Open Estudo sistemático das parábolas

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❯♥✐✈❡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✉❞♦ ❙✐st❡♠át✐❝♦ ❞❛s P❛rá❜♦❧❛s

♣♦r

❍❡❧❞❡r ❘♦❞r✐❣✉❡s ▼❛❝❡❞♦

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❯♥✐✈❡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✉❞♦ ❙✐st❡♠át✐❝♦ ❞❛s P❛rá❜♦❧❛s

†

♣♦r

❍❡❧❞❡r ❘♦❞r✐❣✉❡s ▼❛❝❡❞♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡✲ s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥✲ çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇

† _{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

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M141e Macedo, Helder Rodrigues.

Estudo sistemático das parábolas / Helder Rodrigues Macedo.- João Pessoa, 2015.

71f. : il.

Orientador: Napoleón Caro Tuesta Dissertação (Mestrado) - UFPB/CCEN

1. Matemática. 2. Construção das Cônicas. 3. Apolônio de Perga - matemático. 4. Geometria analítica. 5. Função

quadrática.

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♠ s❛ú❞❡ ❡ ❞✐s♣♦s✐çã♦ ❡ ♣♦r ❤❛✈❡r ❊❧❡ ❡s❝♦❧❤✐❞♦ ♦ ❧✐✈r♦ ❞♦ ❯♥✐✈❡rs♦ ❞❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠

♣r❡s❡♥t❡ ❛♦ ♥♦ss♦ ❞♦♠ ♦♣♦rt✉♥✐③❛♥❞♦ ❛ ♥ós ✉♠ ♣♦✉❝♦ ❞❛ s✉❛ ❛❞♠✐rá✈❡❧ ♦❜r❛✳ ❆❣r❛❞❡ç♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❙✉❡❧❧❡♥ ▼❛❝❡❞♦✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s t❡♠♣♦ q✉❡ ♠❡ ✜③ ❛✉s❡♥t❡ ❛♦ ♠❡ ❞❡❞✐❝❛r ❛♦s ❡st✉❞♦s ♥❛ ✈✐❞❛

❛❝❛❞ê♠✐❝❛ ❞♦ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✜❧❤♦s✱ ▲✉❝❛s ●❛❜r✐❡❧ ❡ ❏ú❧✐❛ ▼❛❝❡❞♦ ♣❡❧❛ ❢❡❧✐❝✐❞❛❞❡ q✉❡ ♠❡ ❞ã♦ ♥❛ ♥❛t✉r❡③❛ ❧í♠♣✐❞❛ q✉❡ ❡❧❡s ♣♦ss✉❡✱ ❛ ♠✐♥❤❛ ♠ã❡ ❱✐❧♠❛ ▼❛❝❡❞♦✱ ❛♦s ♠❡✉s ✐r♠ã♦s ❍✉❣♦ ▼❛❝❡❞♦ ❡ ❍✐❧♠❛ ❱❛s❝♦♥❝❡❧♦s ▼❛❝❡❞♦ q✉❡ ❛♣♦st❛r❛♠ ♠✉✐t♦ ♥♦ ♠❡✉ tr❛❜❛❧❤♦ ❡ ♥❛ ♣❡rs❡✈❡r❛♥ç❛ ♣❛ss❛❞♦ ♣❛r❛ ♠✐♠ ❞✐❛r✐❛♠❡♥t❡ ♥❛ ❥♦r♥❛❞❛ ❛❝❛❞ê♠✐❝❛ q✉❡ ❡s❝♦❧❤✐✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❖r✐❡♥t❛❞♦r✱ ◆❛♣♦❧❡♦♥ ❈❛r♦✱ ♣❡❧♦ ♣r❛③❡r✱ ♣r♦♥t✐❞ã♦✱ ❡♥t✉s✐❛s♠♦

❡♠ ♦r✐❡♥t❛r✲♠❡ ♥❡ss❡ tr❛❜❛❧❤♦ ❛ ❝❛❞❛ ♠♦♠❡♥t♦ ❞❡ ❢♦r♠❛ ♦❜❥❡t✐✈❛ ❡ s❡❣✉r❛ ♥❛s s✉❛s ✐♥❞❛❣❛çõ❡s ❡ ♥❛ s✉❛ ❜r✐❧❤❛♥t❡ ❝♦♠♣❡tê♥❝✐❛ ❛❝❛❞ê♠✐❝❛ q✉❡ ❛t✉❛✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛ ❞✉r❛♥t❡ ❛ ♣ós✲❣r❛❞✉❛çã♦✱ ♣❛ss❛♥❞♦ ♣❛r❛ ♥♦s s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ♠❛♥❡✐r❛

❜r✐❧❤❛♥t❡✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ♣ós✲❣r❛❞✉❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❋❛❜rí❝✐♦ ❞❡ P❛✉❧❛ ❡ ❆♥tô♥✐♦ ▼♦r❛✐s q✉❡ s❡♠❛♥❛❧♠❡♥t❡ ❡st❛✈❛♠ ❝♦♠✐❣♦ ❡st✉❞❛♥❞♦✱

✐♥❝❡♥t✐✈❛♥❞♦✱ ❜r✐❣❛♥❞♦✱ ❜❛t❛❧❤❛♥❞♦ ♣❛r❛ ❝♦♥str✉✐r ❡ss❡ tr❛❜❛❧❤♦✱ ❡ss❛ ♣ós✲❣r❛❞✉❛çã♦✳

❆❣r❛❞❡ç♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋P❇✱ ♣♦r ❛♣♦st❛r ♥❡ss❛ ♣ós✲❣r❛❞✉❛çã♦ ❝♦♠ ❛ ♣❛r❝❡r✐❛ ♥♦ P❘❖❋▼❆❚ ❡ ❛ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

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❉❡❞✐❝❛tór✐❛

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❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❛❜♦r❞❛❣❡♠ q✉❡ ♣❡r♠✐t❡ t❛♥t♦ ❛♦ ♣r♦❢❡ss♦r q✉❛♥t♦ ❛♦ ❛❧✉♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ✉♠ ❡st✉❞♦ ❤✐stór✐❝♦ ❞❛ ❝♦♥str✉çã♦ ❞❛s

❈ô♥✐❝❛s ❞❡s❡♥✈♦❧✈✐❞❛s ♣❡❧♦ ▼❛t❡♠át✐❝♦ ❡ ❆strô♥♦♠♦ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ q✉❡ ❝♦♥tr✐❜✉✐✉ ✐♠❡♥s❛♠❡♥t❡ ❝♦♠ ❛s ❞❡✜♥✐çõ❡s ❤♦❥❡ ❡st✉❞❛❞❛s ♥❛ ▼❛t❡♠át✐❝❛✳ ◆♦ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ❥á ❜❡♠ ♠❛✐s ❞❡✜♥✐❞❛s ❛s ❈ô♥✐❝❛s ♣♦r P✐❡rr❡ ❋❡r♠❛t ♦ ❡st✉❞♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛❜♦r❞❛r ♦ ❝♦♥t❡ú❞♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❝♦♠♦ é ❡♥s✐♥❛❞♦ ♥❛s sér✐❡s ❜ás✐❝❛s ❡ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ❈á❧❝✉❧♦✳ ◆♦ t❡r❝❡✐r♦ ♠♦♠❡♥t♦✱ ❛ ❛❜♦r❞❛❣❡♠

é ❢❡✐t❛ ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❛s ❋✉♥çõ❡s ◗✉❛❞rát✐❝❛s✱ ✉♠❛ r❡✈✐sã♦ ❞❛ ♣r✐♠❡✐r❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳

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❆❜str❛❝t

❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ♦♥❡ ♣r♦♣♦s❛❧ t❤❛t ❛❧❧♦✇s ❍✐❣❤ ❙❝❤♦♦❧ t❡❛❝❤❡rs ❛♥❞ st✉❞❡♥ts ❛ ❤✐st♦r✐❝❛❧ st✉❞② ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❈♦♥✐❝s✱ ❞❡✈❡❧♦♣❡❞ ❜② ❆♣♦❧❧♦♥✐✉s ♦❢ P❡r❣❛✱

t❤❡ ▼❛t❤❡♠❛t✐❝✐❛♥ ❛♥❞ ❆str♦♥♦♠❡r t❤❛t ❝♦♥tr✐❜✉t❡❞ ✐♠♠❡♥s❡❧② ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥s ✇❡ st✉❞② ♥♦✇❛❞❛②s ✐♥ ▼❛t❤❡♠❛t✐❝s✳ ■♥ ❛ s❡❝♦♥❞ ♠♦♠❡♥t✱ ✇✐t❤ ❈♦♥✐❝s

✇❡❧❧ ❞❡✜♥❡❞ ❜② P✐❡rr❡ ❋❡r♠❛t✱ t❤❡ ❣♦❛❧ ♦❢ t❤❡ ✇♦r❦ ✐s t♦ ❛❞❞r❡ss t❤❡ ❝♦♥t❡♥t ♦❢ ❆♥❛❧②t✐❝❛❧ ●❡♦♠❡tr② ❛s t❛✉❣❤t ✐♥ t❤❡ ✐♥✐t✐❛❧ s❝❤♦♦❧ ②❡❛rs ❛♥❞ ❈❛❧❝✉❧✉s ❝♦✉rs❡s✳ ■♥

❛ t❤✐r❞ ♠♦♠❡♥t✱ t❤❡ ❛♣♣r♦❛❝❤ ✐s ❞♦♥❡ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ◗✉❛❞r❛t✐❝ ❋✉♥❝t✐♦♥s✱ ✉s✐♥❣ ❛ r❡✈✐❡✇ ♦❢ t❤❡ ❝♦♥t❡♥t t❛✉❣❤t ✐♥ ❙♦♣❤♦♠♦r❡ ②❡❛r ♦❢ ❍✐❣❤ ❙❝❤♦♦❧✳

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❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❆s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ✶

✷ ❆ P❛rá❜♦❧❛ ♥❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✺

✷✳✶ ❈♦♦r❞❡♥❛❞❛s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❉✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦ ❡ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ ❉❡✜♥✐çã♦ ❞❡ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✺ ❊❧❡♠❡♥t♦s ♣r✐♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✻ ❊q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✼ ❊q✉❛çõ❡s ❈❛♥ô♥✐❝❛s ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ ❱ért✐❝❡ ♥❛ ♦r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✽ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✾ ❘♦t❛çã♦ ❞❡ ❊✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✵ ❈♦♦r❞❡♥❛❞❛s P♦❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✶ ❊q✉❛çã♦ P♦❧❛r ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✷ ❈♦♥str✉çã♦ ❞❛ P❛rá❜♦❧❛ ♣♦r ♠❡✐♦ ❞❡ ❆rt✐❢í❝✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✸✻

✸✳✶ ❖ ●rá✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❋♦r♠❛ ❈❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✸ ❩❡r♦✭s✮ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✹ ❖ ❱ért✐❝❡ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✺ ❆ ■♠❛❣❡♠ ❞❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✻ ❊✐①♦ ❞❡ ❙✐♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✼ ▼♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✽ ❆ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✺

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❉✉♣❧✐❝❛çã♦ ❞♦ ❈✉❜♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ Pá❣✐♥❛ ❞♦ tít✉❧♦ ❞❛ ♣r✐♠❡✐r❛ ❡❞✐çã♦ ✐♠♣r❡ss❛ ❡♠ ❧❛t✐♠ ❞❛s ❈ô♥✐❝❛s

