❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛
Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚
P❛rá❜♦❧❛✿ ❆♥á❧✐s❡✱ ❆❜♦r❞❛❣❡♥s ❡ Pr♦♣♦st❛s
♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✳
▲✉❝❛s ❙✐❧✈❛ ❞❡ ❈❛r✈❛❧❤♦
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ◆❡✇t♦♥ ▲✉✐s ❙❛♥t♦s
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
P❛rá❜♦❧❛✿ ❆♥á❧✐s❡✱ ❆❜♦r❞❛❣❡♥s ❡ Pr♦♣♦st❛s
♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✳
▲✉❝❛s ❙✐❧✈❛ ❞❡ ❈❛r✈❛❧❤♦
❉✐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❡
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ◆❡✇t♦♥ ▲✉✐s ❙❛♥t♦s
1
FICHA CATALOGRÁFICA
Serviço de Processamento Técnico da Universidade Federal do Piauí Biblioteca Setorial do CCN
C331p Carvalho, Lucas Silva de.
Parábola: Análise, abordagens e propostas no ensino básico / Lucas Silva de Carvalho. – Teresina, 2015.
51f. il.: color
Dissertação (Mestrado Profissional) – Pós-Graduação em Matemática, Universidade Federal do Piauí, 2015.
Orientador: Prof. Dr. Newton Luís Santos
1. Geometria. 2. Geometria Analítica Plana. 3. Parábola. 4. Matemática – Estudo e Ensino. I. Título
❆❣r❛❞❡❝✐♠❡♥t♦s
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ é ❞✐s❝✉t✐❞♦ ♦ ❡♥s✐♥♦ ❞❛ ❝ô♥✐❝❛ ♣❛rá❜♦❧❛ ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✳ ❙ã♦ ❛♣r❡s❡♥t❛❞❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ❞❡st❛ ❝ô♥✐❝❛ ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ s✐♠❡tr✐❛✳ ❙ã♦ ❞✐s❝✉t✐✲ ❞❛s té❝♥✐❝❛s ❞❡ ❝♦♥str✉çõ❡s ✉t✐❧✐③❛♥❞♦✲s❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ❝♦♠♦ ✉♠ ❛♣♦✐♦ ❛♦ ❡♥s✐♥♦ ❞❡st❡ tó♣✐❝♦ ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✱ ♣r♦❝✉r❛♥❞♦ ❝♦♠ ✐st♦ t♦r♥❛r ♦ ❡♥s✐♥♦ ❞❡st❡ tó♣✐❝♦ ♠❛✐s ✐♥t❡r❡ss❛♥t❡✱ ❡st✐♠✉❧❛♥t❡ ❡ ❛❝❡ssí✈❡❧✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✐t ✐s ❞✐s❝✉ss❡❞ t❤❡ t❡❛❝❤✐♥❣ ♦❢ t❤❡ ❝♦♥✐❝ ♣❛r❛❜♦❧❛ ✐♥ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥✳ ●❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s ❝♦♥✐❝ ❛♥❞ s②♠♠❡tr② ♣r♦♣❡rt✐❡s ❛r❡ ♣r❡s❡♥t❡❞✳ ❈♦♥str✉❝t✐♦♥ t❡❝❤♥✐q✉❡s ❛r❡ ❞✐s❝✉ss❡❞ ✉s✐♥❣ ❝♦♥❝r❡t❡ ♠❛t❡r✐❛❧s ❛s ❛ s✉♣♣♦rt ❢♦r t❤❡ t❡❛❝❤✐♥❣ ♦❢ t❤✐s t♦♣✐❝ ✐♥ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥ ❛♥❞ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ ♣r❡s❡♥t❡❞✱ ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤❡ t❡❛❝❤✐♥❣ ♦❢ t❤✐s t♦♣✐❝ ♠♦r❡ ✐♥t❡r❡st✐♥❣✱ st✐♠✉❧❛t✐♥❣ ❛♥❞ ❛❝❝❡ss✐❜❧❡✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ▲❛♥t❡r♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❆♥t❡♥❛ P❛r❛❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✶ P❛rá❜♦❧❛ ❝♦♠♦ ❧✉❣❛r ❣❡♦♠❡tr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ P❛rá❜♦❧❛ ❝♦♠ ❝♦r❞❛s ❡ ❧❛t✉s r❡❝t✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✹ ❋✐❣✉r❛ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ q✉❛❧q✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✺ ❘♦t❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✻ ❚r❛♥s❧❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✼ ❍♦♠♦t❡t✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✽ ❘♦t❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✾ ●rá✜❝♦ ❛♣ós r♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✵ ❘❡t❛ t❛♥❣❡♥t❡ ❛ ♣❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✶ ❘❡t❛ ❞✐r❡tr✐③✱ ❢♦❝♦ ❡ ✈ért✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✷ ❘❡t❛ ❞✐r❡tr✐③✱ ❢♦❝♦✱ ✈ért✐❝❡ ❡ r1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✺✳✸ ❘❡t❛ ❞✐r❡tr✐③✱ ❢♦❝♦✱ ✈ért✐❝❡ ❡ r1, r2 ❡ r3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✺✳✹ ❊♥❝♦♥tr❛♥❞♦ ♦s ♣♦♥t♦sPn ❡ Pn′ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✺✳✺ P❛rá❜♦❧❛ ❝♦♥str✉í❞❛ ♣❛ss❛♥❞♦ ♣❡❧♦s ♣♦♥t♦s Pn ❡ Pn′ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✺✳✻ P❛rá❜♦❧❛ ❝♦♥str✉í❞❛ ♣❛ss❛♥❞♦ ♣❡❧♦s ♣♦♥t♦s Pn ❡ Pn′✱ ❡✐①♦ ❢♦❝❛❧✭❡✐①♦ ❞❡
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶✵
✶ ❉❆ ●❊❖▼❊❚❘■❆ ➚❙ ❈Ô◆■❈❆❙✴P❆❘➪❇❖▲❆❙ ✶✷
✷ ❆❇❖❘❉❆●❊▼ ❯❙❯❆▲ ✶✺
✸ P❆❘➪❇❖▲❆✿ ❆❇❖❘❉❆●❊▼ ❆◆❆▲❮❚■❈❆ ✶✾
✸✳✶ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ❊q✉❛çã♦ ●❡r❛❧ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✸ ❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✸✳✶ ❘♦t❛çã♦ ❞❡ ❡✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✸✳✷ ❚r❛♥s❧❛çã♦ ❞❡ ❡✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸✳✸ ❍♦♠♦t❡t✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✹ Pr♦♣♦s✐çã♦ ❞❛s r❡t❛s t❛♥❣❡♥t❡s à ✉♠❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹ ❯❙❖ ❉❊ ▼❆❚❊❘■❆■❙ ❈❖◆❈❘❊❚❖❙ ◆❖ ❊◆❙■◆❖ ❉❆ ▼❆❚❊▼➪✲
❚■❈❆ ✸✸
✺ P❘❖P❖❙❚❆❙ ❉❊ ❆❇❖❘❉❆●❊◆❙ ✸✺
