1 ❡S2 ❝♦♠ ♦ ♣❧❛♥♦

πr❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠c1 ❡c2 ❛ ✐♥t❡rs❡çã♦ ❞♦ ❝♦♥❡C ❝♦♠ ❛s ❡s❢❡r❛sS1 ❡S2 r❡s♣❡❝t✐✈❛✲

♠❡♥t❡✳ ❆✐♥❞❛✱ s❡❥❛♠A=g_∩c1✱ B =g∩c2✱P =g∩π❡AB ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦

❖s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥ ✷✽

❋✐❣✉r❛ ✷✳✸✿ ❊s❢❡r❛s S1 ❡ S2 t❛♥❣❡♥t❡s ❛♦ ♣❧❛♥♦ π ❡ ❛♦ ❝♦♥❡ C ♥♦ ♠❡s♠♦ r❛♠♦

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ t❡♠✲s❡ q✉❡P F1 =P B ❡P F2 =P A ⇒P F1+P F2 =P A+P B =

AB q✉❡ é ❝♦♥st❛♥t❡✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ P✳ ❖ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❝♦♠

❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❝❤❛♠❛❞♦ ❞❡ ❡❧✐♣s❡✳ ❖s ♣♦♥t♦s F1 ❡ F2 sã♦ ❝❤❛♠❛❞♦s ❞❡ ❢♦❝♦s ❞❛

❡❧✐♣s❡✳

❚❡♦r❡♠❛ ✷✳✷ ✭❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ✲ ❍✐♣ér❜♦❧❡✮✳ ❙❡❥❛♠ C ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ❞❡ ❞✉❛s ❢♦❧❤❛s✱ ✉♠ ♣❧❛♥♦π q✉❡ ♦ ✐♥t❡rs❡❝t❛ ❡ ❞✉❛s ❡s❢❡r❛s✱ S1 ❡ S2✱ t❛♥❣❡♥t❡s ❛C ✭✐♥t❡r♥❛♠❡♥t❡✮

❡ ❛ π✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✷✳✹✳ ❙❡ ♦s ♣♦♥t♦s F1 ❡ F2✱ ❞✐st✐♥t♦s✱ sã♦ ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦

❞❛s ❡s❢❡r❛s ❝♦♠ ♦ ♣❧❛♥♦ π✱ ❡♥tã♦ ❞❛❞♦ ✉♠ ♣♦♥t♦ P _{∈ C ∩}π ✱ t❡♠✲s❡ q✉❡P F1−P F2 _♥ã♦

❞❡♣❡♥❞❡ ❞❡ P✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♥❡st❡ ❝❛s♦✱ t♦❞♦ ♣♦♥t♦ ♥❛ ✐♥t❡rs❡❝ã♦ ❞♦ ❝♦♥❡ ❝♦♠ ♦

♣❧❛♥♦ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ s✉❛s ❞✐stâ♥❝✐❛s ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s✱F1 ❡ F2✱ é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ss❡ ♣♦♥t♦ ❡s❝♦❧❤✐❞♦✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛F1 ❡ F2 ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❡s❢❡r❛sS1 ❡S2 ❝♦♠ ♦ ♣❧❛♥♦

π r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ c1 ❡ c2 ❛ ✐♥t❡rs❡çã♦ ❞♦ ❝♦♥❡ C ❝♦♠ ❛s ❡s❢❡r❛s S1 ❡ S2 r❡s♣❡❝t✐✲

✈❛♠❡♥t❡✳ ❆✐♥❞❛✱ s❡❥❛ ✉♠ ❞❛❞♦ ♣♦♥t♦P ❞❛ ✐♥t❡rs❡çã♦ ❞♦ ♣❧❛♥♦ π ❝♦♠ ❛ ❢♦❧❤❛ s✉♣❡r✐♦r ❞♦

❝♦♥❡ q✉❡ ❞❡t❡r♠✐♥❛ ❛ ❣❡r❛tr✐③ q✉❡ ♣❛ss❛ ♣♦rP ❡ V✳ ❉❡ss❛ ❢♦r♠❛✱ ❡ss❛ ❣❡r❛tr✐③ t❛♥❣❡♥❝✐❛

❖s ❚❡♦r❡♠❛s ❞❡ ❉❛♥❞❡❧✐♥✲◗✉❡t❡❧❡t ❡ P✐❡r❝❡ ▼♦rt♦♥ ✷✾

❋✐❣✉r❛ ✷✳✹✿ ❊s❢❡r❛s S1 ❡S2 t❛♥❣❡♥t❡s ❛♦ ♣❧❛♥♦ π ❡ ❛♦ ❝♦♥❡ C ❡♠ r❛♠♦s ❞✐st✐♥t♦s

❋♦♥t❡✿ ❊❧❛❜♦r❛❞❛ ♣❡❧♦ ❆✉t♦r

P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ t❡♠✲s❡ q✉❡P F1 =P A❡P F2 =P B⇒

P F1−P F2

=P A₋P B=

AB q✉❡ é ❝♦♥st❛♥t❡✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ P✳ ❖ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❝♦♠

❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❝❤❛♠❛❞♦ ❞❡ ❤✐♣ér❜♦❧❡✳ ❖s ♣♦♥t♦s F1 ❡ F2 sã♦ ❝❤❛♠❛❞♦s ❞❡ ❢♦❝♦s ❞❛

❤✐♣ér❜♦❧❡✳

❚❡♦r❡♠❛ ✷✳✸ ✭P✐❡r❝❡ ▼♦rt♦♥ ✲ P❛rá❜♦❧❛✮✳ ❙❡❥❛♠C ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ❞❡ ✉♠❛ ❢♦❧❤❛✱π1

✉♠ ♣❧❛♥♦ q✉❡ ♦ ✐♥t❡rs❡❝t❛ ❡ ✉♠❛ ❡s❢❡r❛ S t❛♥❣❡♥t❡ ❛ _C ✭✐♥t❡r♥❛♠❡♥t❡✮ ❡ ❛ π1✱ ❝♦♥❢♦r♠❡

✜❣✉r❛ ✷✳✺✳ ❙❡ F é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛ ❡s❢❡r❛ S ❝♦♠ ♦ ♣❧❛♥♦ π1✱ ❡ d é ❛ r❡t❛ ❞❡

✐♥t❡rs❡❝çã♦ ❞♦ ♣❧❛♥♦ π1 ❝♦♠ ♦ ♣❧❛♥♦ π2 q✉❡ ❝♦♥té♠ ❛ ❝r✐❝✉♥❢❡rê♥❝✐❛ c = S ∩ C✱ ❡♥tã♦

❞❛❞♦ ✉♠ ♣♦♥t♦ P _∈C_∩π1✱ t❡♠✲s❡ q✉❡ ❞(P, F) =❞(P, d)

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡F é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛ ❡s❢❡r❛S❝♦♠ ♦ ♣❧❛♥♦π1✳ ❙❡❥❛♠

c ❛ ✐♥t❡rs❡çã♦ ❞♦ ❝♦♥❡ _C ❝♦♠ ❛ ❡s❢❡r❛ S✱ g ❛ ❣❡r❛tr✐③ ❞♦ ❝♦♥❡ ♣❛r❛❧❡❧❛ ❛♦ ♣❧❛♥♦ π1✱ t❛❧

q✉❡ c_∩g = A✱ ❡ d ❛ ✐♥t❡rs❡çã♦ ❞♦ ♣❧❛♥♦ π1 ❝♦♠ ♦ ♣❧❛♥♦ π2 q✉❡ ❝♦♥té♠ c✳ ❚♦♠❡ B ✉♠