❞❡ ❆♣♦❧ô♥✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❆s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ P✐❧❛r❡s ❞❛ ❝❛t❡❞r❛❧ ❞❡ ❇r❛s✐❧✐❛✳ ❋♦r♠❛ ❞❡ ✉♠ ❤✐♣❡r❜♦❧♦✐❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✺ ❈♦❧✐s❡✉✱ ❡♠ ❘♦♠❛✱ t❡♠ ❢♦r♠❛ ❡❧í♣t✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✻ P♦♥t❡ ❏✉s❝❡❧✐♥♦ ❑✉❜✐ts❝❤❡❦✱ ❡♠ ❇r❛sí❧✐❛✱ t❡♠ ❛ ❢♦r♠❛ ❞❛ ♣❛rá❜♦❧❛✳ ✳ ✹ ✶✳✼ ❚♦rr❡s ❞❡ ❘❡❢r✐❣❡r❛çã♦ ❞❡ ✉s✐♥❛s ♥✉❝❧❡❛r❡s✱ tê♠ ❛ ❢♦r♠❛ ❞♦ ❤✐♣❡r❜♦❧♦✐❞❡✳ ✹ ✷✳✶ ❈♦♦r❞❡♥❛❞❛s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ❉✐stâ♥❝✐❛ ❞❡ ❞♦✐s ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❉✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ à r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ❉❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✺ ❙❡❝çã♦ ❞♦ ❝♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✻ ❘♦t❛çã♦ ❞♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✼ ❖ P♦♥t♦ ❡♠ ❈♦♦r❞❡♥❛❞❛ P♦❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✽ P❛rá❜♦❧❛ ❡♠ ❈♦♦r❞❡♥❛❞❛s P♦❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✾ ❈♦♥str✉✐♥❞♦ ❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✵ P❛rá❜♦❧❛ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✶✶ P❛rá❜♦❧❛ ❞❡s❡♥❤❛❞❛ ❝♦♠ ❛ ré❣✉❛ ❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✶ ❊✐①♦ ❞❡ ❙✐♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷ ▼♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

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■♥tr♦❞✉çã♦

▼♦t✐✈❛❞♦ ♣♦r ♣❡r❣✉♥t❛s ❢❡✐t❛s ♣♦r ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ♥♦s ❝♦♥t❡ú❞♦s ❞❡ ❋✉♥çõ❡s ◗✉❛❞rát✐❝❛s ❡ ♥❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♦ ❝♦♥t❡ú❞♦ ❞❡ P❛rá❜♦❧❛s✱ ♣r♦❝✉r❛♠♦s ❡s❝r❡✈❡r ♥❡ss❡ tr❛❜❛❧❤♦✱ ✉♠ ❊st✉❞♦ ❙✐st❡♠át✐❝♦ ❞❛s P❛rá❜♦❧❛s✱ ❝♦♠❡ç❛♥❞♦ ✉♠❛ ❆❜♦r❞❛❣❡♠ ❙✐♥tét✐❝❛✱ ❛♦ ❞❡st❛❝❛r ♦ ❡s♣❡t❛❝✉❧❛r tr❛❜❛❧❤♦ ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ♣❡❧❛ ❢♦r♠❛ ❜r✐❧❤❛♥t❡ ❝♦♠♦ ✈❡✐♦ ❛ ❞❡st❛❝❛r ♦ ❝♦♥❡✱

❛s s❡❝çõ❡s ♥❡❧❡ ♣r♦❞✉③✐❞❛s ❡ à ❞❡✜♥✐çã♦ q✉❡ ♦ ♠❡s♠♦ ❝❤❛♠♦✉ ❞❡ ✧s✐♥t♦♠❛✧❞❛ ❝ô♥✐❝❛✳ ❖s s✐♥t♦♠❛s ❞❡ ✉♠❛ ❝✉r✈❛ ♣♦❞❡♠✲s❡ ❡st❡♥❞❡r ❝♦♠♦ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦

❛❧❣é❜r✐❝❛ ❞❡ ❞❡❞✉çã♦ ❣❡♦♠étr✐❝❛✱ ✐st♦ é✱ ♥❛ ❧✐♥❣✉❛❣❡♠ ♠♦❞❡r♥❛✱ ✉♠❛ r❡❧❛çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♥tr❡ ❛ ♦r❞❡♥❛❞❛ ❡ ❛ ❛❜s❝✐s❛ ❞❡ ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ❞❛ ❝✉r✈❛✳ ❆♣❡s❛r ❞❡ ❛♥t❡❝❡ss♦r❡s ❝♦♠♦ ▼❡♥❡❝♠♦✱ ❊✉❝❧✐❞❡s✱ ❞❡♥tr❡ ♦✉tr♦s✱ tr❛❜❛❧❤❛r❡♠ ❝♦♠

❛s ❝ô♥✐❝❛s✱ ❛ ❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ ❞❡ss❡ ❝♦♥t❡ú❞♦ ❢♦✐ ❡♥❢❛t✐③❛❞♦ ❛ ♣❛rt✐r ❞♦s tr❛❜❛❧❤♦s ❞❡ ❆♣♦❧ô♥✐♦✱ ❡♠ ❡s♣❡❝✐❛❧ s✉❛ ♦❜r❛ ♣r✐♠❛ ❆s ❈ô♥✐❝❛s ♣♦r s❡r✈✐r ❞❡ ❡st✉❞♦s ♣❛r❛ P✐❡rr❡ ❋❡r♠❛t ✐♥tr♦❞✉③✐r ♥❛ ♠❛t❡♠át✐❝❛ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❙❡❣✉❡✲s❡ ♥♦ s❡❣✉♥❞♦ ♠♦♠❡♥t♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❢❡✐t❛ ♣♦r P✐❡rr❡ ❋❡r♠❛t✱ q✉❛♥❞♦ ♣❛ss♦✉ ❛ ❡s❝r❡✈❡r ❛s s❡❝çõ❡s ❝ô♥✐❝❛s ♣♦r ♠❡✐♦s ❞❡ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ♥❛s ❝♦♦r❞❡♥❛❞❛s ✭①✱ ②✮✳ ❚r❛❜❛❧❤❛♠♦s ❛ ❞❡✜♥✐çã♦✱ ♦s ❡❧❡♠❡♥t♦s ❞❛

P❛rá❜♦❧❛✱ ❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s✱ ❛s ❡q✉❛çõ❡s ❞❡ tr❛♥s❧❛çã♦ ❡ r♦t❛çã♦ ❡ ❛ ❢♦r♠❛ ♣♦❧❛r ❞❛ ♣❛rá❜♦❧❛✳ P♦r ✜♠✱ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ✉t✐❧✐③❛♥❞♦✲❡ ❞❛s ✐♥❢♦r♠❛çõ❡s ♦❜t✐❞❛s✱ ❝♦♥str✉í♠♦s ❛ ♣❛rá❜♦❧❛ ❛tr❛✈és ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✱ ♠♦str❛♠♦s ❝♦♠♦ ❝♦♥str✉✐r ❛

♣❛rá❜♦❧❛ ♣♦r ♠❡✐♦ ❞❡ ❞♦❜r❛❞✉r❛s ❝♦♠ ♣❛♣❡❧✱ ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ ♣❛r❛ ❛♣❧✐❝❛çã♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♣❡❧♦ ❞❡s❡♥❤♦ ❣❡♦♠étr✐❝♦ ❛tr❛✈és ❞❛ ré❣✉❛ ❚ ❡ ✉♠❛ ♣r❛♥❝❤❡t❛ ❞❡

❞❡s❡♥❤♦✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❛❣♦r❛ ❥á ❜❡♠ ❞❡✜♥✐❞❛ ❛ ♣❛rá❜♦❧❛✱ ✜③❡♠♦s ✉♠ ❡st✉❞♦ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ❞❡s❞❡ ❛ ❝♦♥str✉çã♦ ❞♦ s❡✉ ❣rá✜❝♦ ♣♦r ♠❡✐♦ ❞❡ ❛tr✐❜✉✐çã♦ à ✈❛r✐á✈❡❧x✱ s❡✉s ❝♦♥❝❡✐t♦s ✐♥✐❝✐❛✐s✱ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❡✱ ❛tr❛✈és ❞❡❧❛ ♦s ③❡r♦s ❞❛

❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❜❡♠ ❝♦♠♦ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❇❤ás❦❛r❛✳ ❆✐♥❞❛ ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ❛♣r❡s❡♥t❛♠♦s ♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ s❡✉ ✈❛❧♦r ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦✱ ❛ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦✱ s❡✉ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ ❡ ✉♠ ❡st✉❞♦ ❞❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡

♣❛r❛ ♠♦str❛r ♦ ♠♦♠❡♥t♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡ ❞❡❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛✳ ❋✐♥❛❧♠❡♥t❡✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ♣❛rá❜♦❧❛ ♣♦r ♠❡✐♦ ❞❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ❞❡

s❡❣✉♥❞❛ ♦r❞❡♠✱ ❝♦♥❝r❡t✐③❛♥❞♦ ❛✐♥❞❛ ♠❛✐s ♦ ❡st✉❞♦ ❞❡ss❡ ❝♦♥t❡ú❞♦✳

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❈❛♣ít✉❧♦ ✶

❆s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛

❆♣♦❧ô♥✐♦ ♥❛s❝❡✉ ❡♠ P❡r❣❛✱ ❝✐❞❛❞❡ ❛♦ s✉❧ ❞♦ q✉❡ ❤♦❥❡ é ❛ ❚✉rq✉✐❛ ✭✷✻✷✲✶✾✵ ❛✳❈✳✮✱ ❝♦♥❤❡❝✐❞♦ ❥á ♥❛ ❆♥t✐❣✉✐❞❛❞❡ ❝♦♠♦ ♦ ●r❛♥❞❡ ●❡ô♠❡tr❛✱ ❢♦✐ ✉♠ ❝é❧❡❜r❡ ♠❛t❡♠át✐❝♦

❡ ❛strô♥♦♠♦ q✉❡ ✈✐✈❡✉ ❣r❛♥❞❡ ♣❛rt❡ ❞❛ s✉❛ ✈✐❞❛ ❡♠ ❆❧❡①❛♥❞r✐❛✱ ♣r✐♠❡✐r♦ ❝♦♠♦ ❞✐s❝í♣✉❧♦ ❡ ♠❛✐s t❛r❞❡ ❝♦♠♦ ♣r♦❢❡ss♦r ♥❛ ❡s❝♦❧❛ ❞♦s s✉❝❡ss♦r❡s ❞❡ ❊✉❝❧✐❞❡s✳ ❙❡✐s ❞❛s ♦❜r❛s ❞❡ ❆♣♦❧ô♥✐♦ sã♦ r❡❢❡r✐❞❛s ♥♦ ❚❡s♦✉r♦ ❞❛ ❆♥á❧✐s❡✱ ❞❛ ❈♦❧❡çã♦ ▼❛t❡♠át✐❝❛ ❞❡ P❛♣✉s✿ ❙♦❜r❡ ❛ ❙❡❝çã♦ ❞✉♠❛ ❘❛③ã♦✱ ❙♦❜r❡ ❛ ❙❡❝çã♦ ❞✉♠❛ ➪r❡❛✱ ❙♦❜r❡ ❛ ❙❡❝çã♦ ❉❡t❡r♠✐♥❛❞❛✱ ❙♦❜r❡ ❛s ❚❛♥❣ê♥❝✐❛s✱ ❙♦❜r❡ ❛s ■♥❝❧✐♥❛çõ❡s ❡ ❙♦❜r❡ ♦s ▲✉❣❛r❡s P❧❛♥♦s✳