✺✳✶ P❘❖P❖❙❚❆ ✵✶✿ ❈♦♥str✉çã♦ ●❡♦♠étr✐❝❛ ❞❛ P❛rá❜♦❧❛ ❝♦♠ ❘é❣✉❛ ❡ ❈♦♠♣❛ss♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✺✳✷ P❘❖P❖❙❚❆ ✵✷✿ ❈♦♥str✉çã♦ ●❡♦♠étr✐❝❛ ❞❛ P❛rá❜♦❧❛ ❝♦♠ ❉♦❜r❛❞✉r❛s ✸✽ ✺✳✸ P❘❖P❖❙❚❆ ✵✸✿ P❛rá❜♦❧❛ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞♦ ❝♦♥❡ ❞❡ ♣❛♣❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✹ P❘❖P❖❙❚❆ ✵✹✿ ❈♦♥str✉çã♦ ❞❡ P❛r❛❜♦❧ó✐❞❡s ✲ ❋♦❣ã♦ ❙♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✺ P❘❖P❖❙❚❆ ✵✺✿ ❈♦♥str✉çã♦ ❞❡ P❛r❛❜♦❧ó✐❞❡ ✲ P❛r❛❜♦❢♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✻ ❆P▲■❈❆➬Õ❊❙ ❉❆ P❆❘➪❇❖▲❆ ❊▼ ❉■❱❊❘❙❆❙ ➪❘❊❆❙ ✹✺
✼ ❈❖◆❈▲❯❙➹❖ ✹✾
❘❡❢❡rê♥❝✐❛s ✹✾
✶✵
■♥tr♦❞✉çã♦
❖ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ❡♥tr❡ ♦s ♠✉✐t♦s ❞❡s❛✜♦s q✉❡ ❡♥❢r❡♥t❛✱ ❡stá ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s s❡✉s ❝♦♥❝❡✐t♦s é ❞❡ ❝❛rát❡r ❛❜str❛t♦ q✉❡ ❞✐✜❝✉❧t❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❝♦♥t❡ú❞♦s ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s✳ ❉❡ss❛ ❢♦r♠❛ é ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞❛ ❡♥tr❡ ♦s ❡❞✉❝❛❞♦✲ r❡s ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦s ❞❛❞♦s ❞❛ r❡❛❧✐❞❛❞❡ ❝♦♥❝r❡t❛ ♥❛ ❤♦r❛ ❞❡ ❛❜♦r❞❛r ❛ss✉♥t♦s ♠❛✐s ❝♦♠♣❧❡①♦s ❡ ❛❜str❛t♦s✳
➱ ❝♦♠ ❜❛s❡ ♥❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❛ ♣❛rt✐r ❞❡ ❡①♣❡r✐ê♥❝✐❛s ♥❛ ❞♦❝ê♥❝✐❛ q✉❡ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ s❡ ♣r♦♣õ❡ ❛ ❛♥❛❧✐s❛r ❡ ❞✐s❝✉t✐r ♣r♦♣♦st❛s ❞❡ ❛❜♦r❞❛❣❡♠ ♥♦ ❡st✉❞♦ ❞❛ ♣❛rá❜♦❧❛ ❞❡♥tr♦ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ✉♠❛ ✈❡③ s❡r ❡ss❡ ✉♠ t❡♠❛ r✐❝♦ ❡♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s q✉❡✱ ♥♦ ❡♥t❛♥t♦✱ sã♦ ♣♦✉❝♦ ❡①♣❧♦r❛❞❛s✳ ◆♦s ❊♥s✐♥♦s ❋✉♥❞❛♠❡♥t❛❧ ❡ ▼é❞✐♦ ✭❡①❝❡t✉❛♥❞♦ ♦ ✸♦
❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✮ ❛ P❛rá❜♦❧❛ ❛♣❛r❡❝❡✱ ✈✐❛ ❞❡ r❡❣r❛✱ ❞❡♥tr♦ ❞♦ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ❝♦♠♦ ❣rá✜❝♦ ❞❡st❛✱ s❡♥❞♦ ♣♦✉❝♦ ❡①♣❧♦r❛❞♦ ❛ ♥♦çã♦ ❞❡ P❛rá❜♦❧❛ ❝♦♠♦ ✉♠❛ ❝ô♥✐❝❛ ❝♦♠ ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❡ ❢❛❝✐❧♠❡♥t❡ ♦❜s❡r✈á✈❡✐s ♥♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❛❧✉♥♦s✳
❆ ♣r❡s❡♥t❡ ❛♥á❧✐s❡ ❡ ❞✐s❝✉ssã♦ t❡♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♣r♦♣♦r ❢♦r♠❛s ❞❡ ❛❜♦r❞❛❣❡♠ ❞♦ ❊st✉❞♦ ❞❛ P❛rá❜♦❧❛✱ s✉❛ ❝♦♥str✉çã♦ ❡ q✉❡ ❡①♣❧♦r❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❝♦t✐❞✐❛♥❛s t♦r♥❛♥❞♦ ❛ss✐♠ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ s❡✉ ❡st✉❞♦ ❡ ❞❡ ♠❛✐s ❢á❝✐❧ ❛♣r❡❡♥sã♦✱ ❜❡♠ ❝♦♠♦ ❞❡ ♦✉tr♦s ❛s✲ s✉♥t♦s ❛ ❡❧❛ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠♦ é ♦ ❝❛s♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳
P❛r❛ ❛❜♦r❞❛r ♦ t❡♠❛ ♣r✐♠❡✐r❛♠❡♥t❡ s❡rá ❛♣r❡s❡♥t❛❞♦ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❡ ♦r✐❣❡♠ ❞♦s ❡st✉❞♦s ❞❛ ❣❡♦♠❡tr✐❛✱ ♥♦♠❡s q✉❡ s❡ ❞❡st❛❝❛r❛♠ ❡ ♣r✐♠❡✐r♦s ✉s♦s✳ ❊ ❞❡♥tr♦ ❞❡ss❡ ❤✐stór✐❝♦ ♦s ♥♦♠❡s ❡ ❝♦♠♦ s❡ ❞❡✉ ♦ ❡st✉❞♦ ❞❛s ❝ô♥✐❝❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ P❛rá❜♦❧❛✳
❖ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ ❛♣r❡s❡♥t❛ ❛ ❛❜♦r❞❛❣❡♠ ✉s✉❛❧ ♣❛r❛ ♦ ❊♥s✐♥♦ ❞❛s P❛rá❜♦❧❛s ❡♥✲ ❝♦♥tr❛❞❛ ♥♦s ❧✐✈r♦s ❞❡ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❡ ▼é❞✐♦✳
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ é ❢❡✐t♦ ♦ ❡st✉❞♦ ❛❧❣é❜r✐❝♦ ❡ ❣❡♦♠étr✐❝♦ ❞❛ P❛rá❜♦❧❛✱ ❛s ❡q✉❛✲ çõ❡s✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❡st✉❞♦ ❞❛ r♦t❛çã♦ ❡ tr❛♥s❧❛çã♦ ❞♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✱ ❛❧é♠ ❞❡ ❤♦♠♦t❡t✐❛✳
✶✶
♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ♥♦ ❝♦♥t❡①t♦ ❞♦ ❊♥s✐♥♦ ❞❡ P❛rá❜♦❧❛s✱ ❛s ♦r✐❣❡♥s ❞❡ss❛ ❞✐s❝✉ssã♦ ❡ ❛s ✈❛♥t❛❣❡♥s ❞♦ ✉s♦ ❞❡ss❛ ♠❡t♦❞♦❧♦❣✐❛✳
◆♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ é ❞❡st✐♥❛❞♦ à ❛♣r❡s❡♥t❛çã♦ ❞❡ ♣r♦♣♦st❛s ❞❡ ❛❜♦r❞❛❣❡♥s ♣❛r❛ ♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡♠ ❝♦♥s♦♥â♥❝✐❛ ❝♦♠ ❛ ♣❡rs♣❡❝t✐✈❛ ✐♥✐❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦✱ ♦✉ s❡❥❛✱ ❛ ♣❛rt✐r ❞♦ ✉s♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ❡ ✴♦✉ ✐♥s❡rçã♦ ♥❛ ✈✐❞❛ ❝♦t✐❞✐❛♥❛✳
✶ ❉❆ ●❊❖▼❊❚❘■❆ ➚❙
❈Ô◆■❈❆❙✴P❆❘➪❇❖▲❆❙
❙❡❣✉♥❞♦ ❇♦②❡r ✭✶✾✾✻✮ ❛ ❣❡♦♠❡tr✐❛ t❡♠ s✉❛s ♦r✐❣❡♥s✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❛✐♥❞❛ ❝♦♠ ♦ ❤♦♠❡♠ ♣ré✲❤✐stór✐❝♦ ✉♠❛ ✈❡③ q✉❡
s❡✉s ❞❡s❡♥❤♦s ❡ ✜❣✉r❛s s✉❣❡r❡♠ ✉♠❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ r❡❧❛çõ❡s ❡s♣❛❝✐❛✐s q✉❡ ❛❜r✐✉ ❝❛♠✐♥❤♦ ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛✳ ❙❡✉s ♣♦t❡s✱ t❡❝✐❞♦s ❡ ❝❡st❛s ♠♦str❛♠ ❡①❡♠♣❧♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡ s✐♠❡tr✐❛✱ q✉❡ ❡♠ ❡ssê♥❝✐❛ sã♦ ♣❛rt❡s ❞❛ ❣❡♦♠❡tr✐❛ ❡❧❡♠❡♥t❛r✳ ❬✺❪