P❛r❛ ❛ ♠❛✐♦r✐❛ ❞♦s ❤✐st♦r✐❛❞♦r❡s ❞❛ ▼❛t❡♠át✐❝❛✱ ❛s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❝♦♥st✐t✉❡♠ ✉♠ tr❛t❛❞♦ ❞❡ ❣r❛♥❞❡ ❛♠♣❧✐t✉❞❡ ❡ ♣r♦❢✉♥❞✐❞❛❞❡✳ ❚r❛t❛✲s❡ ❞❡ ✉♠❛ ♦❜r❛

❡①t❡♥s❛ ❡ r✐❣♦r♦s❛✱ ❝♦♠♣❛rá✈❡❧ ❛♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s ♣❡❧♦ s❡✉ ❝♦♥t❡ú❞♦ ❡ r✐❣♦r✳ ❊ss❛ ♦❜r❛ ♣❛ss♦✉ ❛ s❡r ♦ tr❛t❛❞♦ ♣♦r ❡①❝❡❧ê♥❝✐❛ s♦❜r❡ ❛s s❡❝çõ❡s ❝ô♥✐❝❛s ❡

s✉❜st✐t✉✐✉ ♦s ❡s❝r✐t♦s ❛♥t❡r✐♦r❡s ❞❡ ▼❡♥❡❝♠♦✱ ❆r✐st❡✉ ❡ ❊✉❝❧✐❞❡s✱ ♣♦✐s ❆♣♦❧ô♥✐♦ ❡st✉❞♦✉ tã♦ ♣r♦❢✉♥❞❛♠❡♥t❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❝ô♥✐❝❛s q✉❡ t♦r♥♦✉ ❛s ♦❜r❛s

❛♥t❡r✐♦r❡s ♦❜s♦❧❡t❛s✳

❆ ♦r✐❣❡♠ ❞❛s ❝ô♥✐❝❛s ❡stá r❡❧❛❝✐♦♥❛❞❛ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✉♣❧✐❝❛çã♦ ❞♦ ❝✉❜♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ❝♦♥str✉✐r✱ ❝♦♠ ♦ ✉s♦ ❞❡ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❞❛❞❛ ❛ ❛r❡st❛ ❞❡ ✉♠ ❝✉❜♦✱ ❛

❛r❡st❛ ❞❡ ✉♠ s❡❣✉♥❞♦ ❝✉❜♦ ❝✉❥♦ ✈♦❧✉♠❡ é ♦ ❞♦❜r♦ ❞♦ ❛♥t❡r✐♦r ✭❋✐❣✉r❛ ✶✳✶✮ ➱ ❞✐❢í❝✐❧ ❛t✉❛❧♠❡♥t❡ ❝♦♠♣r❡❡♥❞❡r ❝♦♠♦ ❆♣♦❧ô♥✐♦ ♣♦❞✐❛ ❞❡s❝♦❜r✐r ❡ ♣r♦✈❛r ❝❡♥t❡♥❛s

❞❡ ❜❡❧♦s ❡ ❞✐❢í❝❡✐s t❡♦r❡♠❛s s❡♠ ♦ s✐♠❜♦❧✐s♠♦ ❛❧❣é❜r✐❝♦ ♠♦❞❡r♥♦✱ ♥♦ ❡♥t❛♥t♦✱ ❡❧❡ ❢❡③✳ ❊♠❜♦r❛ s❡ tr❛t❡ ❞❡ ✉♠❛ ♦❜r❛ ❡①t❡♥s❛ ❡ ❞❡ ❞✐❢í❝✐❧ ❧❡✐t✉r❛✱ ♣❛r❛ q✉❡♠ ❡stá ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ♦s ♠ét♦❞♦s ♠♦❞❡r♥♦s✱ ❡st❡ tr❛t❛❞♦ ❡stá ❛ ♣❛r ❞♦ q✉❡ ♠❛✐s

❜r✐❧❤❛♥t❡ ❝♦♥❤❡❝❡♠♦s ❞❛ ❣❡♦♠❡tr✐❛ ❛♥t✐❣❛✳

❆♣♦❧ô♥✐♦ ❞❡✜♥✐✉ ❛s s❡❝çõ❡s ❝ô♥✐❝❛s ❞❡ ♠♦❞♦ ❞✐❢❡r❡♥t❡ ❞♦ q✉❡ t✐♥❤❛ s✐❞♦ ❛❞♦t❛❞♦ ♣❡❧♦s s❡✉s ❛♥t❡❝❡ss♦r❡s✳ ❈♦♠ ✉♠❛ ✈✐sã♦ ❣❡♦♠étr✐❝❛ ✐♥♦✈❛❞♦r❛✱ ❞❡♠♦♥str♦✉ q✉❡

♥ã♦ ❡r❛ ♥❡❝❡ssár✐❛ ❛ r❡str✐çã♦ ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r✐❞❛❞❡ ❞♦ ♣❧❛♥♦ ❞❡ ✐♥t❡rs❡❝çã♦ à ❣❡r❛tr✐③ ❞♦ ❝♦♥❡✱ ❡ ❛✐♥❞❛ q✉❡ ❞❡ ✉♠ ú♥✐❝♦ ❝♦♥❡ ❡r❛ ♣♦ssí✈❡❧ ♦❜t❡r ❛s três ❡s♣é❝✐❡s

❞❡ s❡❝çõ❡s ❝ô♥✐❝❛s ✈❛r✐❛♥❞♦ ❛♣❡♥❛s ❛ ✐♥❝❧✐♥❛çã♦ ❞♦ ♣❧❛♥♦ ❞❡ s❡❝çã♦✳ ❊st❡ ❢♦✐ ✉♠ ♣❛ss♦ ✐♠♣♦rt❛♥t❡✱ ♣♦✐s ❝♦♥tr✐❜✉✐✉ ♣❛r❛ r❡❧❛❝✐♦♥❛r ♦s três t✐♣♦s ❞❡ ❝✉r✈❛s✿ ❛ ❡❧✐♣s❡✱

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❋✐❣✉r❛ ✶✳✶✿ ❉✉♣❧✐❝❛çã♦ ❞♦ ❈✉❜♦✳

❋✐❣✉r❛ ✶✳✷✿ Pá❣✐♥❛ ❞♦ tít✉❧♦ ❞❛ ♣r✐♠❡✐r❛ ❡❞✐çã♦ ✐♠♣r❡ss❛ ❡♠ ❧❛t✐♠ ❞❛s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦✳

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❛ ♣❛rá❜♦❧❛ ❡ ❛ ❤✐♣ér❜♦❧❡✳ ❖✉tr❛ ❣❡♥❡r❛❧✐③❛çã♦ ✐♠♣♦rt❛♥t❡ ❝♦♥t✐❞❛ ♥❡st❡ tr❛t❛❞♦ é ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❝♦♥❡ ♥ã♦ ♣r❡❝✐s❛r s❡r r❡t♦❀ ❆♣♦❧ô♥✐♦ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❣❡ô♠❡tr❛ ❛ ♠♦str❛r q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❛s ❝✉r✈❛s ♥ã♦ ❡r❛♠ ❞✐❢❡r❡♥t❡s ❝♦♥❢♦r♠❡ ❢♦ss❡♠ ❝♦rt❛❞♦s ♦s ❝♦♥❡s ♦❜❧íq✉♦s ♦✉ r❡t♦s✳ ❚❛♠❜é♠ ❛❧❛r❣♦✉ ♦s s❡✉s ❡st✉❞♦s ❛♦ s✉❜st✐t✉✐r

♦ ❝♦♥❡ ❞❡ ✉♠❛ só ❢♦❧❤❛ ♣♦r ✉♠ ❞✉♣❧♦✱ ♦ q✉❡ ❢❡③ ❝♦♠ q✉❡ ❛♣r❡s❡♥t❛ss❡ ❛s ❝✉r✈❛s ❛♥t✐❣❛s ❞❡ ❢♦r♠❛ ♠❛✐s ♣ró①✐♠❛ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ✉s❛❞❛ ❤♦❥❡✳

❆ ♦❜r❛ ❆s ❈ô♥✐❝❛s✱ ❞❡ ❆♣♦❧ô♥✐♦✱ ❢♦✐ ❞✉r❛♠❡♥t❡ ❝r✐t✐❝❛❞❛ ♣♦r ❛❧❣✉♥s sá❜✐♦s ❞❡ s✉❛ é♣♦❝❛✱ q✉❡ ♥ã♦ ✈✐❛♠ ♥❡ss❡ ❡st✉❞♦ ♥❡♥❤✉♠❛ ❛♣❧✐❝❛çã♦ ♥♦ ♠✉♥❞♦ r❡❛❧✳ ▼❛s ♦ t❡♠♣♦

s❡ ✐♥❝✉♠❜✐✉ ❞❡ ♠♦str❛r q✉❡ ❡ss❡s sá❜✐♦s ❡st❛✈❛♠ ❡♥❣❛♥❛❞♦s✿ ♣♦r ✈♦❧t❛ ❞❡ ✶✻✵✺✱ ♦ ❛strô♥♦♠♦ ❛❧❡♠ã♦ ❏♦❤❛♥♥❡s ❑❡♣❧❡r ❞❡s❝♦❜r✐✉ q✉❡ ♦s ♣❧❛♥❡t❛s ❞❡s❝r❡✈❡♠ ór❜✐t❛s ❡❧í♣t✐❝❛s ❡♠ t♦r♥♦ ❞♦ ❙♦❧❀ ❡♠ ✶✻✸✷✱ ●❛❧✐❧❡✉ ●❛❧✐❧❡✐ ❞❡s❝r❡✈❡✉ ❝♦♠♦ ♣❛r❛❜ó❧✐❝❛ ❛

tr❛❥❡tór✐❛ ❞❡ ♣r♦❥ét❡✐s❀ ❡♠ ✶✻✻✷✱ ❘♦❜❡rt ❇♦②❧❡ ❞❡s❝♦❜r✐✉ q✉❡✱ s♦❜ t❡♠♣❡r❛t✉r❛ ❝♦♥st❛♥t❡✱ ❛ ❢✉♥çã♦ q✉❡ ❡①♣r❡ss❛ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♠❛ss❛ ✜①❛ ❞❡

❣ás ❡ ❛ ♣r❡ssã♦ ❡①❡r❝✐❞❛ s❡❜r❡ ❡❧❛ é ❤✐♣❡r❜ó❧✐❝❛✳

❋✐❣✉r❛ ✶✳✸✿ ❆s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✳

❆♣❡s❛r ❞❛ ♥♦t❛❜✐❧✐❞❛❞❡ ❞❛s ❈ô♥✐❝❛s✱ t❛♥t♦ ♣❡❧♦ ❝♦♥t❡ú❞♦ ❝♦♠♦ ♣❡❧❛ ✐♥♦✈❛çã♦ ❞♦s ♠ét♦❞♦s✱ ❆♣♦❧ô♥✐♦ ✜❝♦✉ ♥♦s ❡st✉❞♦s ❞❛s ❝✉r✈❛s ♠✉✐t♦ ❛q✉é♠ ❞❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❡ ❞❛