◆♦ ❡♥t❛♥t♦ ❞❡✈✐❞♦ ❛ ❛✉sê♥❝✐❛ ❞❡ ♣r♦✈❛s ❤✐stór✐❝❛s s♦❜r❡ ❡ss❛ ❝♦♥❥❡❝t✉r❛ ♦ ❛✉t♦r ❝♦♥❝❡♥tr❛✲s❡ ♥❛ ❤✐stór✐❛ ❞❛ ●❡♦♠❡tr✐❛ ❡♥❝♦♥tr❛❞❛ ❡♠ ❞♦❝✉♠❡♥t♦s ❡s❝r✐t♦s q✉❡ ❝❤❡❣❛✲ r❛♠ ❛té ❛ ❛t✉❛❧✐❞❛❞❡✳ ❊st❡s ♠♦str❛♠ q✉❡ ❛ ♦r✐❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ❡st❛r✐❛ ♥❛ ❝✐✈✐❧✐③❛çã♦ ❡❣í♣❝✐❛✱ s❡❣✉♥❞♦ ❍❡ró❞♦t♦✱ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ♣rát✐❝❛ ❞❡ ❢❛③❡r ♥♦✈❛s ♠❡❞✐❞❛s ❞❡ t❡rr❛s ❛♣ós ❝❛❞❛ ✐♥✉♥❞❛çã♦ ❛♥✉❛❧ ♥♦ ✈❛❧❡ ❞♦ r✐♦ ❡ s❡❣✉♥❞♦ ❆r✐stót❡❧❡s ❡❧❛ ❡st❛✈❛ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❧❛③❡r ❞❡ ✉♠❛ ❝❧❛ss❡ s❛❝❡r❞♦t❛❧✳
❊❣í♣❝✐♦s ❡ ❜❛❜✐❧ô♥✐♦s✱ ❤á ♠❛✐s ❞❡ ✸✵✵✵ ❛♥♦s✱ ❥á ✉t✐❧✐③❛✈❛♠ ❛ ❣❡♦♠❡tr✐❛ ♥❛s r❡❣✐õ❡s ✐♥✉♥❞á✈❡✐s ❞♦s r✐♦s ◆✐❧♦✱ ❚✐❣r❡ ❡ ❊✉❢r❛t❡s✱ ♥❛ ❞❡♠❛r❝❛çã♦ ❞❛s t❡rr❛s ❛✜♠ ❞❡ ♦r❣❛♥✐③❛r ♦ ♣❧❛♥t✐♦ ❡ ❢❛❝✐❧✐t❛r ❛ ❝♦❜r❛♥ç❛ ❞❡ ✐♠♣♦st♦s✳ ❙❡❣✉♥❞♦ ♦ ❤✐st♦r✐❛❞♦r ❣r❡❣♦ ❍❡ró❞♦t♦ ❝♦♠ ♦ ❛♣❛❣❛♠❡♥t♦ ❞❛s ❞❡♠❛r❝❛çõ❡s ❞❡ t❡rr❛s ♣❡❧❛s ✐♥✉♥❞❛çõ❡s ❞♦ ◆✐❧♦ t♦r♥♦✉✲s❡ ♥❡❝❡ssár✐♦ ❛ ❝r✐❛çã♦ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ tr❛❜❛❧❤❛❞♦r❡s✱ ♦s ❝❤❛♠❛❞♦s ✏❡st✐r❛❞♦r❡s ❞❡ ❝♦r❞❛✑ q✉❡ ❢❛③✐❛♠ ❛s ♠❡♥s✉r❛çõ❡s ♥♦✈❛♠❡♥t❡✳ ❬✺❪✳ ◗✉❛♥❞♦ Pt♦❧♦♠❡✉ ❛ss✉♠✐✉ ♦ ❣♦✈❡r♥♦ ❞♦ ❊❣✐t♦✱ ❡s❝♦❧❤❡✉ ❆❧❡①❛♥❞r✐❛ ❝♦♠♦ s✉❛ ❝❛♣✐t❛❧ ❡ ❧á ❝♦♥str✉✐✉ ❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛ ❛ q✉❛❧ t✐♥❤❛ ❊✉❝❧✐❞❡s ❝♦♠♦ ♦ ❝❤❡❢❡ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ◆❡ss❛ ✉♥✐✈❡rs✐❞❛❞❡ ♦✉tr♦ ♥♦♠❡ q✉❡ s❡ ❞❡st❛❝❛ é ♦ ❞❡ ❆♣♦❧ô♥✐♦✱ q✉❡ ❢♦✐ ✉♠ ❞♦s ♠❛✐♦r❡s ❡st✉❞✐♦s♦s ❞❛s ❝ô♥✐❝❛s✱ ✉♠ r❛♠♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❬✶✼❪✳
❆ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❝♦♥❝❡❜✐❞❛ ♣♦r ❊✈❡s ✭✷✵✵✹✮ ✏❝♦♠♦ ✉♠ ♣♦❞❡r♦s♦ ♠ét♦❞♦ ♣❛r❛ ❡♥❢r❡♥t❛r ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s✑ t❡♠ s✉❛ ♦r✐❣❡♠ ✐♥❝❡rt❛ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ q✉❡♠ ❛ ✐♥✈❡♥t♦✉ ❡ ❡♠ q✉❡ é♣♦❝❛ ❛❝♦♥t❡❝❡✉✳ P❛r❛ ❇♦✉r❜❛❦✐ ✭✶✾✼✹✮ ❛ ú❧t✐♠❛ ❝♦♥tr✐❜✉✐çã♦ ❡ss❡♥❝✐❛❧ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛ ❢♦✐ à ❚❡♦r✐❛ ❞❛s ❈ô♥✐❝❛s✳ ❖ ❛✉t♦r r❡ss❛❧t❛ q✉❡ ♠❡s♠♦ ♦s ❣r❡❣♦s ♥ã♦ t❡♥❞♦ ✐❞❡✐❛ ❞♦s ♣r✐♥❝í♣✐♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❡❧❡s ❢❛③✐❛♠ ✉s♦ ❞❡ ✏❝♦♦r❞❡♥❛❞❛s✑ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✜❣✉r❛s ♣❛rt✐❝✉❧❛r❡s✱ ❡♠ r❡❧❛çã♦ ❛ ❞♦✐s ❡✐①♦s ♥♦ ♣❧❛♥♦✳
✶✸
❆♣♦❧ô♥✐♦✱ ❝♦♠♦ ❥á ❢♦✐ ❝✐t❛❞♦✱ ❞❡s❡♥✈♦❧✈❡✉ ❛ s✉❛ ❣❡♦♠❡tr✐❛ ❞❛s s❡❝çõ❡s ❝ô♥✐❝❛s ✏❛ ♣❛rt✐r ❞❡ ❡q✉✐✈❛❧❡♥t❡s ❣❡♦♠étr✐❝♦s ❞❡ ❝❡rt❛s ❡q✉❛çõ❡s ❝❛rt❡s✐❛♥❛s ❞❡ss❛s ❝✉r✈❛s✱ ✉♠❛ ✐❞é✐❛ q✉❡ ♣❛r❡❝❡ t❡r✲s❡ ♦r✐❣✐♥❛❞♦ ❝♦♠ ▼❡♥❛❡❝♠♦s✑ ❬✶✶❪✳
❋♦✐✱ ♣♦rt❛♥t♦✱ ✉♠❛ r❡❛❧✐③❛çã♦ ✐♠♣♦rt❛♥t❡ ❞❡ ▼❡♥❛❡❝♠✉s t❡r ❞❡s❝♦❜❡rt♦ q✉❡ ❝✉r✈❛s ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡s❡❥❛❞❛ ❡st❛✈❛♠ à ❞✐s♣♦s✐çã♦✳ ◆❛ ✈❡r❞❛❞❡✱ ❤❛✈✐❛ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s ❛❞❡q✉❛❞❛s✱ q✉❡ ♣♦❞✐❛♠ s❡r ♦❜t✐❞❛s ❞❡ ✉♠❛ ♠❡s♠❛ ❢♦♥t❡✱ ❝♦rt❛♥❞♦ ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ♣♦r ✉♠ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❡✳ ■st♦ é✱ ♣❛r❡❝❡ t❡r ❞❡s❝♦❜❡rt♦ q✉❡ ♠❛✐s t❛r❞❡ ❢♦r❛♠ ❝❤❛♠❛❞❛s✱ ❡❧✐♣s❡✱ ♣❛rá❜♦❧❛ ❡ ❤✐♣ér❜♦❧❡✳ ❬✺❪✳ ❊♠ ✉♠ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ♣♦r ❏❛❝✐r ❏✳ ❱❡♥t✉r✐✱ ♠♦str❛ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❡ ❝♦♥✜r♠❛ ❛ ♣❛rt✐❝✐♣❛çã♦ ❞❡ ▼❡♥❛❡❝♠✉s ♥❛ ❞❡s❝♦❜❡rt❛ ❞❛s ❝ô♥✐❝❛s✳
❈♦♥t❛ ✉♠❛ ❧❡♥❞❛ q✉❡✱ ❡♠ ✹✷✾ ❛✳❈✳✱ ❞✉r❛♥t❡ ♦ ❝❡r❝♦ ❡s♣❛rt❛♥♦ ♥❛ ●✉❡rr❛ ❞♦ P❡❧♦♣♦♥❡s♦✱ ✉♠❛ ♣❡st❡ ❞✐③✐♠♦✉ ✉♠ q✉❛rt♦ ❞❛ ♣♦♣✉❧❛çã♦ ❞❡ ❆t❡♥❛s✱ ♠❛t❛♥❞♦ ✐♥❝❧✉s✐✈❡ Pér✐❝❧❡s✱ ❡ q✉❡ ✉♠❛ ♣❧ê✐❛❞❡ ❞❡ sá❜✐♦s ❢♦r❛ ❡♥✈✐❛❞❛ ❛♦ ♦rá❝✉❧♦ ❞❡ ❆♣♦❧♦✱ ❡♠ ❉❡❧❢♦s✱ ♣❛r❛ ✐♥q✉✐r✐r ❝♦♠♦ ❛ ♣❡st❡ ♣♦❞❡r✐❛ s❡r ❡❧✐♠✐♥❛❞❛✳ ❖ ♦rá❝✉❧♦ r❡s✲ ♣♦♥❞❡✉ q✉❡ ♦ ❛❧t❛r ❝ú❜✐❝♦ ❞❡ ❆♣♦❧♦ ❞❡✈❡r✐❛ s❡r ❞✉♣❧✐❝❛❞♦✳ ❖s ❛t❡♥✐❡♥s❡s ❝❡✲ ❧❡r❡♠❡♥t❡ ❞♦❜r❛r❛♠ ❛s ♠❡❞✐❞❛s ❞❛s ❛r❡st❛s ❞♦ ❝✉❜♦✳ ❆ ♣❡st❡✱ ❡♠ ✈❡③ ❞❡ s❡ ❛♠❛✐♥❛r✱ r❡❝r✉❞❡s❝❡✉✳ ◗✉❛❧ ♦ ❡rr♦❄ ❊♠ ✈❡③ ❞❡ ❞♦❜r❛r✱ ♦s ❛t❡♥✐❡♥s❡s ♦❝t✉♣❧✐❝❛✲ r❛♠ ♦ ✈♦❧✉♠❡ ❞♦ ❛❧t❛r✱ ♣♦✐s✿ ♣❛r❛a= 1t❡♠♦s q✉❡Vcubo= 13= 1❡ ♣❛r❛a= 2
t❡♠♦s q✉❡ Vcubo = 23 = 8✳ ❆ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛ ❞❡✈❡✲s❡ ❛♦ ❢❛t♦ ❞❡
q✉❡ ♦s ❣r❡❣♦s ♣r♦❝✉r❛✈❛♠ ✉♠❛ s♦❧✉çã♦ ❣❡♦♠étr✐❝❛✱ ✉s❛♥❞♦ ré❣✉❛ ✭s❡♠ ❡s❝❛❧❛✮ ❡ ❝♦♠♣❛ss♦✳ ❆✐♥❞❛ ♥♦ sé❝✉❧♦ ■❱ ❛✳❈✳✱ ♦ ❣❡ô♠❡tr❛ ❣r❡❣♦ ▼❡♥❛❡❝♠✉s r❡s♦❧✈❡✉ ♦ ♣r♦❜❧❡♠❛ ❝♦♠ ♦ tr❛ç❛❞♦ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ ❡ ❞❡ ✉♠❛ ❤✐♣ér❜♦❧❡✳ ❍♦❞✐❡r♥❛♠❡♥t❡✱ t❛❧ s♦❧✉çã♦ é ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r❡❡♥sí✈❡❧ ♣♦r ♠❡✐♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✿ ▼❡♥❛✲ ❡❝♠✉s ♦❜t❡✈❡ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛ ♣❛rá❜♦❧❛x2
= 2y ❝♦♠
❛ ❤✐♣ér❜♦❧❡xy= 1✳ ❆ s♦❧✉çã♦ éx=√3
2✳ ❋♦✐ r❛③♦á✈❡❧ ♦ s✉❝❡ss♦ ❞❡ ▼❡♥❛❡❝♠✉s
❡♥tr❡ ♦s s❡✉s ❝♦♠♣❛tr✐♦t❛s✱ ❥á q✉❡ ♥ã♦ s❡ ✈❛❧❡✉ ❛♣❡♥❛s ❞❡ ré❣✉❛ ✭s❡♠ ❡s❝❛❧❛✮ ❡ ❝♦♠♣❛ss♦✳ ❬✷✶❪✳
❖✉tr♦ ❡st✉❞✐♦s♦ q✉❡ ❡♥tr❛ ♥❡ss❛ ❞✐s♣✉t❛ é ◆✐❝♦❧❡ ❖r❡s♠❡ ✭✶✸✷✸✲✶✸✽✷✮✱ ✉♠❛ ✈❡③ q✉❡ ❛♥t❡❝✐♣♦✉ ♦✉tr♦s ❛s♣❡❝t♦s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❛♦ r❡♣r❡s❡♥t❛r ❣r❛✜❝❛♠❡♥t❡ ❝❡rt❛s ❧❡✐s✿ ✏❝♦♥❢r♦♥t❛♥❞♦ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ✭❧❛t✐t✉❞♦✮ ❝♦♠ ❛ ✐♥❞❡♣❡♥❞❡♥t❡ ✭❧♦♥❣✐t✉❞♦✮✱ à ♠❡❞✐❞❛ q✉❡ s❡ ♣❡r♠✐t❛ q✉❡ ❡st❛ ú❧t✐♠❛ s♦❢r❡ss❡ ♣❡q✉❡♥♦s ❛❝rés❝✐♠♦s✑✳ ❬✶✶❪✳
❆ ❡ssê♥❝✐❛ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❝♦♠♦ ♦❜s❡r✈❛ ❊✈❡s ✭✷✵✵✹✮ é ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ tr❛♥s❢❡r✐r ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ❛❧❣é❜r✐❝❛✳ ▼❛s ❛♥t❡s ❞❛ ●❡♦♠❡tr✐❛ s❡ ❞❡s❡♥✈♦❧✈❡r ♣❧❡♥❛♠❡♥t❡ ❡ ♣♦❞❡r ❡①❡r❝❡r ❡ss❡ ♣❛♣❡❧✱ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s ♣♦r ♠❡✐♦s ❛❧❣é❜r✐❝♦s✱ ❢♦✐ ♥❡❝❡ssár✐♦ ❡s♣❡r❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ á❧❣❡❜r❛✱ ♠♦t✐✈♦ ♣❡❧♦ q✉❛❧ ❊✈❡s ✭✷✵✵✹✮ ❝♦♥s✐❞❡r❛ s❡r ♠❛✐s ❝♦rr❡t♦ ❝♦♥❝♦r❞❛r ❝♦♠ ❛ ♠❛✐♦r✐❛ ❞♦s ❤✐st♦r✐❛❞♦r❡s✱ ♦s q✉❛✐s ❞❡❢❡♥❞❡♠ q✉❡ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❢❡✐t❛s ♣❡❧♦s ♠❛t❡♠át✐❝♦s ❢r❛♥❝❡✲ s❡s ❘❡♥é ❉❡s❝❛rt❡s ❡ P✐❡rr❡ ❞❡ ❋❡r♠❛t t❡r✐❛♠ ❞❡ ❢❛t♦ ❞❛❞♦ ❛♦ ❛ss✉♥t♦✳
✶✹
✳✳✳❋❡r♠❛t ❛❝r❡s❝❡♥t♦✉ ✏❆ ❙♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s ❙ó❧✐❞♦s ♣♦r ♠❡✐♦ ❞❡ ▲✉❣❛r❡s✑✱ ❡♠ q✉❡ ♦❜s❡r✈❛ q✉❡ ❡q✉❛çõ❡s ❞❡t❡r♠✐♥❛❞❛s ❝ú❜✐❝❛s ♦✉ q✉árt✐❝❛s ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ♣♦r ♠❡✐♦ ❞❡ ❝ô♥✐❝❛s✱ ♦ t❡♠❛ q✉❡ t♦♠❛✈❛ tã♦ ❣r❛♥❞❡s ♣r♦♣♦çõ❡s ♥❛ ❣❡♦♠❡tr✐❛ ❞❡ ❉❡s❝❛rt❡s✳ ❬✺❪
✷ ❆❇❖❘❉❆●❊▼ ❯❙❯❆▲
◆❡ss❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❝♦♠♦ ♦ ❊st✉❞♦ ❞❛s P❛rá❜♦❧❛s é ❢❡✐t♦✱ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❛ ♣❛rt✐r ❞❡ ♣❡sq✉✐s❛ ❢❡✐t❛ ❝♦♠ ♦♥③❡ ❧✐✈r♦s ❞❡ss❡ ❝✐❝❧♦ ❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛ ♣❛rt✐r ❞❡ ♣❡sq✉✐s❛ ❡♠ ♦✉tr♦s s❡✐s ❧✐✈r♦s✱ ❛❧é♠ ❞❡ ✐♥❢♦r♠❛çõ❡s ❛❞✈✐♥❞❛s ❞❛ ♣ró♣r✐❛ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳
◆♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ♦ ❊st✉❞♦ ❞❛s ❈ô♥✐❝❛s s❡ r❡str✐♥❣❡ ❛♣❡♥❛s ❛♦ ❊st✉❞♦ ❞❛ P❛rá❜♦❧❛✳ ◆♦ ú❧t✐♠♦ ❛♥♦ ❞❡ss❡ ❝✐❝❧♦✱ ❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛❜♦r❞❛ ♦ t❡♠❛ P❛rá❜♦❧❛ ❛♣❡♥❛s ❝♦♠♦ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❛❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❞♦s s❡✉s ♣♦♥t♦s ♥♦tá✈❡✐s ❝♦♠♦ ③❡r♦s✱ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦✱ ❝♦♥❝❛✈✐❞❛❞❡ ❡ ♦ ❡st✉❞♦ ❞♦ s✐♥❛❧✳ ❉❛♥t❡ ✭✷✵✶✷✮ ❡♠ s❡✉ ❧✐✈r♦ ❞✐❞át✐❝♦✱ ✐♥✐❝✐❛ ♦ ❡st✉❞♦ ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ✷♦
❣r❛✉ ❛✜r♠❛♥❞♦ q✉❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ❛q✉❡❧❛ ❝✉❥❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦ ♣♦❞❡ s❡r ✐♥❞✐❝❛❞❛ ♣♦r✿ y = ax2
+bx+c ✱ ❝♦♠ a✱ b ❡ c r❡❛✐s ❡ a 6= ✵✳ ❆♣ós ❛❧❣✉♥s ❡①❡♠♣❧♦s ❛♣r❡s❡♥t❛ ♦s ③❡r♦s ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❝❛s♦s ❡♠ q✉❡ ♦s ✈❛❧♦r❡s r❡❛✐s ❞❡ x ❛♥✉❧❛♠ y ♥❛ ❢✉♥çã♦ ❛♣r❡s❡♥t❛❞❛ ❛❝✐♠❛✱ ❞❡♣♦✐s ❡♥s✐♥❛ ❝♦♠♦ ❝♦♥str✉✐r ♦ ❣rá✜❝♦ ❛tr✐❜✉✐♥❞♦ ✈❛❧♦r❡s ❡♠ x✱ ❡♥❝♦♥tr❛♥❞♦ ♦s ✈❛❧♦r❡s ❞❡y ❝♦rr❡s♣♦♥❞❡♥t❡s ❡ ❧✐❣❛♥❞♦ ♦s ♣♦♥t♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡✲ s✐❛♥♦✳ ❊♠ s❡❣✉✐❞❛ ♠♦str❛ ❛ ✜❣✉r❛ ❞❡ ✉♠ ❝♦♥❡ s❡♥❞♦ s❡❝❝✐♦♥❛❞♦ ♣♦r ✉♠ ♣❧❛♥♦ ❡ ✉♠❛ ❛♥t❡♥❛ ♣❛r❛❜ó❧✐❝❛✱ ❛❝r❡s❝❡♥t❛♥❞♦ q✉❡ é ❞❛ ♣❛❧❛✈r❛ ♣❛rá❜♦❧❛ q✉❡ ✈❡♠ ♦ ♥♦♠❡ ❛♥t❡♥❛ ♣❛r❛❜ó❧✐❝❛✳ ▼❡♥❝✐♦♥❛ ❛✐♥❞❛ q✉❡ ❡ss❛ ❝✉r✈❛ ❛♣❛r❡❝❡ q✉❛♥❞♦ ❢❛③❡♠♦s✱ ❞❡ ❞❡t❡r♠✐♥❛❞❛ ♠❛♥❡✐r❛✱ ❛ s❡❝çã♦ ❞❡ ✉♠ ❝♦♥❡ ♣♦r ✉♠ ♣❧❛♥♦✳ ❆♣ós ✐ss♦ tr❛❜❛❧❤❛ ❝♦♠ ♦s ❝♦❡✜❝✐❡♥t❡s a✱ b ❡c ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ♠♦str❛♥❞♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ s✉❛s ❢✉♥çõ❡s✱ ❡♠ s❡❣✉✐❞❛ ♦s ❝❛s♦s ❞❡ ✐♥t❡rs❡❝çõ❡s ❝♦♠ ♦s ❡✐①♦s x ❡ y✱ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛ ❡ ✈❛❧♦r ♠á①✐♠♦ ♦✉ ✈❛❧♦r ♠í♥✐♠♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳
❖✉tr♦ ❛ss✉♥t♦ ♠❡♥♦s ✉s✉❛❧✱ ♠❛s ✐❣✉❛❧♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ❡♠ ❧✐✈r♦s ❞♦ ✾♦ ❛♥♦ ❞♦
❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ é ♦ ❢♦❝♦ ❡ ❛ r❡t❛ ❞✐r❡tr✐③✳ ❚❡♠✲s❡ ❛✐♥❞❛ ❛ ❡q✉❛çã♦ ❣❡r❛❧ q✉❡✱ ❞❡ ♠♦❞♦ ❛❧❣✉♠✱ é tr❛❜❛❧❤❛❞❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳
❊♥t❡♥❞❡✲s❡ q✉❡ tr❛❜❛❧❤❛r ❝♦♠ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ♣♦❞❡ s❡r ❝♦♠♣❧❡①♦ ♥❡ss❡ ♥í✈❡❧ ❞♦ ❡♥s✐♥♦✱ ♥♦ ❡♥t❛♥t♦ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ ❢♦❝♦ ❡ r❡t❛ ❞✐r❡tr✐③✱ ♣♦r s❡r❡♠ ♣❛rt❡s ❢✉♥❞❛♠❡♥✲ t❛✐s ❡ ❝♦♥str✉t♦r❛s ❞❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛✱ ❛❝r❡❞✐t❛✲s❡ q✉❡ ❞❡✈❡r✐❛♠ ❛♣❛r❡❝❡r ❝♦♠ ♠❛✐s ❢r❡q✉ê♥❝✐❛ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ♠❡s♠♦ q✉❡ ❛♣r❡s❡♥t❛❞♦s ♣♦r ♠❡✐♦ ❞❡ s✉❛s
✶✻
❝❛r❛❝t❡ríst✐❝❛s ♦✉ ❞❡ s✉❛ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡✱ ✉♠❛ ✈❡③ q✉❡ s❡♠ ❡❧❡s ♥ã♦ s❡ ❝♦♠♣r❡❡♥❞❡ ❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❞❛ ♣❛rá❜♦❧❛✳
❉♦s ❧✐✈r♦s ♣❡sq✉✐s❛❞♦s ❛♣❡♥❛s ❇♦♥❥♦r♥♦ ✫ ❆②rt♦♥ ✭✷✵✵✻✮✱ ♠❡♥❝✐♦♥❛ ❢♦❝♦ ❡ r❡t❛ ❞✐r❡tr✐③✳ ❙❡❣✉♥❞♦ ♦ ❛✉t♦r
❛ P❛rá❜♦❧❛ é ✉♠❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ✭❡stã♦ à ♠❡s♠❛ ❞✐stâ♥❝✐❛✮ ❞❡ ✉♠❛ r❡t❛ r ❡ ❞❡ ✉♠ ♣♦♥t♦ ❋ ❞❛❞♦s✳ ❱❡❥❛ ❛ ✜❣✉r❛ ✷✳✶ ❡ ❝♦♥✜r❛
❋✐❣✉r❛ ✷✳✶✿ P❛rá❜♦❧❛