❣❡♥❡r❛❧✐❞❛❞❡ ❞❡ tr❛t❛♠❡♥t♦ ❞❛❞♦ ♠❛✐s t❛r❞❡ ♣♦r ❘❡♥ê ❉❡s❝❛rt❡s ✭✶✺✾✻✲✶✻✺✾✮ ❡ P✐❡rr❡ ❋❡r♠❛t ✭✶✻✵✶✲✶✻✻✺✮✳ ❆s ❧✐♠✐t❛çõ❡s ❞❛ ➪❧❣❡❜r❛ ●❡♦♠étr✐❝❛ ❡ ❛ ❛✉sê♥❝✐❛ ❞❡

s✐♠❜♦❧✐s♠♦ ❛❧❣é❜r✐❝♦ ❢♦r❛♠ ♦s ❢❛t♦r❡s ❞❡❝✐s✐✈♦s q✉❡ ✐♠♣❡❞✐r❛♠ ❆♣♦❧ô♥✐♦ ❞❡ t❡r s✐❞♦ ♦ ❝r✐❛❞♦r ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❋❛t♦ ❡ss❡ ❞❛❞♦ ❛ ❋❡r♠❛t ❡ ❉❡s❝❛rt❡s✳ ❍♦❥❡✱ ♣♦❞❡♠♦s ❝♦♥st❛t❛r ❛ ♣r❡s❡♥ç❛ ❞❛s ❝ô♥✐❝❛s ❡♠ ♠✉✐t❛s ♦✉tr❛s s✐t✉❛çõ❡s ❞♦

♠✉♥❞♦ r❡❛❧✱ ❝♦♠♦ ♥❛ ❝♦♥str✉çã♦ ❞❛s ❛♥t❡♥❛s ♣❛r❛❜ó❧✐❝❛s✱ ❡s♣❡❧❤♦s ❡ ❧❡♥t❡s ♣❛r❛❜ó❧✐❝♦s ♦✉ ❤✐♣❡r❜ó❧✐❝♦s❀ ♥❛s tr❛❥❡tór✐❛s ❡❧í♣t✐❝❛s✱ ♣❛r❛❜ó❧✐❝❛s ♦✉ ❤✐♣❡r❜ó❧✐❝❛s

❞❡ ❛str♦s ❝❡❧❡st❡s✱ ❞❡♥tr❡ ♦✉tr♦s✳

❱❡❥❛ ❛❧❣✉♠❛s ❝ô♥✐❝❛s q✉❡ ❛♣❛r❡❝❡♠ ♥❛s ✜❣✉r❛s ❛❜❛✐①♦✿

❉❛s ❈ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ s✐st❡♠át✐❝♦ r❡❧❛t✐✈♦ às P❛rá❜♦❧❛s ♥♦

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❋✐❣✉r❛ ✶✳✹✿ P✐❧❛r❡s ❞❛ ❝❛t❡❞r❛❧ ❞❡

❇r❛s✐❧✐❛✳ ❋♦r♠❛ ❞❡ ✉♠ ❤✐♣❡r❜♦❧♦✐❞❡✳ ❋✐❣✉r❛ ✶✳✺✿ ❈♦❧✐s❡✉✱ ❡♠ ❘♦♠❛✱ t❡♠❢♦r♠❛ ❡❧í♣t✐❝❛✳

❋✐❣✉r❛ ✶✳✻✿ P♦♥t❡ ❏✉s❝❡❧✐♥♦ ❑✉❜✐ts✲ ❝❤❡❦✱ ❡♠ ❇r❛sí❧✐❛✱ t❡♠ ❛ ❢♦r♠❛ ❞❛ ♣❛✲ rá❜♦❧❛✳

❋✐❣✉r❛ ✶✳✼✿ ❚♦rr❡s ❞❡ ❘❡❢r✐❣❡r❛çã♦ ❞❡ ✉s✐♥❛s ♥✉❝❧❡❛r❡s✱ tê♠ ❛ ❢♦r♠❛ ❞♦ ❤✐✲ ♣❡r❜♦❧♦✐❞❡✳

❛❞✈❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❡ ❛s ❋✉♥çõ❡s ◗✉❛❞rát✐❝❛s✱ ❡st❡✱ ♥♦ ❝♦♥t❡ú❞♦ ❞❡ ❋✉♥çõ❡s✳ ❆tr❛✈és ❞❛s ♥♦t❛çõ❡s ♠♦❞❡r♥❛s ❛♣r❡s❡♥t❛❞❛s ♣♦r ❋❡r♠❛t ❡ ❉❡s❝❛rt❡s ✐r❡♠♦s ❞❡s❡♥✈♦❧✈❡r ❛s t❡♦r✐❛s s♦❜r❡ ❛s ❝ô♥✐❝❛s ♥❛s ♣ró①✐♠❛s s❡çõ❡s✱ ♣♦ré♠✱ ♠✉✐t♦

♣ró①✐♠❛s ❞♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♣♦r ❆♣♦❧ô♥✐♦ ♥❛q✉❡❧❛ é♣♦❝❛✳

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❈❛♣ít✉❧♦ ✷

❆ P❛rá❜♦❧❛ ♥❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛

❯♠ ❞♦s ♠❛✐♦r❡s ❝♦♥tr✐❜✉✐♥t❡s ♣❛r❛ ❝♦♥str✉çã♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧✱ ❞♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ❡ ❞❛ ❚❡♦r✐❛ ❞❛s Pr♦❜❛❜✐❧✐❞❛❞❡ ❢♦✐ P✐❡rr❡ ❋❡r♠❛t✳

❙❡♠ s♦♠❜r❛ ❞❡ ❞ú✈✐❞❛✱ ❡❧❡ ❢♦✐ ♦ ❣r❛♥❞❡ ♥♦♠❡ ❞❛ ❢❛s❡ ✐♥✐❝✐❛❧ ❞❛ ♠♦❞❡r♥❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❋❡r♠❛t ❝✉rs♦✉ ❉✐r❡✐t♦ ❡♠ ❚♦✉❧♦✉s❡ ♥❛ ❋r❛♥ç❛ ❡♠ ❝✉❥♦ ♣❛r❧❛♠❡♥t♦

❝♦♠❡ç♦✉ ❛ tr❛❜❛❧❤❛r ♥♦ ❛♥♦ ❞❡ ✶✻✸✶✱ ♣r✐♠❡✐r♦ ❝♦♠♦ ❛❞✈♦❣❛❞♦ ❡✱ ♣♦st❡r✐♦r♠❡♥t❡ ❝♦♠♦ ❝♦♥s❡❧❤❡✐r♦✳ ❈♦♠♦ ❞♦♠✐♥❛✈❛ ❛s ❞✉❛s ❛t✐✈✐❞❛❞❡s✱ ♥✐♥❣✉é♠ ♣♦❞❡r✐❛ ✐♠❛❣✐♥❛r q✉❡ s✉❛ ✈♦❝❛çã♦ ❝✉❧t✐✈❛❞❛ ❝♦♠ ❣r❛♥❞❡ t❛❧❡♥t♦ ♥❛s ❤♦r❛s ❞❡ ❧❛③❡r ❡r❛ ❛ ▼❛t❡♠át✐❝❛✳

P❡❧❛ s✉❛ ❝♦♥❞✐çã♦ ❞❡ ❛♠❛❞♦r ♥❛ ár❡❛ ❞❛ ▼❛t❡♠át✐❝❛ ❋❡r♠❛t✱ r❡❝✉s❛✈❛✲s❡ s✐st❡♠❛t✐❝❛♠❡♥t❡ ❛ ♣✉❜❧✐❝❛r s✉❡s tr❛❜❛❧❤♦s ❡✱ s❡ ❡ss❡s sã♦ r❡❝♦♥❤❡❝✐❞♦s ❤♦❥❡✱ é

♣♦rq✉❡ ✜❝❛r❛♠ r❡❣✐str❛❞♦s ❡♠ ♠❛r❣❡♥s ❞❡ ❧✐✈r♦s✱ ❢♦❧❤❛s ❛✈✉❧s❛s ❡ ❝❛rt❛s✳ ❆ ●❡♦♠❡tr✐❛ ❞❡ P✐❡rr❡ ❋❡r♠❛t ❛ ♣❛rt✐r ❞❡ ✶✻✷✾ ❡♠♣r❡❡♥❞❡✉ ❛ r❡❝♦♥str✉✐r ❛ ♦❜r❛

❞❡ ❆♣♦❧ô♥✐♦ ▲✉❣❛r❡s P❧❛♥♦s✱ ♠❡❞✐❛♥t❡ ❛ r❡❢❡rê♥❝✐❛s ❝♦♥t✐❞❛s ♥❛ ❈♦❧❡çã♦ ▼❛t❡♠át✐❝❛✱ ❞❡ P❛♣✉s✳ ❋♦✐ ✉♠ ♣❡q✉❡♥♦ tr❛t❛❞♦ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✻✼✾✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ■♥tr♦❞✉çã♦ ❛♦s ❧✉❣❛r❡s ♣❧❛♥♦s ❡ só❧✐❞♦s✳ ◆❡ss❡ tr❛t❛❞♦✱ ❛♦ ❛♥✉♥❝✐❛r q✉❡ ❞❛❞❛

✉♠❛ ❡q✉❛çã♦ ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s✱ ✉♠❛ ❞❡st❛s ❞❡s❝r❡✈❡ ✉♠❛ r❡t❛ ♦✉ ✉♠❛ ❝✉r✈❛✱ r❡✈❡❧❛✈❛ ❡❧❡ ❡♥tã♦✱ ❡stá ❞❡ ♣♦ss❡ ❞♦ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ♥♦✈♦ ♠ét♦❞♦✳ ❊❧❡

♣ró♣r✐♦ ♠♦str♦✉ q✉❡ ✉♠❛ ❡q✉❛çã♦ ❣❡r❛❧ ax+by =c ❡♠ q✉❡ a₆= 0 ♦✉ b₆= 0 r❡♣r❡s❡♥t❛✈❛ ✉♠❛ r❡t❛❀ ❡ q✉❡ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡♠ ❞✉❛s ✈❛r✐á✈❡✐s ♣♦❞❡♠

s❡r ❝ír❝✉❧♦s✱ ❡❧✐♣s❡s✱ ♣❛rá❜♦❧❛s ♦✉ ❤✐♣ér❜♦❧❡s✳

❱❡r❡♠♦s ❡♥tã♦ ❞❛s ❝ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ✉♠ ❡st✉❞♦ ❆♥❛❧ít✐❝♦ ❞❛ P❛rá❜♦❧❛✱ ❛ ❝♦♠❡ç❛r ♣♦r ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♣r❡❧✐♠✐♥❛r❡s✿

✷✳✶ ❈♦♦r❞❡♥❛❞❛s ♥♦ P❧❛♥♦

❉❛❞♦ ✉♠ ♣♦♥t♦ P ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❝❤❛♠❛✲s❡ ❞❡ ✧♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ P s♦❜r❡ ✉♠ ❞♦s ❡✐①♦s Ox ♦✉Oy✧❛ ✐♥t❡rs❡❝çã♦ ❞❡ss❡ ❡✐①♦ ❝♦♠ ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛

❡❧❡✱ tr❛ç❛❞♦ ♣♦r P✳ ❖♥❞❡✿

(17)

✷✳✷✳ ❉■❙❚➶◆❈■❆ ❊◆❚❘❊ ❉❖■❙ P❖◆❚❖❙

❋✐❣✉r❛ ✷✳✶✿ ❈♦♦r❞❡♥❛❞❛s ♥♦ P❧❛♥♦

• P1 é ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ P s♦❜r❡ ♦ ❡✐①♦ Ox✳

• P2 é ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ P s♦❜r❡ ♦ ❡✐①♦ Oy✳

• ❉✐③❡♠♦s q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P sã♦ ♦s ♥ú♠❡r♦s ❛ss♦❝✐❛❞♦s ❛ P1 ❡ P2 ♥♦s ❡✐①♦ Ox ❡ Oy✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦rt❛♥t♦✱ r❡♣r❡s❡♥t❛♠♦s ♦ ♣♦♥t♦ P ❞❛

❢♦r♠❛ P = (P1, P2)✳

✷✳✷ ❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s

◆❛ ✜❣✉r❛ ✭✷✳✷✮✱ t❡♠♦s ✉♠ s❡❣♠❡♥t♦ AB ♥ã♦ ♣❛r❛❧❡❧♦ ❛ ♥❡♥❤✉♠ ❞♦s ❡✐①♦s

❝♦♦r❞❡♥❛❞♦s✳

❋✐❣✉r❛ ✷✳✷✿ ❉✐stâ♥❝✐❛ ❞❡ ❞♦✐s ♣♦♥t♦s

❖ tr✐â♥❣✉❧♦ ❆❇❈✱ ❝♦♠ A= (xA, yA), B = (xB, yB) ❡ C = (xC, yC)✱ é r❡tâ♥❣✉❧♦ ❡♠

(18)

✷✳✸✳ ❉■❙❚➶◆❈■❆ ❊◆❚❘❊ P❖◆❚❖ ❊ ❘❊❚❆

❈✳

AC é ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ Ox✳ ❆ss✐♠✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❆❈ é ✐❣✉❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛

♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ AC s♦❜r❡ ♦ ❡✐①♦ Ox✱ ♦✉ s❡❥❛✱AC =_|xB−xA|. ❆♥❛❧♦❣❛♠❡♥t❡✱ CB é ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ Oy❀ ❧♦❣♦✿ CB =_|yB−yA|.

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ t❡♠♦s✿ (AB)2

= (AC)2

+ (CB)2

⇒(AB)2

=_|xB−xA|2+|yB−yA|2

∴(AB)2

= (xB−xA)2+ (yB−yA)2 ∴AB =± p

(xB−xA)2+ (yB−yA)2. ❈♦♠♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ♥ã♦ ♣♦❞❡ s❡r ♥❡❣❛t✐✈♦✱ t❡♠♦s q✉❡✿

AB =p(xB−xA)2+ (yB−yA)2.

■st♦ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿

❉❡✜♥✐çã♦ ✷✳✶ ❆ ❞✐stâ♥❝✐❛ dAB ❡♥tr❡ ❞♦✐s ♣♦♥t♦s A = (xA, yA) ❡ B = (xB, yB) é ❞❛❞❛ ♣♦r✿

❞AB = p

(xB−xA)2+ (yB−yA)2

✷✳✸ ❉✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦ ❡ r❡t❛

❱❡❥❛♠♦s ❛❣♦r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ✉♠ ♣♦♥t♦ P ❡ ✉♠❛ r❡t❛ r✳ ❊ss❛ ❞✐stâ♥❝✐❛ é ❞❡✜♥✐❞❛ ❛ s❡❣✉✐r✳

❙❡❥❛ P ✉♠ ♣♦♥t♦ ❡ r ✉♠❛ r❡t❛✳ ❆ ❞✐stâ♥❝✐❛ dP r✱ ❞♦ ♣♦♥t♦ P à r❡t❛ r✱ é ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ P P′✱ ♦♥❞❡ _P′ _{é ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡}_P _{s♦❜r❡ r✳}

❚❡♦r❡♠❛ ✷✳✶ ❆ ❞✐stâ♥❝✐❛ d❡♥tr❡ ✉♠ ♣♦♥t♦P = (x0, y0)❡ ✉♠❛ r❡t❛r :ax+by+c= 0 é ❞❛❞❛ ♣♦r✿

(19)

✷✳✸✳ ❉■❙❚➶◆❈■❆ ❊◆❚❘❊ P❖◆❚❖ ❊ ❘❊❚❆

❞ ❂ |ax0√+by0+c|

a2₊_b2

❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ ❝❛s♦✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣♦♥t♦P = (x0, y0) ❡ ✉♠❛ r❡t❛ r ❞❡ ❡q✉❛çã♦ ❣❡r❛❧ax+by+c= 0✱ t❛✐s q✉❡ P /_∈r ❡r ♥ã♦ é ✈❡rt✐❝❛❧ ♥❡♠

❤♦r✐③♦♥t❛❧✿

❋✐❣✉r❛ ✷✳✸✿ ❉✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ à r❡t❛

E é ♦ ♣♦♥t♦ ❞❛ r❡t❛r ❞❡ ❛❜s❝✐ss❛ x0✳ ❙✉❜st✐t✉✐♥❞♦ x=x0 ♥❛ ❡q✉❛çã♦ r✱ t❡♠♦s✿

ax0+by+c= 0_⇒y=₋ax0+c

b .

▲♦❣♦✱ E =

x0,₋ax0+c

b

✳

F é ♦ ♣♦♥t♦ ❞❛ r❡t❛r ❞❡ ♦r❞❡♥❛❞❛ y0✳ ❙✉❜st✐t✉✐♥❞♦y =y0 ♥❛ ❡q✉❛çã♦ ❞❡ r✱ t❡♠♦s✿

ax+by0+c= 0 _⇒x=₋by0+c

a .

▲♦❣♦✱ F =

− by0_a+c, y0

✳

▲❡♠❛✿ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ♦ ♣r♦❞✉t♦ ❞❛ ♠❡❞✐❞❛ ❞❛ ❤✐♣♦t❡♥✉s❛ ♣❡❧❛ ♠❡❞✐❞❛ ❞❡ s✉❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛ ❡ss❛ ❤✐♣♦t❡♥✉s❛ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛s ♠❡❞✐❞❛s ❞♦s

❝❛t❡t♦s✳ ❆ss✐♠ s❡♥❞♦✱ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ P EF✱ t❡♠♦s✿

(20)

✷✳✸✳ ❉■❙❚➶◆❈■❆ ❊◆❚❘❊ P❖◆❚❖ ❊ ❘❊❚❆

EF _·d=P E_·P F

⇒

s

x0+by0+c

a

2

+

y0+ax0+c

b

2

·d = s

y0+ax0+c

b

2

·

s

x0+by0 +c

a

2

⇒

r

(ax0+by0+c)2

a2 +

(by0+ax0+c)2

b2 · d =

r

(by0+ax0+c)2

b2 ·

r

(ax0+by0+c)2 a2

⇒

s

(ax0+by0+c)2

a2

+b2 a2_b2

· d = r

(by0+ax0+c)2

b2 ·

r

(ax0+by0+c)2 a2

⇒ |ax0+by0+c_|

√

a2₊_b2

|a_{| · |}b_| · d =

|by0+ax0+c_|

|b_| ·

|ax0+by0+c_|

|a_| .

❖ ♣♦♥t♦ P ♥ã♦ ♣❡rt❡♥❝❡ ❛ r❀ ❡♥tã♦ ax0+by0+c₆= 0✳ ▲♦❣♦✱ t❡♠♦s √a2₊_b2

· d=_|ax0+by0+c_{| ⇒} d= |ax0√+by0+c|

a2₊_b2 .

❙❡❣✉♥❞♦ ❝❛s♦✿ ✭❈❛s♦s ♣❛rt✐❝✉❧❛r❡s✮ ✶◦_✮ _P _∈_r

❙❡ ♦ ♣♦♥t♦ P = (x0, y0) ♣❡rt❡♥❝❡ à r❡t❛ r:ax+by+c= 0✱ ❡♥tã♦ t❡♠♦s q✉❡ ax0+by0+c= 0 ❡ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛r é ✐❣✉❛❧ ❛ ③❡r♦✳ ◆♦t❡✱ ♣♦rt❛♥t♦✱ q✉❡ ❛

❢ór♠✉❧❛ ❞❡❞✉③✐❞❛ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦ ✈❛❧❡ t❛♠❜é♠ ♣❛r❛ ❡ss❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ♣♦✐s✿

d= |ax0_√+by0+c|

a2₊_b2 =

|0_|

√

a2₊_b2 = 0.

✷◦_✮_r _{é ✈❡rt✐❝❛❧}

❙❡r é ✈❡rt✐❝❛❧✱ ❡♥tã♦ s✉❛ ❡q✉❛çã♦ ❣❡r❛❧ é ❞❛ ❢♦r♠❛ax+c= 0✱ ❝♦♠ a₆= 0✳ ❆ ❞✐stâ♥❝✐❛ d é ❞❛❞❛ ♣♦r✿

d= x0+ c a =

ax0+c

a

=

|ax0+c_|

|a_| .

◆♦t❡ q✉❡ ❛ ❢ór♠✉❧❛ ♦❜t✐❞❛ ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ✈❛❧❡ t❛♠❜é♠ ♣❛r❛ ❡ss❡ ❝❛s♦✱ ♣♦✐s✿

d= |ax0√+ 0y0+c|

a2_{+ 0}2 =

|ax0+c_|

|a_| .