❋♦♥t❡✿ ❇♦♥❥♦r♥♦ ✫ ❆②rt♦♥ ✭✷✵✵✻✮
❖ ♣♦♥t♦ ❋ é ❞❡♥♦♠✐♥❛❞♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛ ❡ t❡♠ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♣rát✐❝❛ ❡♠ ♦❜❥❡t♦s q✉❡ ❛♣r❡s❡♥t❛♠ ❛ ❢♦r♠❛ ♣❛r❛❜ó❧✐❝❛✱ ❝♦♠♦ ❡s♣❡❧❤♦s ❡ ❛♥t❡♥❛s✳✮ ❖ ❛✉t♦r ❛❝r❡s❝❡♥t❛ ❛♦ ❡st✉❞♦ ♦✉tr❛s ✐♥❢♦r♠❛çõ❡s ♣rát✐❝❛s✱ ❞❡ ❝♦♠♦ ✉♠❛ ❧â♠✲ ♣❛❞❛ ❝♦❧♦❝❛❞❛ ♥♦ ❢♦❝♦ ❞❡ ✉♠ ❡s♣❡❧❤♦ ❞❡ s✉♣❡r❢í❝✐❡ ♣❛r❛❜ó❧✐❝❛ r❡✢❡t❡ s❡✉s r❛✐♦s ❞❡ ❧✉③ ♣❛r❛❧❡❧❛♠❡♥t❡✳ ❖ q✉❡ ❛❝♦♥t❡❝❡✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❢❛r♦❧ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧✳
❋✐❣✉r❛ ✷✳✷✿ ▲❛♥t❡r♥❛
✶✼
❋✐❣✉r❛ ✷✳✸✿ ❆♥t❡♥❛ P❛r❛❜ó❧✐❝❛
❋♦♥t❡✿ ❇♦♥❥♦r♥♦ ✫ ❆②rt♦♥ ✭✷✵✵✻✮
◆✉♠❛ ❛♥t❡♥❛ ♣❛r❛❜ó❧✐❝❛✱ ❛s ♦♥❞❛s ❝❛♣t❛❞❛s r❡✢❡t❡♠✲s❡ ♥❛ s✉♣❡r❢í❝✐❡ ♣❛r❛❜ó❧✐❝❛ ❡ ❞✐r✐❣❡♠✲s❡ ♣❛r❛ ♦ ❢♦❝♦ ♦♥❞❡ ❡stá ❧♦❝❛❧✐③❛❞♦ ♦ r❡tr❛♥s♠✐ss♦r✳ ❬✹❪✳
❊ss❡s ❡ ♦✉tr♦s ❡①❡♠♣❧♦s ♥♦s ♣❡r♠✐t❡♠ ❛✜r♠❛r q✉❡ ♣❡❧♦ ❢❛t♦ ❞❛ P❛rá❜♦❧❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ s❡r ❛❜♦r❞❛❞❛ ❞❡♥tr♦ ❞♦ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛♣❡♥❛s ✐❧✉str❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡❧❛ ♣❛r❛ ♦ ♠✉♥❞♦ ❡♠ ❢❛❝❡ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s r❡✢❡t♦r❛s✳ ❊ss❛ ♠❡t♦❞♦❧♦❣✐❛ ♣♦❞❡ r❡♣r❡s❡♥t❛r ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ♥❛ ❛♣r❡❡♥sã♦ ❞♦ ❛ss✉♥t♦ ♣♦r ♣❛rt❡ ❞♦ ❛❧✉♥♦✱ ♣♦✐s ❡st❡ ♣♦❞❡ ❛té ✈✐s✉❛❧✐③❛r ❛s ♣❛rá❜♦❧❛s ❡♠ ❛♥t❡♥❛s✱ ❢❛ró✐s✱ s❛✲ té❧✐t❡s✱ ♣♦♥t❡s ❡ tr❛❥❡tór✐❛s ❞❡ ♦❜❥❡t♦s✱ ♠❛s✱ ♣r♦✈❛✈❡❧♠❡♥t❡ ♥ã♦ ❝♦♠♣r❡❡♥❞❡rá ✧❝♦♠♦ ❢✉♥❝✐♦♥❛♠✧✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❡st✉❞♦✉ ♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛✱ ❡❧❡♠❡♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦ s✉❝❡ss♦ ❞❡ t❛✐s ❝r✐❛çõ❡s ❤✉♠❛♥❛s✳
❖❜s❡r✈❛♥❞♦ ♦s ❛♥♦s s❡❣✉✐♥t❡s t❡♠♦s q✉❡ ♥♦ ✶♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛ P❛rá❜♦❧❛
é tr❛t❛❞❛ ♣r❛t✐❝❛♠❡♥t❡ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❏á ♥♦ ✷♦ ❛♥♦
♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ♥❡♠ ❛❜♦r❞❛♠ ❛ t❡♠át✐❝❛ ❞❛s P❛rá❜♦❧❛s✳ ❙ó ♥♦ ✸♦ ❛♥♦ ❞♦ ❊♥s✐♥♦
▼é❞✐♦ ❛ P❛rá❜♦❧❛ é ✈✐st❛ ❝♦♠♦ ♣❛rt❡ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛rt❡ ❞♦ ❊st✉❞♦ ❞❡ ❈ô♥✐❝❛s✳ ◆❡st❛ ❡t❛♣❛✱ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❢❛③❡♠ ✉♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠♦❞♦ ♠❛✐s ❛❜r❛♥❣❡♥t❡✱ ❛♣r❡s❡♥t❛♥❞♦ s✉❛ ♦r✐❣❡♠✱ ❞❡✜♥✐çã♦✱ ❡❧❡♠❡♥t♦s ✭❢♦❝♦✱ ❞✐r❡tr✐③✱ ✈ért✐❝❡✱ ❡✐①♦ ❞❡ s✐♠❡tr✐❛✱ ♣❛râ♠❡tr♦✮ ❡ ❡q✉❛çõ❡s ♦❜t✐❞❛s ❝♦♠ r❡❧❛çã♦ ❛♦s ❡✐①♦s ❡ ♦r✐❣❡♠✳
◗✉❛r❛♥t❛ ❡t ❛❧✳ ✭✷✵✵✼✮ ❛♣✉❞ P❡r❡✐r❛ ✭✷✵✶✸✮✱ ♥♦s ❝♦♥✜r♠❛ ❡ss❛s ♦❜s❡r✈❛çõ❡s ❛♦ ❡①♣♦r q✉❡ ♦ ❊♥s✐♥♦ ❞❛s ❈ô♥✐❝❛s ✜❝♦✉ r❡str✐t♦ ❛♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛♣❡s❛r ❞❡ t❡r ✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ❤✐stór✐❝❛✳ ❙❡♥❞♦ ❛❜♦r❞❛❞♦✱ ♥♦ ❡♥t❛♥t♦✱ ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛ ❡ tr❛❜❛❧❤❛❞♦ s♦♠❡♥t❡ ❝♦♠ ♠❛♥✐♣✉❧❛çã♦ ❡ ♠❡♠♦r✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s✱ ❧❡✈❛♥❞♦ ♦s ❛❧✉♥♦s ❡ ❛té ♦s ♣r♦✲ ❢❡ss♦r❡s ❛ ♥ã♦ q✉❡r❡r❡♠ tr❛❜❛❧❤❛r ❝♦♠ ❛s ❝ô♥✐❝❛s✳ ❙❡❣✉♥❞♦ ❛ ❛✉t♦r❛ ❡♠ ✉♠❛ ♣❡sq✉✐s❛ ❝♦♠ ✈ár✐♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ✈❡r✐✜❝♦✉ ✏q✉❡ ❛❧❣✉♥s tr❛③❡♠ ✉♠ ♣❡q✉❡♥♦ r❡s✉♠♦ ❤✐stór✐❝♦ ❡ tr❛t❛♠ ❛s ❝ô♥✐❝❛s ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛ r❡s✉♠✐♥❞♦✲s❡ à ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❢ór♠✉❧❛s✑ ❬✶✼❪✳
✶✽
❡s♣❡❝í✜❝❛ ❞♦ ❡st✉❞♦ ❞❡ ♣❛rá❜♦❧❛s✱ ❞✐❢❡r❡♥t❡ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ✶♦ ❛♥♦ ❞♦ ❊♥s✐♥♦
▼é❞✐♦ ❡♠ q✉❡ s❡ ✉s❛ ❛♣❡♥❛s ❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦ ❢✉♥çã♦f(x) =ax2
✸ P❆❘➪❇❖▲❆✿ ❆❇❖❘❉❆●❊▼
❆◆❆▲❮❚■❈❆
✸✳✶ ❉❡✜♥✐çã♦
❙❡❣✉♥❞♦ ❙t❡✐♥❜r✉♥❝❤ ✫ ❲✐♥t❡r❧❡ ✭✶✾✾✼✮✱ ❝♦♥s✐❞❡r❛♥❞♦ ❡♠ ✉♠ ♣❧❛♥♦ ✉♠❛ r❡t❛d ❡ ✉♠ ♣♦♥t♦ F ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ d✳ ❉❡✜♥❡✲s❡ ❛ P❛rá❜♦❧❛ ❞❡ ❢♦❝♦ F ❡ r❡t❛ ❞✐r❡tr✐③ d✱ ❝♦♠♦ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ sã♦ ❡q✉✐❞✐st❛♥t❡s ❞❡ F ❡ d✳
❋✐❣✉r❛ ✸✳✶✿ P❛rá❜♦❧❛ ❝♦♠♦ ❧✉❣❛r ❣❡♦♠❡tr✐❝♦
❋♦♥t❡✿ ❆rq✉✐✈♦ ♣ró♣r✐♦
❙❡♥❞♦ P′ ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡ ✉♠ ♣♦♥t♦ P ❞♦ ♣❧❛♥♦ s♦❜r❡ ❛ r❡t❛ d✱
❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ P ♣❡rt❡♥❝❡ à ♣❛rá❜♦❧❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ D(P, F) = d(P, P′)✱ ♦✉ t❛♠❜é♠✿
P F❂P P′
❙❡❣✉♥❞♦ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮✱ ✉♠❛ P❛rá❜♦❧❛ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❡ ✉♠ ♣♦♥t♦ q✉❡ s❡ ♠♦✈❡ ♥✉♠ ♣❧❛♥♦ ❞❡ ♠❛♥❡✐r❛ q✉❡ s✉❛ ❞✐stâ♥❝✐❛ ❛ ✉♠❛ r❡t❛ ✜①❛ ♥♦ ♣❧❛♥♦ é s❡♠♣r❡ ✐❣✉❛❧ à s✉❛ ❞✐stâ♥❝✐❛ ❛ ✉♠ ♣♦♥t♦ ✜①♦ ♥♦ ♣❧❛♥♦ ❡ ♥ã♦ s✐t✉❛❞♦ s♦❜r❡ ❛ r❡t❛✳
❖ ♣♦♥t♦ ✜①♦ é ❞❡♥♦♠✐♥❛❞♦ ❢♦❝♦ ❡ ❛ r❡t❛ ✜①❛ é ❞❡♥♦♠✐♥❛❞❛ ❞✐r❡tr✐③ ❞❛ ♣❛rá❜♦❧❛✳ ❆ ❞❡✜♥✐çã♦ ❡①❝❧✉✐ ♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❢♦❝♦ s❡ ❡♥❝♦♥tr❡ s♦❜r❡ ❛ ❞✐r❡tr✐③✳