✸◦_✮_r _{é ❤♦r✐③♦♥t❛❧}

❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ s❡❣✉♥❞♦ ❝❛s♦✱ ♣r♦✈❛✲s❡ q✉❡ ❛ ❢ór♠✉❧❛ ♦❜t✐❞❛ ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ✈❛❧❡ t❛♠❜é♠ ♣❛r❛ r ❤♦r✐③♦♥t❛❧✳

(21)

✷✳✹✳ ❉❊❋■◆■➬➹❖ ❉❊ P❆❘➪❇❖▲❆

✷✳✹ ❉❡✜♥✐çã♦ ❞❡ P❛rá❜♦❧❛

❉❡✜♥✐çã♦ ✷✳✷ ❉❛❞♦ ✉♠ ♣♦♥t♦ F ❡ ✉♠❛ r❡t❛d ❞❡ ✉♠ ♣❧❛♥♦ α✱ ❝♦♠F /_∈d✱ ❝❤❛♠❛✲

s❡ ♣❛rá❜♦❧❛ P ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❢♦r♠❛❞♦s ♣❡❧♦s ♣♦♥t♦s ❞❡ss❡ ♣❧❛♥♦ ❡q✉✐❞✐st❛♥t❡ ❞❡ d ❡ F✳

✷✳✺ ❊❧❡♠❡♥t♦s ♣r✐♥❝✐♣❛✐s

❈♦♠ ❜❛s❡ ♥❛ ❞❡✜♥✐çã♦ ✹✳✶✱ ♦ ♣♦♥t♦ F ❡ ❛ r❡t❛ d sã♦ ♦ ❢♦❝♦ ❡ ❛ ❞✐r❡tr✐③ ❞❛

♣❛rá❜♦❧❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r F ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à ❞✐r❡tr✐③ dé

♦ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ ✭❊✐①♦ e✮ ❞❛ ♣❛rá❜♦❧❛✳ ❖ ♣♦♥t♦ V✱ ✐♥t❡rs❡❝çã♦ ❞❛ ♣❛rá❜♦❧❛

❝♦♠ ♦ ❡✐①♦ ❞❡ s✐♠❡tr✐❛✱ é ♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛✳ ❆ ❞✐stâ♥❝✐❛ p❞♦ ❢♦❝♦ F à ❞✐r❡tr✐③ d é ❝❤❛♠❛❞♦ ❞❡ ♣❛râ♠❡tr♦ ❞❛ ♣❛rá❜♦❧❛✳

◆♦t❡ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ ✈ért✐❝❡ V ❡ ♦ ❢♦❝♦ F é ♠❡t❛❞❡ ❞♦ ♣❛râ♠❡tr♦ p✱ ♣♦✐sV

♣❡rt❡♥❝❡ à ♣❛rá❜♦❧❛ ❡✱ ♣♦rt❛♥t♦✱ V ❡q✉✐❞✐st❛ ❞❡ F ❡d✳ ❆ss✐♠✱ t❡♠♦s✿

V F =V d

V F +V d=p

⇒ ❱❋ ❂ p₂ ❡ ❱❞ ❂ p₂

❆❜❛✐①♦✱ t❡♠♦s ❛ ✜❣✉r❛ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ ♦❜t✐❞❛ ❛tr❛✈és ❞❛ s❡❝çã♦ tr❛♥s✈❡rs❛❧ ♥♦ ❝♦♥❡✳

(22)

✷✳✺✳ ❊▲❊▼❊◆❚❖❙ P❘■◆❈■P❆■❙

❋✐❣✉r❛ ✷✳✹✿ ❉❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛

❋✐❣✉r❛ ✷✳✺✿ ❙❡❝çã♦ ❞♦ ❝♦♥❡

(23)

✷✳✻✳ ❊◗❯❆➬➹❖ ❉❆ P❆❘➪❇❖▲❆

✷✳✻ ❊q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛

❙❡❥❛♠ F = (x0, y0) ❡ d:ax+by+c= 0 ♦ ❢♦❝♦ ❡ ❛ ❞✐r❡tr✐③ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖❜té♠✲s❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ss❛ ♣❛rá❜♦❧❛ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠ ♣♦♥t♦ ❣❡♥ér✐❝♦P = (x, y) ❡ ✐♠♣♦♥❞♦ q✉❡ P F =P d✱ ♦✉ s❡❥❛✿

p

(x₋x0)2_{+ (}_y

−y0)2 ₌ |ax_√+by+c|

a2₊_b2 .

❱❛♠♦s ♠♦str❛r ❛ s❡❣✉✐r q✉❡ t♦❞❛s ❛s ❝ô♥✐❝❛s ♥ã♦ ❞❡❣❡♥❡r❛❞❛s✱ ❝♦♠ ❡①❝❡çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣♦❞❡♠ s❡r ❞❡s❝r✐t❛s ❞❡ ✉♠❛ ♠❡s♠❛ ♠❛♥❡✐r❛✳

Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡❥❛ d ✉♠❛ r❡t❛ ✜①❛ ✭❞✐r❡tr✐③✮ ❡ F ✉♠ ♣♦♥t♦ ✜①♦ ✭❢♦❝♦✮ ♥ã♦

♣❡rt❡♥❝❡♥t❡ ❛ d✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ P ❂ ✭①✱ ②✮ t❛✐s q✉❡

❞✐st✭P✱ ❋✮❂ ❡ · dist(P, d) ✭✷✳✶✮

(24)

✷✳✻✳ ❊◗❯❆➬➹❖ ❉❆ P❆❘➪❇❖▲❆

❡♠ q✉❡ e >0 é ✉♠❛ ❝♦♥st❛♥t❡ ✜①❛ ❞❡♥♦♠✐♥❛❞❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ ✶✱ ✭❛✮ s❡ e= 1✱ ❡♥tã♦ ❛ ❝ô♥✐❝❛ é ✉♠❛ ♣❛rá❜♦❧❛✳

✭❜✮ s❡ 0< e <1✱ ❡♥tã♦ ❛ ❝ô♥✐❝❛ é ✉♠❛ ❡❧✐♣s❡✳ ✭❝✮ s❡ e >1✱ ❡♥tã♦ ❛ ❝ô♥✐❝❛ é ✉♠❛ ❤✐♣ér❜♦❧❡✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ t♦❞❛ ❝ô♥✐❝❛ q✉❡ ♥ã♦ s❡❥❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛ ✷✳✶✳

❉❡♠♦♥str❛çã♦✿ ❙❡ e= 1 ❛ ❡q✉❛çã♦ ✷✳✶ é ❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❛ ♣❛rá❜♦❧❛✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ❝❛s♦ ❡♠ q✉❡e >0✱ ❝♦♠ e₆= 1✳ ❙❡❥❛ D=dist(F, d)✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡♠♦s t♦♠❛r ♦ ❢♦❝♦ ❝♦♠♦ s❡♥❞♦ ♦ ♣♦♥t♦ F = (p,0)❡ ❛ ❞✐r❡tr✐③

❝♦♠♦ s❡♥❞♦ ❛ r❡t❛ ✈❡rt✐❝❛❧ d:x= p

e2✱ ❡♠ q✉❡ p=

De2

1₋e2 s❡ ❛ r❡t❛ d ❡st✐✈❡r ❛

❞✐r❡✐t❛ ❞♦ ❢♦❝♦ F ✭❋✐❣✉r❛s ✹✳✻ ❡ ✹✳✼✮ ❡ p= De

2

e2

−1 s❡ ❛ r❡t❛ d ❡st✐✈❡r ❛ ❡sq✉❡r❞❛ ❞♦ ❢♦❝♦ F ✭❋✐❣✉r❛s ✹✳✽ ❡ ✹✳✾✮✳

❋✐❣✉r❛ ✹✳✻✿ ❊❧✐♣s❡✱ ✉♠ ❞♦s s❡✉s ❢♦❝♦s ❡ ❛ r❡t❛ ❞✐r❡tr✐③ à ❞✐r❡✐t❛✳

❋✐❣✉r❛ ✹✳✼✿ ❍✐♣ér❜♦❧❡✱ ✉♠ ❞♦s s❡✉s ❢♦❝♦s ❡ ❛ r❡t❛ ❞✐r❡✲ tr✐③ à ❞✐r❡✐t❛✳

✶_{❊①❝❡♥tr✐❝✐❞❛❞❡ é ✉♠ ♣❛râ♠❡tr♦ ❛ss♦❝✐❛❞♦ ❛ q✉❛❧q✉❡r ❝ô♥✐❝❛✱ q✉❡ ♠❡❞❡ ♦ s❡✉ ❞❡s✈✐♦ ❡♠}

r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

(25)

✷✳✻✳ ❊◗❯❆➬➹❖ ❉❆ P❆❘➪❇❖▲❆

❋✐❣✉r❛ ✹✳✽✿ ❊❧✐♣s❡✱ ✉♠ ❞♦s s❡✉s ❢♦❝♦s ❡ ❛ r❡t❛ ❞✐r❡tr✐③ à ❡sq✉❡r❞❛✳

❋✐❣✉r❛ ✹✳✾✿ ❍✐♣ér❜♦❧❡✱ ✉♠ ❞♦s s❡✉s ❢♦❝♦s ❡ ❛ r❡t❛ ❞✐r❡✲ tr✐③ à ❡sq✉❡r❞❛✳

❆ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s P = (x, y) t❛✐s q✉❡

dist(P, F) = e_·dist(P, d),

♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s P = (x, y) t❛✐s q✉❡

p

(x₋p)2₊_y2 ₌ _e

x−

p e2

✱

❡❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ ❡ s✐♠♣❧✐✜❝❛♥❞♦✱ ♦❜t❡♠♦s

(1₋e2

)x2

+y2

= p2

1

e2 −1

✱

q✉❡ ♣♦❞❡ ❛✐♥❞❛ s❡r ❡s❝r✐t❛ ❝♦♠♦

x2 p2 e2

+ y

2

p2

(1₋e2

)

e2

✳ ✭✷✳✷✮

❙❡ 0< e < 1✱ ❡st❛ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ ❡❧✐♣s❡✳ ❙❡ e >1✱ é ❛ ❡q✉❛çã♦ ❞❡ ✉♠❛

❤✐♣ér❜♦❧❡✳ P❛r❛ ♠♦str❛r ❛ r❡❝í♣r♦❝❛✱ ❝♦♥s✐❞❡r❡ ✉♠❛ ❡❧✐♣s❡ ♦✉ ❤✐♣ér❜♦❧❡ ❝♦♠ ❡①❝❡♥tr✐❝✐❞❛❞❡ e >0 ❡ ✉♠ ❞♦s ❢♦❝♦s F = (p,0)✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ ✷✳✷ é ❛ ❡q✉❛çã♦ ❞❡st❛ ❝ô♥✐❝❛ ❡ ♣♦rt❛♥t♦ ✷✳✶ t❛♠❜é♠ ♦ é✱ ❝♦♠ ❛ r❡t❛ ❞✐r❡tr✐③ s❡♥❞♦

d:x= p

e2✳

(26)

✷✳✼✳ ❊◗❯❆➬Õ❊❙ ❈❆◆Ô◆■❈❆❙ ❉❆ P❆❘➪❇❖▲❆ ❈❖▼ ❱➱❘❚■❈❊ ◆❆ ❖❘■●❊▼

✷✳✼ ❊q✉❛çõ❡s ❈❛♥ô♥✐❝❛s ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ ❱ért✐❝❡

♥❛ ♦r✐❣❡♠

❱❡❥❛♠♦s ❡♥tã♦ ❝♦♠♦ ❞❡t❡r♠✐♥❛r ❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s ♦✉ ❡q✉❛çõ❡s r❡❞✉③✐❞❛s ❞❛s ♣❛rá❜♦❧❛s ❡st✉❞❛♥❞♦✲❛s ❝❛s♦ ❛ ❝❛s♦✳

✶♦ _{❈❛s♦✿ ❖ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❡✐①♦ ①}

◆❛ ✜❣✉r❛ ❛♦ ❧❛❞♦ t❡♠✲s❡ ✉♠❛ ♣❛rá❜♦❧❛ ❝♦♠ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ r❡♣r❡s❡♥t❛❞❛ ♥♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ①❖② ❡ ❝✉❥❛ ❞✐r❡tr✐③ t❡♠ ❡q✉❛çã♦ x=₋p

2✳ ❚❡♠♦s t❛♠❜é♠ q✉❡✿

P = (x, y) é ✉♠ ♣♦♥t♦ ❣❡♥ér✐❝♦ ❞❛ ♣❛rá✲ ❜♦❧❛✳

F =

p

2,0

é ♦ ❢♦❝♦✳

P′ ₌

− p₂, y

é ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞♦ ♣♦♥t♦ P s♦❜r❡ ❛ ❞✐r❡tr✐③✳