❉❡✜♥✐çã♦ ✷✵
❙❡❥❛♠ ❞❡s✐❣♥❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❡❧♦ ♣♦♥t♦ F ❡ ♣❡❧❛ r❡t❛ l✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♦ ❢♦❝♦ ❡ ❛ ❞✐r❡tr✐③ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ ❡ s❡❥❛ A ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡ F ❛ l✳ ❆ r❡t❛ a q✉❡ ♣❛ss❛ ♣♦r F ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ l é ❞❡♥♦♠✐♥❛❞❛ ❊✐①♦ ❋♦❝❛❧✳ ❊♥tã♦✱ s❡ V é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ r❡t✐❧í♥❡♦ AF✱ ♣❡❧❛ ❞❡✜♥✐çã♦✱ V ♣❡rt❡♥❝❡ à P❛rá❜♦❧❛✱ t❛❧ ♣♦♥t♦ é ❞❡♥♦♠✐♥❛❞♦ ♦ ✈ért✐❝❡ ❞❛ P❛rá❜♦❧❛✳ ❖ s❡❣♠❡♥t♦ r❡t✐❧í♥❡♦ t❛❧ ❝♦♠♦ BB′✱ ❧✐❣❛♥❞♦ q✉❛✐sq✉❡r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s s♦❜r❡ ❛ P❛rá❜♦❧❛ é ❞❡♥♦♠✐♥❛❞♦ ❝♦r❞❛❀ ❡♠
♣❛rt✐❝✉❧❛r✱ ✉♠❛ ❝♦r❞❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❢♦❝♦✱ t❛❧ ❝♦♠♦ CC′✱ é ❞❡♥♦♠✐♥❛❞❛ ❝♦r❞❛ ❢♦❝❛❧✳
❆ ❝♦r❞❛ ❢♦❝❛❧ LL′ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ é ❞❡♥♦♠✐♥❛❞❛ ❧❛t✉s r❡❝t✉♠✳ ❙❡ P é q✉❛❧q✉❡r
♣♦♥t♦ s♦❜r❡ ❛ P❛rá❜♦❧❛✱ ❛ r❡t❛ F P✱ tr❛ç❛❞❛ ❞❡s❞❡ ♦ ❢♦❝♦ F ❛ P✱ é ❞❡♥♦♠✐♥❛❞❛ r❛✐♦ ❢♦❝❛❧ ❞❡ P✳
❋✐❣✉r❛ ✸✳✷✿ P❛rá❜♦❧❛ ❝♦♠ ❝♦r❞❛s ❡ ❧❛t✉s r❡❝t✉♠
❋♦♥t❡✿ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮
❆ ❞❡✜♥✐çã♦ ❞❡ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮ ♥♦s ❞á ♦ ♠❡s♠♦ ❝♦♥t❡ú❞♦ ❞❡ ✐♥❢♦r♠❛çã♦ q✉❡ ❙t❡✐♥✲ ❜r✉♥❝❤ ✫ ❲✐♥t❡r❧❡ ✭✶✾✾✼✮✱ ❛❝r❡s❝❡♥t❛♥❞♦ ♥❛ ❞❡✜♥✐çã♦ ❡❧❡♠❡♥t♦s ❝♦♠♦ ❝♦r❞❛✱ ❝♦r❞❛ ❢♦❝❛❧ ❡ ❧❛tt✉s r❡❝t✉♠✳
❙❡❣✉♥❞♦ ❨♦✉ss❡❢ ✭✷✵✵✺✮✱ ♣♦❞❡✲s❡ t❛♠❜é♠ ❞❡✜♥✐r ♣❛rá❜♦❧❛ ❝♦♠♦ ✉♠ ❧✉❣❛r ❣❡♦♠é✲ tr✐❝♦ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦β q✉❡ sã♦ ❡q✉✐❞✐st❛♥t❡s ❞❡ ✉♠ ♣♦♥t♦F ❡ ❞❡ ✉♠❛ r❡t❛r❞❡ss❡ ♣❧❛♥♦✳ ❱❡❥❛ q✉❡ ♥❛ ❋✐❣✉r❛ ✸✳✸ é ♣♦ssí✈❡❧ ♣❡r❝❡❜❡r ❛ P❛rá❜♦❧❛ ❡ ❛❧❣✉♥s ❞❡ s❡✉s ♣♦♥t♦s ❡ ❛s ❞✐stâ♥❝✐❛s ❞❡ss❡s ♣♦♥t♦s ❞❛ ♣❛rá❜♦❧❛ à F ❡ à r✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❋✐❣✉r❛ ✸✳✸ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ sã♦✿
❛✮ F é ♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛❀
❜✮ ❆ r❡t❛ r é ❛ ❞✐r❡tr✐③ ❞❛ ♣❛rá❜♦❧❛❀
❝✮ ❆ r❡t❛ e✱ ♣❡r♣❡♥❞✐❝✉❧❛r ❛r✱ ♣❛ss❛♥❞♦ ♣❡❧♦ ❢♦❝♦✱ é ♦ ❡✐①♦ ❞❛ ♣❛rá❜♦❧❛✱ t❛♠❜é♠ ❝❤❛✲ ♠❛❞♦ ❞❡ ❡✐①♦ ❞❡ s✐♠❡tr✐❛❀
❞✮ ❆ ❞✐stâ♥❝✐❛ p ❡♥tr❡ ♦ ❢♦❝♦ ❡ ❛ ❞✐r❡tr✐③ é ♦ ♣❛râ♠❡tr♦ ❞❛ ♣❛rá❜♦❧❛❀
❡✮ ❖ ♣♦♥t♦ V é ♦ ✈ért✐❝❡ ❞❛ P❛rá❜♦❧❛ ❡ ❡stá ♥♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ♦ ❢♦❝♦ ❡ ♦ ♣♦♥t♦ M✱ ❡♠ q✉❡ ♦ ❡✐①♦ ❝r✉③❛ ❛ ❞✐r❡tr✐③ ❞❛ ♣❛rá❜♦❧❛✳ ❈♦♠♦V é ✉♠ ♣♦♥t♦ ❞❛ P❛rá❜♦❧❛✱ t❡♠✲s❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦✿ V F ❂V M ❂ p
❊q✉❛çã♦ ●❡r❛❧ ❞❛ P❛rá❜♦❧❛ ✷✶
❋✐❣✉r❛ ✸✳✸✿ P❛rá❜♦❧❛
❋♦♥t❡✿ ❆rq✉✐✈♦ Pró♣r✐♦
❈❤❛♠❛✲s❡ ❊①❝❡♥tr✐❝✐❞❛❞❡ ❞❡ ✉♠❛ P❛rá❜♦❧❛ ❛♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ q✉❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ q✉♦❝✐❡♥t❡ ❡♥tr❡ ❛ ❞✐stâ♥❝✐❛ ❞❛ ❞✐r❡tr✐③ ❛ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ P❛rá❜♦❧❛ ❡ ❛ ❞✐stâ♥❝✐❛ ❞❡st❡ ♣♦♥t♦ ❞❛ P❛rá❜♦❧❛ ❛♦ ❢♦❝♦✳ ❈♦♠♦ ♥❛ P❛rá❜♦❧❛ ❡st❛s ❞✐stâ♥❝✐❛s sã♦ s❡♠♣r❡ ✐❣✉❛✐s✱ t❡♠✲s❡ q✉❡ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ é s❡♠♣r❡ ✐❣✉❛❧ ❛ ✶✳
❆ ❞❡✜♥✐çã♦ ❞❡ ❨♦✉ss❡❢ ✭✷✵✵✺✮ t❛♠❜é♠ tr❛③ ♦ ♠❡s♠♦ ❝♦♥t❡ú❞♦ ❞❡ ✐♥❢♦r♠❛çã♦ q✉❡ ❙t❡✐♥❜r✉♥❝❤ ✫ ❲✐♥t❡r❧❡ ✭✶✾✾✼✮✱ ❛❝r❡s❝❡♥t❛♥❞♦ q✉❡ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ ❞❛ ♣❛rá❜♦❧❛ é s❡♠✲ ♣r❡ ✐❣✉❛❧ ❛ ✶✳✭◆❛s ♦✉tr❛s ❈ò♥✐❝❛s✱ ❝❤❛♠❛♥❞♦ ❡①❝❡♥tr✐❝✐❞❛❞❡ ❞❡ e✱ ♥❛ ❊❧✐♣s❡ t❡r❡♠♦s 0< e <1 ❡ ♥❛ ❍✐♣ér❜♦❧❡ t❡r❡♠♦s e >1✮✳
✸✳✷ ❊q✉❛çã♦ ●❡r❛❧ ❞❛ P❛rá❜♦❧❛
❋✐❣✉r❛ ✸✳✹✿ ❋✐❣✉r❛ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ q✉❛❧q✉❡r
❋♦♥t❡✿ ❆rq✉✐✈♦ ♣ró♣r✐♦
❊q✉❛çã♦ ●❡r❛❧ ❞❛ P❛rá❜♦❧❛ ✷✷
s❛❜✐❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦Q(u, v)❛ r❡t❛dé ❞❛❞❛ ♣♦rD(Q, d) = |au√+bv+c| a2
+b2 ✱
❛ss✐♠ s❡ P é ✉♠ ♣♦♥t♦ ❞❛ P❛rá❜♦❧❛✱ t❡♠♦s✿
D(F, P) = D(P, d)
q
(xf −x)2+ (yf −y)2 = |
ax+by+c|
√
a2
+b2
(xf −x)
2
+ (yf −y)
2
= (ax+by+c)
2
a2
+b2 , resolvendo:
Ax2
+By2
+Cxy+Dx+Ey+F = 0
♦♥❞❡✿ A=b2
B =a2
C =−2ab D=−2(a2
+b2
)xf −2ac
E =−2(a2
+b2
)yf −2bc
F = (a2
+b2
)(x2
f +y
2
f)−c
2
❈♦♠ ❛ ❡q✉❛çã♦ ❞♦ t✐♣♦Ax2
+By2
+Cxy+Dx+Ey+F = 0✱ ♣♦❞❡♠♦s ♥♦t❛r q✉❡ ♣❛r❛ s❡r ❡q✉❛çã♦ ❞❛ P❛rá❜♦❧❛ t❡♠✲s❡ q✉❡ ♦❜❡❞❡❝❡r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
C2
−4AB = 0
Pr♦♣♦s✐çã♦ ✸✳✶✳ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛s ❈ô♥✐❝❛s
P❛rt✐♥❞♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ P❛rá❜♦❧❛ ❡♥❝♦♥tr❛♠♦s ♦ ❞✐s❝r✐♠✐♥❛♥t❡ C2
−4AB = 0 q✉❡
❝❛r❛❝t❡r✐③❛ ♥ã♦ só ❛ P❛rá❜♦❧❛✱ ♠❛s t♦❞❛s ❛s ❝ô♥✐❝❛s✳ ❊ss❛ ❝❛r❛❝t❡r✐③❛çã♦ é ♠♦str❛❞❛ ♣♦r ❘❡✐s✭✶✾✾✻✮ ❡ ❙❛t♦✭✷✵✵✺✮✳
❉❛❞❛ ❛ ❡q✉❛çã♦
Ax2 +By2
+Cxy+Dx+Ey+F = 0,(I)
♦ ♥ú♠❡r♦
∆ =C2−4AB
é ✐♥✈❛r✐❛♥t❡ ♣♦r r♦t❛çã♦ ♦✉ tr❛♥s❧❛çã♦✱ ✐st♦ é✱ s❡
A1x 2 1+B1y
2
1+C1x1y1+D1x1+E1y1+F1= 0
é ❛ ❡q✉❛çã♦ q✉❡ s❡ ♦❜té♠ ❞❡ ✭■✮ ❡❢❡t✉❛♥❞♦✲s❡ r♦t❛çã♦ ♦✉ tr❛♥s❧❛çã♦ ❞❡ ❡✐①♦s✱ ❡♥tã♦
∆ =C1 2
−4A1B1=C 2
−4AB.
❊ ❛✐♥❞❛ ❝♦♥❢♦r♠❡ ∆ s❡❥❛ ♠❡♥♦r✱ ♠❛✐♦r ♦✉ ✐✉❛❧ ❛ ③❡r♦✱ ♦ ❣rá✜❝♦ ❞❡ ✭■✮ é✱
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✸
❙❡❣✉♥❞♦ ❙❛t♦✭✷✵✵✺✮ t❡♠♦s q✉❡✿
✐✮ ❙❡ ∆<0t❡♠♦s q✉❡A❡B ♣♦ss✉❡♠ s✐♥❛✐s ✐❣✉❛✐s✱ ❡ tr❛t❛✲s❡ ❞❡ ✉♠❛ ❝ô♥✐❝❛ ❞♦ ❣ê♥❡r♦ ❡❧✐♣s❡❀
✐✐✮ ❙❡ ∆>0 t❡♠♦s q✉❡ A ❡ B ♣♦ss✉❡♠ s✐♥❛✐s ❝♦♥trár✐♦s✱ ❡ tr❛t❛✲s❡ ❞❡ ✉♠❛ ❝ô♥✐❝❛ ❞♦ ❣ê♥❡r♦ ❤✐♣ér❜♦❧❡❀
✐✐✐✮ ❙❡ ∆ = 0 t❡♠♦s q✉❡C2
= 4AB✱ ❡ tr❛t❛✲s❡ ❞❡ ✉♠❛ ❝ô♥✐❝❛ ❞♦ ❣ê♥❡r♦ ♣❛rá❜♦❧❛✳
✸✳✸ ❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s
◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ❛♣r❡s❡♥t❛❞♦ ✉♠ ♠❡✐♦ ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡st✉❞♦ ❛♥❛❧ít✐❝♦ ❞❛ ♣❛✲ rá❜♦❧❛✳ ❆ tr❛♥s❧❛çã♦ ♦✉ r♦t❛çã♦ ❞♦s ❡✐①♦s ♥♦s ♣❡r♠✐t❡ ❡s❝♦❧❤❡r ♠✉✐t❛s ✈❡③❡s ♦s ❡✐①♦s ❝❡rt♦s q✉❡ t♦r♥❛♠ ❛ ❡q✉❛çã♦ ❛ s❡r tr❛❜❛❧❤❛❞❛ ♠❛✐s s✐♠♣❧❡s q✉❛♥❞♦ s❡ tr❛❜❛❧❤❛ ❝♦♠ ❝ô♥✐❝❛s✳
P❛r❛ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ é ❛ ❞❡✲ t❡r♠✐♥❛çã♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✈ár✐❛s ❝✉r✈❛s ❡ ❝♦♥✜❣✉r❛çõ❡s ❣❡♦♠étr✐❝❛s✳ ❊♥tr❡t❛♥t♦✱ à ♠❡❞✐❞❛ q✉❡ ❛♣r♦❢✉♥❞❛♠♦s ♥♦ss♦ ❡st✉❞♦✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❛s ❝✉r✈❛s ❡ s✉❛s ❡q✉❛çõ❡s s❡ t♦r♥❛♠ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛s ❡ ♠❛✐s ❞✐❢í❝❡✐s ❞❡ s❡r❡♠ ❛♥❛❧✐s❛❞❛s✳ ❆ss✐♠ é ♠❛✐s ❝♦♥✈❡✲ ♥✐❡♥t❡ ✐♥tr♦❞✉③✐r ❛ ♥♦çã♦ ❞❡ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ✉♠ r❡❝✉rs♦ q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛ s✐♠♣❧✐✜❝❛r ❛s ❡q✉❛çõ❡s ❞❡ ♠✉✐t❛s ❝✉r✈❛s✳
❆✐♥❞❛ ♣❛r❛ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ é ✉♠❛ ♦♣❡r❛çã♦ ♣♦r ♠❡✐♦ ❞❛ q✉❛❧ ✉♠❛ r❡❧❛çã♦✱ ❡①♣r❡ssã♦ ♦✉ ✜❣✉r❛ s❡ tr❛♥s❢♦r♠❛ ❡♠ ♦✉tr❛ s❡❣✉✐♥❞♦ ✉♠❛ ❧❡✐ ❞❛❞❛✳ ❆♥❛✲ ❧✐t✐❝❛♠❡♥t❡✱ ❛ ❧❡✐ s❡ ❡①♣r❡ss❛ ♣♦r ✉♠❛ ♦✉ ♠❛✐s ❡q✉❛çõ❡s ❝❤❛♠❛❞❛s ❡q✉❛çõ❡s ❞❡ tr❛♥s✲ ❢♦r♠❛çõ❡s✳
✸✳✸✳✶ ❘♦t❛çã♦ ❞❡ ❡✐①♦s
❈♦♥s✐❞❡r❡ ✉♠ ♣❧❛♥♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ①❖②✳ ●✐r❛♥❞♦ ♦s ❡✐①♦s ❞❡ ✉♠ â♥❣✉❧♦θ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ ❖✱ ❡♥❝♦♥tr❛r❡♠♦s ✉♠ ♥♦✈♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s x′Oy′✱ ❝♦♠
✐ss♦✱ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r P ♥♦ ♣❧❛♥♦ t❡rá ❝♦♦r❞❡♥❛❞❛s (x, y)❡(x′, y′) ❛♥t❡s ❡ ❞❡♣♦✐s ❞❛
r♦t❛çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✸✳✺✳
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✹
❋✐❣✉r❛ ✸✳✺✿ ❘♦t❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s
❋♦♥t❡✿ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮
\
P OA′ =φ ✱ ♣♦❞❡♠♦s ♦❜t❡r ❛♥❛❧✐s❛♥❞♦ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦P OA✿
x = rcos(θ+φ)
= [(cosθ)(cosφ)−(senθ)(senφ)]
= r(cosθ)(cosφ)−r(senθ)(senφ)(1),
❡ ❛✐♥❞❛✱
y = rsen(θ+φ)
= [(senθ)(cosφ) + (senφ)(cosθ)]
= r(senθ)(cosφ) +r(senφ)(cosθ)(2)
❊ ❛♥❛❧✐s❛♥❞♦ ♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦P OA′✿
x′ =rcosφ ❡y′ =rsenφ ✭✸✮
❙✉❜st✐t✉✐♥❞♦ ✭✸✮ ❡♠ ✭✶✮ ❡ ✭✷✮✱ ♦❜t❡♠♦s ❛ tr❛♥s❢♦r♠❛çã♦ ❞❛s ❛♥t✐❣❛s ♣❛r❛ ❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s✱ ❞❛❞❛s ♣♦r✿
x=x′cosθ−y′senθ ❡ y=x′senθ+y′cosθ ✭✯✮
P❛r❛ ▲❡❤♠❛♥♥ ✭✶✾✾✽✮✱ ♦ ♣r✐♥❝✐♣❛❧ ❡♠♣r❡❣♦ ♣❛r❛ ❛ r♦t❛çã♦ ❞❡ ❡✐①♦s é ❛ r❡♠♦çã♦ ❞♦ t❡r♠♦ ❡♠ xy ❞❛s ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞♦ ❣r❛✉✳
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✺
Ax2
+By2
+Cxy+Dx+Ey+F = 0
A(x′cosθ−y′senθ)2
+B(x′senθ+y′cosθ)2
+
+C(x′cosθ−y′senθ)(x′senθ+y′cosθ) +
+D(x′cosθ−y′senθ) +E(x′senθ+y′cosθ) +F = 0
❈♦❧♦❝❛♥❞♦ ❡♠ ❡✈✐❞ê♥❝✐❛ ♦s t❡r♠♦s x′ ❡y′✱ ♦❜t❡r❡♠♦s✿
[Acos2
θ+Bsen2
θ+C(cosθ)(senθ)]x′2
+
+Asen2
θ+Bcos2
θ−C(senθ)(cosθ)]y′2
+
+[−2A(cosθ)(senθ) + 2B(senθ)(cosθ) +Ccos2
θ−Csen2
θ]x′y′+
+(Dcosθ+Esenθ)x′+
+(−Dsenθ+Ecosθ)y′+F = 0
❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s✿ A=b2
B =a2
C =−2ab D=−2(a2
+b2
)xf −2ac
E =−2(a2
+b2
)yf −2bc
F = (a2
+b2
)(x2
f +y
2
f)−c
2✱ t❡♠♦s✿
(bcosθ−asenθ)2
x′2
+
+(bcosθ+asenθ)2
y′2
+
+[(−2b2
+ 2a2
)(cosθ)(senθ)−2abcos2
θ+ 2absen2
θ]x′y′+
+[−2(a2
xf +b
2
xf +ac)cosθ−2(a
2
yf +b
2
yf +bc)senθ]x′+
+[2(a2
xf +b
2
xf +ac)senθ−2(a
2
yf +b
2
yf +bc)cosθ]y′+
+(a2
x2
f +a
2
y2
f +b
2
x2
f +b
2
y2
f −c
2
) = 0
❆ss✐♠ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ A′x′2
+B′y′2
+C′x′y′+D′x′+E′y′ +F = 0 ✭✯✮✱ ❝♦♠✿
A′ =bcosθ−asenθ
B′ =bcosθ+asenθ
C′ = (−2b2
+ 2a2
)(cosθ)(senθ)−2abcos2
θ+ 2absen2
θ D′ =−2(a2
xf +b
2
xf +ac)cosθ−2(a
2
yf +b
2
yf +bc)senθ
E′ = 2(a2
xf +b
2
xf +ac)senθ−2(a
2
yf +b
2
yf +bc)cosθ
F =a2
x2
f +a
2
y2
f +b
2
x2
f +b
2
y2
f −c
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✻