P♦r ❞❡✜♥✐çã♦✱ t❡♠♦s q✉❡✿

d(P, F) = d(P, P′)✱ ❞❛í

s

x₋ p

2 2

+ (y₋0)2

= s

x+p

2 2

+ (y₋y)2

❊❧❡✈❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❛♦ q✉❛❞r❛❞♦✱ t❡♠♦s q✉❡✿

x2

−px+ p

2

4 +y

2

=x2

+px+p

2

4 ❉♦♥❞❡✿ y2

= 2px✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡ ♥❛

♦r✐❣❡♠ ❡ ❝✉❥♦ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ é ♦ ❡✐①♦ x✳

◆❛ ❡q✉❛çã♦ y2

= 2px✱ ♣♦❞❡♠♦s t❡r ❛s s❡❣✉✐♥t❡s s✐t✉❛çõ❡s✿

(27)

✷✳✼✳ ❊◗❯❆➬Õ❊❙ ❈❆◆Ô◆■❈❆❙ ❉❆ P❆❘➪❇❖▲❆ ❈❖▼ ❱➱❘❚■❈❊ ◆❆ ❖❘■●❊▼

❙❡p > 0✱ ❛ ♣❛rá❜♦❧❛ t❡♠ ❝♦♥✲

❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❛ ❞✐✲ r❡✐t❛✱

❙❡p <0✱ ❛ ♣❛rá❜♦❧❛ t❡♠ ❝♦♥✲

❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❛ ❡s✲ q✉❡r❞❛✳

✷♦ _{❈❛s♦✿ ❖ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❡✐①♦ ②}

❆ ✜❣✉r❛ ❛♦ ❧❛❞♦ t❡♠✲s❡ ✉♠❛ ♣❛rá❜♦❧❛ ❝♦♠ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ r❡♣r❡s❡♥t❛❞❛ ♥♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ①❖② ❡ ❝✉❥❛ ❞✐r❡tr✐③ t❡♠ ❡q✉❛çã♦ y=₋p

2✳ ❚❡♠♦s t❛♠❜é♠ q✉❡✿

P = (x, y)✳

F =

0,p

2

✳

P′ ₌

x,₋p

2

✳

(28)

✷✳✼✳ ❊◗❯❆➬Õ❊❙ ❈❆◆Ô◆■❈❆❙ ❉❆ P❆❘➪❇❖▲❆ ❈❖▼ ❱➱❘❚■❈❊ ◆❆ ❖❘■●❊▼

P♦r ❞❡✜♥✐çã♦✱ t❡♠♦s q✉❡✿

d(P, F) = d(P, P′₎_{✱ ❞❛í}

s

(x₋0)2₊

y₋ p

2 2

= s

(x₋x)2₊

y+ p

2 2

❊❧❡✈❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❛♦ q✉❛❞r❛❞♦✱ t❡♠♦s q✉❡✿

x2

+y2

−py+p

2

4 =y

2

+py+ p

2

4 ❉♦♥❞❡✿ x2

= 2py✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡ ♥❛

♦r✐❣❡♠ ❡ ❝✉❥♦ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ é ♦ ❡✐①♦ y✳

◆❛ ❡q✉❛çã♦ x2

= 2py✱ ♣♦❞❡♠♦s t❡r ❛s s❡❣✉✐♥t❡s s✐t✉❛çõ❡s✿

❙❡p >0✱ ❛ ♣❛rá❜♦❧❛ t❡♠ ❝♦♥❝❛✈✐❞❛❞❡

✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛✳ ❙❡✈♦❧t❛❞❛ ♣❛r❛ ❜❛✐①♦✳p <0✱ ❛ ♣❛rá❜♦❧❛ t❡♠ ❝♦♥❝❛✈✐❞❛❞❡ ❊①❡♠♣❧♦ ✶✮ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ ❞❡ ❢♦❝♦F = (0,₋5) ❡ ❞✐r❡tr✐③

d:y= 5✳

❘❡s♦❧✉çã♦✿ ❆♦ ♠❛r❝❛r♠♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ♦ ❢♦❝♦ F = (0,₋5) ❡ ❞✐r❡tr✐③

y= 5✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♦ ✈ért✐❝❡ ❡st❛rá ♥❛ ♦r✐❣❡♠✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳

(29)

✷✳✼✳ ❊◗❯❆➬Õ❊❙ ❈❆◆Ô◆■❈❆❙ ❉❆ P❆❘➪❇❖▲❆ ❈❖▼ ❱➱❘❚■❈❊ ◆❆ ❖❘■●❊▼

❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛✱ t❡♠♦s✿ ❞✭P✱ ❋✮ ❂ ❞✭P✱ ❞✮

⇒ px2_{+ (}_y_{+ 5)}2 ₌p₍_x

−x)2_{+ (}_y

−5)2

⇒ x2

+y2

+ 10y+ 25 =y2

−10y+ 25

⇒ x2

=₋20y

❊①❡♠♣❧♦ ✷✮ ❉❡t❡r♠✐♥❡ ♦ ❢♦❝♦ ❡ ❛ ❞✐r❡tr✐③ ❞❛ ♣❛rá❜♦❧❛ ❞❡ ❡q✉❛çã♦y2

= 5x✳

❘❡s♦❧✉çã♦✿ ❆ ❡q✉❛çã♦ y2

= 5x é ✉♠❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛ y2

= 2px✱ ❝♦♠ p > 0✳

❈♦♠♣❛r❛♥❞♦ ❛s ❡q✉❛çõ❡s✱ t❡♠♦s✿ 2p= 5 _⇒ p= 5

2 ⇒

p

2 = 5 4 ❉❛í✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❢♦❝♦ sã♦ F =

5 4,0

❡ ❛ r❡t❛ ❞✐r❡tr✐③ éx=₋5 4✳

(30)

✷✳✽✳ ❚❘❆◆❙▲❆➬➹❖ ❉❊ ❊■❳❖❙

✷✳✽ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s

❯♠❛ ✈❡③ ❝♦♥❤❡❝✐❞❛ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠ ♣♦♥t♦ ♦✉ ❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝❡rt♦ s✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s

❞❡ss❡ ♣♦♥t♦ ♦✉ ❞❛ ♥♦✈❛ ❡q✉❛çã♦ ❞❛ ❝✉r✈❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ♥♦✈♦ s✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛✳ ❆ss✐♠✱ ❛ ❝✉r✈❛ ❝✉❥❛ ❡q✉❛çã♦ f(x, y) = 0 q✉❛♥❞♦ r❡❢❡r✐❞❛ ❛ ✉♠ s✐st❡♠❛

❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ①❖② tr❛♥s❢♦r♠❛r✲s❡✲á ♥✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦

F(x′_{, y}′_{) = 0✱ q✉❛♥❞♦ r❡❢❡r✐❞❛ ❛ ✉♠ ♥♦✈♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s}

①✬❖✬②✬✳

❊ss❡ ♥♦✈♦ s✐st❡♠❛ é ♦❜t✐❞♦ ❛tr❛✈és ❞❡ ✉♠❛ tr❛♥s❧❛çã♦ ❞❡ ❡✐①♦s ❡✴♦✉ ✉♠❛ r♦t❛çã♦ ❞❡ ❡✐①♦s✳ ❊♥❢❛t✐③❡✲s❡ q✉❡ ♥✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✭♠❡❞✐❛♥t❡ ✉♠❛ tr❛♥s❧❛çã♦ ♦✉ r♦t❛çã♦✮ ♥ã♦ é ❛❢❡t❛❞❛ ❛ ❢♦r♠❛ ❞❛ ❝✉r✈❛ ♦✉ ♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛✳ ◆♦

❡♥t❛♥t♦✱ ❤á ❛❧t❡r❛çã♦ ♥❛ ❡q✉❛çã♦ ❞❡ss❛ ❝✉r✈❛✳

◆♦ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ①❖②✱ ❝♦♥s✐❞❡r❡ ✉♠ ♣♦♥t♦ ❖✬ ❂ (x0, y0)✳ ■♥tr♦❞✉③✐♥❞♦ ✉♠

♥♦✈♦ s✐st❡♠❛ ①✬❖✬②✬ t❛❧ q✉❡ ❖✬ s❡❥❛ ❛ ♥♦✈❛ ♦r✐❣❡♠ ❡ ♦ ❡✐①♦ ❖✬①✬ t❡♥❤❛ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ❞❡ ❖① ❡ ❖✬②✬ t❡♥❤❛ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ❞❡ ❖②✳ ❉✐③❡♠♦s q✉❡ ♦ ♥♦✈♦ s✐st❡♠❛ ①✬❖✬②✬ ❢♦✐ ♦❜t✐❞♦ ♣♦r ✉♠❛ tr❛♥s❧❛çã♦ ❞♦ ❛♥t✐❣♦ s✐st❡♠❛ ①❖②✳ ❊♠ ❛♠❜♦s ♦s s✐st❡♠❛s s❡ ❝♦♥s❡r✈❛♠ ❛s ✉♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r ♦ ♥♦✈♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡♥tr♦ ❞♦ ♣❧❛♥♦

①❖②✳

❉❡ss❛ ❢♦r♠❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P ❡♠ r❡❧❛çã♦ ❛♦s s✐st❡♠❛ ①❖② ❡ ①✬❖✬②✬ sã♦

r❡❧❛❝✐♦♥❛❞❛s ♣❡❧❛s ❢ór♠✉❧❛s✿

x=x0+x′

y=y0+y′

❱❡❥❛♠♦s ❡♥tã♦ ❝♦♠♦ ❞❡t❡r♠✐♥❛r ❛s ❢♦r♠❛s ❝❛♥ô♥✐❝❛s ❞❛s ♣❛rá❜♦❧❛s tr❛♥s❧❛❞❛❞❛s ❡st✉❞❛♥❞♦ ♦s ❝❛s♦ ♣r✐♥❝✐♣❛✐s✳

✶♦ _{❈❛s♦✿ P❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡} _V _{= (}_{x0, y0}₎ _{❡ r❡t❛ ❢♦❝❛❧ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦}

❖①✳

(31)

✷✳✽✳ ❚❘❆◆❙▲❆➬➹❖ ❉❊ ❊■❳❖❙

P❛r❛ ♦❜t❡r♠♦s ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❛ ♣❛rá❜♦❧❛ ❞❡ ✈ért✐❝❡ ♥♦ ♣♦♥t♦ V = (x0, y0)❡ r❡t❛ ❢♦❝❛❧ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❖①✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s ①✬❖✬②✬✱ ❝♦♠ ♦r✐❣❡♠ O′ ₌_V _{= (}_{x0, y0}₎ _{❡ ❡✐①♦s ❖✬①✬ ❡ ❖✬②✬ ❝♦♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡}

s❡♥t✐❞♦ ❞♦s ❡✐①♦s ❖① ❡ ❖②✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❛❜❡♠♦s q✉❡✱ ♥♦ s✐st❡♠❛ ①✬❖✬②✬✱ ❛ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ é P ✿ y′2

= 2px′_{❀ ♦ ❢♦❝♦ é}

F′ ₌

p

2,0

❀ ♦ ✈ért✐❝❡ éV′ _{= (0}_,_{0)❀ ❛ ❞✐r❡tr✐③} _d′ _:_x′ ₌₋p

2 ❡ ❛ r❡t❛ ❢♦❝❛❧ é y

′ _{= 0✳}

❙✉❜st✐t✉✐♥❞♦✱ x = x0 +x′ _❡ _y ₌ _y0 ₊_y′_{✱ ♥❛}

❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ y′2

= 2px′_{✱ t❡♠♦s✿}

P : (y₋y0)2

= 2p(x₋x0)

❖ ♣❛râ♠❡tr♦ p s❡rá ♣♦s✐t✐✈♦ ♦✉ ♥❡❣❛t✐✈♦ s❡✱

r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ ❝♦♥❝❛✈✐❞❛❞❡ ❞❛ ♣❛rá❜♦❧❛ ❡st✐✈❡r ✈♦❧t❛❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ ♦✉ ♣❛r❛ ❛ ❡sq✉❡r❞❛✳

❊ s❡✉s ❡❧❡♠❡♥t♦s ♥❡ss❡ ❝❛s♦✱ sã♦✿

❖ ❋♦❝♦✿ F =

x0 +p

2, y0

❖ ❱ért✐❝❡✿ V = (x0, y0) ❆ ❉✐r❡tr✐③✿ d:x=x0₋ p

2

❆♦ ❞❡s❡♥✈♦❧✈❡r ❛ ❡q✉❛çã♦ (y₋y0)2

= 2p(x₋x0) ❡ ✐s♦❧❛♥❞♦ ❛ ✈❛r✐á✈❡❧ x✱ ♦❜t❡♠♦s✿

(32)

✷✳✽✳ ❚❘❆◆❙▲❆➬➹❖ ❉❊ ❊■❳❖❙

x= 1

2p |{z} a y2 − y0 p |{z} b

y+ y

2

0 + 2px0

2p

| {z } c

✭✶✮

♦✉ x=ay2

+by+c ✭✷✮

❈♦♠♣❛r❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✭✶✮ ❡ ✭✷✮✱ ♦❜s❡r✈❡ q✉❡✿

a= 1

2p ⇒ ♣ ❂

1 2a

b=₋y0

p ⇒ y0 =−bp ⇒ ②0 =−

b

2a

❊st❛ ú❧t✐♠❛ ❢ór♠✉❧❛ ❡♠ ❞❡st❛q✉❡ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ❛ ♦r❞❡♥❛❞❛ ❞♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛ ✭②0).

✷♦ _{❈❛s♦✿ P❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡} _V _{= (}_{x0, y0}₎ _{❡ r❡t❛ ❢♦❝❛❧ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦}

❖②✳

❈♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s ①✬❖✬②✬✱ ❝♦♠ ♦r✐❣❡♠O′ ₌_V _{= (}_{x0, y0}₎ _{❡ ❡✐①♦s ❖✬①✬ ❡ ❖✬②✬ q✉❡ tê♠ ❛ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦}

❞♦s ❡✐①♦s ❖① ❡ ❖②✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ♦❜t❡r ❛s ❡q✉❛çõ❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❛s ♣❛rá❜♦❧❛s ❝♦♠ ✈ért✐❝❡ V = (x0, y0) ❡ r❡t❛ ❢♦❝❛❧ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❖②✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ s✉❜st✐t✉✐♥❞♦x=x0+x′ _❡ _y₌_y0 ₊_y′_{✱ ♥❛ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛}

x′2

= 2py′_{✱ t❡♠♦s✿}

P : (x₋x0)2

= 2p(y₋y0) ✭✶✮

❆♥❛❧♦❣❛♠❡♥t❡✱ ❛ ♣❛rá❜♦❧❛ ❞❡ ❝♦♥❝❛✈✐✲ ❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ ✭q✉❛♥❞♦ p > 0✮

♦✉ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❜❛✐①♦ ✭q✉❛♥❞♦

p <0✮ t❡♠ ❛ ❢♦r♠❛✿

◆♦ ❡♥t❛♥t♦✱ ❞❡s❡♥✈♦❧✈❡♥❞♦ ✭✶✮ ❡ ✐s♦❧❛♥❞♦ ❛ ✈❛r✐á✈❡❧y✱ t❡♠♦s✿

y= 1

2p

|{z} a

x2

− x0_p

|{z} b

x+ x

2

0+ 2py0

2p

| {z } c

✭✷✮

♦✉ y=ax2

+bx+c ✭✸✮

(33)

✷✳✽✳ ❚❘❆◆❙▲❆➬➹❖ ❉❊ ❊■❳❖❙

❙✐♠✐❧❛r♠❡♥t❡ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❛ ❝♦♠♣❛r❛çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✭✷✮ ❡ ✭✸✮ ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r q✉❡✿

a= 1

2p ⇒ ♣ ❂

1 2a

b =₋x0

p ⇒ x0 =−bp ⇒ ①0 =−

b

2a

❊①❡♠♣❧♦✿ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❛ ♣❛rá❜♦❧❛(x₋1)2

=₋12(y₋3)✳ ❘❡s♦❧✉çã♦✿

❖❜s❡r✈❡ q✉❡ ❛ ❡q✉❛çã♦ ❞❛❞❛ é ❞❛ ❢♦r♠❛ (x₋x0)2

= 2p(y₋y0) P❡❧❛ ❝♦♠♣❛r❛çã♦ ❞❛s ❢ór♠✉❧❛s✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿

✲ ♦ ✈ért✐❝❡ é ♦ ♣♦♥t♦ V = (1,3)

✲ ❛ ♣❛rá❜♦❧❛ t❡♠ ♦ ❡✐①♦ ❞❡ s✐♠❡tr✐❛ ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ ② ✭✈❛r✐á✈❡❧ ❞♦ ✶♦ _{❣r❛✉✮}

✲ ❛ ❝♦♥❝❛✈✐❞❛❞❡ ❞❛ ♣❛rá❜♦❧❛ é ♣❛r❛ ❜❛✐①♦ ✭s❡❣✉♥❞♦ ♠❡♠❜r♦ ❞❛ ❡q✉❛çã♦ é ♥❡❣❛t✐✈♦✮ P♦❞❡♠♦s ❡♥tã♦ ❡s❜♦ç❛r ♦ ❣rá✜❝♦ ❞❡ss❛ ♣❛rá❜♦❧❛ ❛ ♠❡♥♦s ❞♦ s❡✉ ❢♦❝♦ ❡ ❞❡ s✉❛

❞✐r❡tr✐③✳

• ❈♦♦r❞❡♥❛❞❛s ❞♦ ❢♦❝♦✿

|2p_|= 12 _{⇒ |}p_|= 6 _⇒

p

2 = 3 ◆♦t❡ q✉❡ xF =x0 = 1

yF =y0−

p

2

= 3− |3|= 0

F = (1,0)

(34)

✷✳✾✳ ❘❖❚❆➬➹❖ ❉❊ ❊■❳❖❙

• ❊q✉❛çã♦ ❞❛ ❞✐r❡tr✐③✿

yd =y0+

p

2

= 3 +|3|= 6

d:y₋6 = 0

✷✳✾ ❘♦t❛çã♦ ❞❡ ❊✐①♦s

❙❡❥❛♠ ❖① ❡ ❖② ♦s ❡✐①♦s ♣r✐♠✐t✐✈♦s✱ ❞♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ♦rt♦❣♦♥❛❧ ❝♦♠ ♦r✐❣❡♠ ❡♠ ❖❂✭✵✱✵✮✳ ❙❡❥❛♠ ❖✬①✬ ❡ ❖✬②✬ ♦s ♥♦✈♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ❞❡♣♦✐s q✉❡ ♦ s✐st❡♠❛ ♣r✐♠✐t✐✈♦ ❢♦✐ r♦t❛❝✐♦♥❛❞♦ ♣♦r ✉♠ â♥❣✉❧♦θ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ ❖ _≡❖✬✳ ▲♦❣♦✱ θ é ♦

â♥❣✉❧♦ ❢♦r♠❛❞♦ ❡♥tr❡ ♦s ❡✐①♦s ❖① ❡ ❖✬①✬✳ ❙❡❥❛ P = (x, y) ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ s✐st❡♠❛ ♣r✐♠✐t✐✈♦✳ P♦rt❛♥t♦✱ ♦ ♠❡s♠♦ ♣♦♥t♦ P t❡rá ❝♦♦r❞❡♥❛❞❛s P = (x′_{, y}′₎_{✱ ❡♠}

r❡❧❛çã♦ ❛♦ ♥♦✈♦ s✐st❡♠❛✳ P❡❧❛ ✜❣✉r❛ ❛❝✐♠❛✱ t❡♠♦s✿

x=OM ₋N M

y= N Q+QP

◆♦ tr✐â♥❣✉❧♦ ❖▼❘✱ t❡♠♦s✿ M R=N Q✱

❝♦sθ = OM

x′ ⇒ OM =x

′_· _❝♦s_θ_✱

(35)

✷✳✾✳ ❘❖❚❆➬➹❖ ❉❊ ❊■❳❖❙

❋✐❣✉r❛ ✷✳✻✿ ❘♦t❛çã♦ ❞♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✳

s❡♥ θ = M R

x′ ⇒ M R= N Q =x

′_· _s❡♥ _θ_✳

◆♦ tr✐â♥❣✉❧♦ P◗❘✱ t❡♠♦s✿ QR=N M✱

s❡♥ θ = QR

y′ ⇒ QR =N M =y

′_· _s❡♥ _θ_✱

❝♦sθ = QP

y′ ⇒ QP =y

′_· _❝♦s _θ_✳

P♦rt❛♥t♦✱

x=OM ₋N M

y= N Q+QP ⇒

x= x′_·❝♦s θ₋y′_· s❡♥ θ

y= x′_· _s❡♥ _θ₊_y′_· _❝♦s_θ

❆ss✐♠✱ ❛♣ós ✉♠❛ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦θ ✭♣♦s✐t✐✈♦✱ ❛ ♣❛rt✐r ❞♦ ❡✐①♦ x ♥♦ s❡♥t✐❞♦

❛♥t✐✲❤♦rár✐♦✮ t❡♠♦s ❛s ❡q✉❛çõ❡s q✉❡ r❡❧❛❝✐♦♥❛♠ q✉❛❧q✉❡r ♣♦♥t♦P = (x, y)❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ s✐st❡♠❛ xOy ❝♦♠ s✉❛s ❝♦♦r❞❡♥❛❞❛s ♥♦ s✐st❡♠❛x′_O′_y′_{✳ ❈❤❛♠❛❞❛s}

❡q✉❛çõ❡s ❞❡ r♦t❛çã♦ R2_✳

x= x′_· _❝♦s_θ₋_y′_· _s❡♥ _θ

y= x′_·_s❡♥ _θ₊_y′_· _❝♦s _θ ou, equivalente,

x′ ₌ _x_· _❝♦s _θ₊_y_· _s❡♥ _θ

y′ ₌ ₋_x_· _s❡♥ _θ₊_y_· _❝♦s_θ

❊ss❛s ❡q✉❛çõ❡s ♣♦❞❡♠ t❛♠❜é♠ s❡r❡♠ ❡s❝r✐t❛s ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❛❜❛✐①♦✿

x y

=

cosθ ₋senθ

senθ cosθ

·

x′

y′

♦♥❞❡

Mθ =

cosθ ₋senθ

senθ cosθ

é ❝❤❛♠❛❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦ θ✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ♠❛tr✐③ ❞❡