❆♥✉❧❛♥❞♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡x′y′ ❡♥❝♦♥tr❛♠♦s✿
(−2b2
+2a2
)(cosθ)(senθ)−2abcos2
θ+2absen2
θ = 0✱ ✉s❛♥❞♦(cosθ)(senθ) = 1 2sen2θ ❡ cos2
θ−sen2
θ =cos2θ✱ ❡♥❝♦♥tr❛♠♦s✿
(−2b2
+ 2a2
)1
2sen2θ−2abcos2θ = 0 (−b2
+a2
)sen2θ−2abcos2θ = 0 sen2θ
cos2θ =
2ab (−b2
+a2
)
tg2θ = 2ab (−b2
+a2
)
tg2θ = C A−B
❝♦♠ A 6=B✱ ❛ss✐♠ t❡♠♦s✿ θ = 1 2arctg
C A−B
❖❜t❡r❡♠♦s ❛❣♦r❛ ♦ ✈❛❧♦r ❞♦ senθ ❡ cosθ✱ ♣❛r❛ s✉❜st✐t✉✐r♠♦s ♥❛ ❡q✉❛çã♦ ❡ ❝❤❡❣❛r♠♦s à r♦t❛çã♦✳
❙❛❜❡♥❞♦ q✉❡✱
tg2θ = 2ab (−b2
+a2
) = C
A−B, temos :
tg2θ = 2tgθ 1−tg2
θ =
2ab (−b2
+a2
)assim,
ab(tgθ)2
+(−b2
+a2
)tgθ−ab= 0✱ r❡s♦❧✈❡♥❞♦ ❡ss❛ ❡q✉❛çã♦ ❡♥❝♦♥tr❛♠♦s✱ ♣❛r❛0< θ < π 2
tgθ= b
a✱ ❛ss✐♠✿ senθ
cosθ = b
a✱ ❡ ❛✐♥❞❛senθ = b
acosθ✱ ❛♣❧✐❝❛♥❞♦ ♥❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ tr✐❣♦♥♦♠❡tr✐❛✿ sen2
θ+cos2
θ = 1
b a
2 cos2
θ+cos2
θ = 1 ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ♣❛r❛0< θ < π
2✱ t❡♠♦s✿ cosθ = √ a
b2
+a2 ❡ senθ=
b
√
b2
+a2✱ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✯✮✿
B′y′2
+D′x′+E′y′ +F = 0✭✯✯✮✱ ♦♥❞❡✿
A′ = 0
B′ = √ 2ba
b2
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✼
C′ = 0
D′ = −2a(a
2
xf +b
2
xf +ac)−2b(a
2
yf +b
2
yf +ac)
√
b2
+a2
E′ = 2b(a
2
xf +b
2
xf +ac)−2a(a
2
yf +b
2
yf +ac)
√
b2+a2
F =a2
x2
f +a
2
y2
f +b
2
x2
f +b
2
y2
f −c
2
◆❛ ❡q✉❛çã♦ B′y′2
+D′x′+E′y′ +F = 0✱ ❝♦♠♣❧❡t❛♥❞♦ ♦s q✉❛❞r❛❞♦s ❡♥❝♦♥tr❛r❡♠♦s✿
(y′−y
0) 2
= 2p(x′−x
0) ✭■✮✱ ♦♥❞❡✿
y0 =−
E′
B′❀2p=−
D′
B′ ❡ x0 =−
F D′ +
E′2
4B′D′
✸✳✸✳✷ ❚r❛♥s❧❛çã♦ ❞❡ ❡✐①♦s
❈♦♥s✐❞❡r❛♥❞♦ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s Ox ❡ Oy✱ ❞❛❞♦ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡ss❡ ♣❧❛♥♦ O′ ❞❡ ❝♦♦r❞❡♥❛❞❛s (a, b)✱ ❝♦♥str✉✐r❡♠♦s ✉♠ ♥♦✈♦ s✐st❡♠❛ ♦r❞❡✲
♥❛❞♦ ❝❛rt❡s✐❛♥♦ ❞❡ ❡✐①♦s O′x′ ❡ O′y′✱ ♦✉ s❡❥❛✱ ✉♠ s✐st❡♠❛ x′O′y′✳ ❈♦♥s✐❞❡r❡ ♦s ❡✐①♦s
O′x′ ❡ O′y′ ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s Ox ❡ Oy✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ ♠❡s♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡✲
❞✐❞❛✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✳ ❉❡st❛ ♠❛♥❡✐r❛ ♣♦❞❡♠♦s ♦❜t❡r ✉♠ s✐st❡♠❛ ❛ ♣❛rt✐r ❞♦ ♦✉tr♦ r❡❛❧✐③❛♥❞♦ ❛ tr❛♥s✐çã♦ ❞♦s ❡✐①♦s✳
❋✐❣✉r❛ ✸✳✻✿ ❚r❛♥s❧❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s
❋♦♥t❡✿ ❆rq✉✐✈♦ ♣ró♣r✐♦
❙❡❥❛ ✉♠ ♣♦♥t♦P q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s✿ x ❡ y ❡♠ xOy ❡ x′ ❡y′ ❡♠ x′O′y′✳
❆♥❛❧✐s❛♥❞♦ ❛ ✜❣✉r❛ ❛❝✐♠❛✱ t❡♠♦s q✉❡✿
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✽
❊♠ ✭✐✮ s❡ ❡♥❝♦♥tr❛♠ ❛s r❡❧❛çõ❡s ❞❡ tr❛♥s❧❛çã♦ q✉❡ ♥♦s ♣❡r♠✐t❡ tr❛♥s❢♦r♠❛r ❝♦♦r❞❡✲ ♥❛❞❛s ❞❡ ✉♠ s✐st❡♠❛ ♣❛r❛ ♦ ♦✉tr♦✳
❱♦❧t❛♥❞♦ ❛ ❡q✉❛çã♦ ✭■✮ ❞❛ s❡❝çã♦ ✸✳✸✳✶✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛❣♦r❛ ✉♠❛ tr❛♥s❧❛çã♦ ❞❡ ❢❛t♦r ✭♣❡❧♦ ✈❡t♦r −→v = (−x0,−y0)✮✱ t❡r❡♠♦s q✉❡✿
x′ =x
o+x ❡ ❛✐♥❞❛y′ =yo+y ♦✉x=x′−x0 ❡ ❛✐♥❞❛ y=y′−y0✱ ♦♥❞❡x ❡ y✱ sã♦
❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❛♣ós ❛ tr❛♥s❧❛çã♦ ❞❡x′Oy′ ♣❛r❛ xOy✱ ❝♦♠ O = (x
0, y0)
❉❡ss❡ ♠♦❞♦ ❛ ❡q✉❛çã♦ ✭■✮ ❞❛ s❡❝çã♦ ✸✳✸✳✶✱ ❛♣ós ❛ tr❛♥s❧❛çã♦ ✜❝❛rá✿
(y′−y
0) 2
= 2p(x′−x
0)
y2
= 2px ✭■■✮
✸✳✸✳✸ ❍♦♠♦t❡t✐❛
❙❡❣✉♥❞♦ ❙❤✐♥❡ ✭✷✵✵✽✮ ❤♦♠♦t❡t✐❛ ❞❡ ✉♠❛ ✜❣✉r❛ F ❝♦♠ ❝❡♥tr♦ O ❡ r❛③ã♦ k✱ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✱ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ P ❞❡ F ♦ ♣♦♥t♦P′ s♦❜r❡ ❛ s❡♠✐✲r❡t❛ −→OP✱ ❞❡ ♦r✐❣❡♠ O✱ t❛❧ q✉❡ OP′ =kOP
❋✐❣✉r❛ ✸✳✼✿ ❍♦♠♦t❡t✐❛
❙❤✐♥❡ ✭✷✵✵✽✮
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✷✾
❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❤♦♠♦t❡t✐❛ ❞❡ ❢❛t♦rλ >0✱ ♥❛ ❡q✉❛çã♦ ✭■■✮ ❞❛ s❡❝çã♦ ✸✳✸✳✷✱ ❝♦♠✿
x=λx ❡y=λy ✱ ❡♥❝♦♥tr❛r❡♠♦s✿ (y)2
= 2p(x) (λy)2
= 2p(λx)
y2
= 2p
λ x✱ ❛ss✐♠ ❡♥❝♦♥tr❛r❡♠♦s✿ y2
=x ✭■■■✮✱ ♦♥❞❡ 2p
λ = 1✱ ♦✉ s❡❥❛✱λ = 2p
❖❜s❡r✈❛çã♦ ✸✳✶✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ♦ ♣❛râ♠❡tr♦ ♣ é ♥❡❣❛t✐✈♦✱ p < 0✱ t♦♠❛♥❞♦ ❝♦♠♦ ❢❛t♦r ❞❡ ❤♦♠♦t❡t✐❛ λ = −2p ❡ tr❛♥s❢♦r♠❛♠♦s ❛ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ y2
= 2px ❡♠ y2
=−x✳ ❊♠ s❡❣✉✐❞❛ ❜❛st❛ ❛♣❧✐❝❛♠♦s ✉♠❛ r❡✢❡①ã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s
(x, y)→(−x, y)
❖ ❝♦♥❝❡✐t♦ ❞❡ ❤♦♠♦t❡t✐❛ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ s❡♠❡❧❤❛♥ç❛✱ ♥❛ ❣❡♦♠❡tr✐❛ ❊✉✲ ❝❧✐❞✐❛♥❛ ♣❧❛♥❛ ❡ q✉❡ q✉❛❧q✉❡r ♣❛rá❜♦❧❛ q✉❡ ❡st❡❥❛ ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛y2
= 2px s❡♠♣r❡ ♣♦❞❡ s❡r ❝♦♥✈❡rt✐❞❛ ❛♦ ❢♦r♠❛t♦ y2
=x ♦✉y=x2✳
❆ s❡❝çã♦ ✸✳✸ ♥♦s ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r q✉❡ t♦❞❛s ❛s ♣❛rá❜♦❧❛s sã♦ s❡♠❡❧❤❛♥t❡s✱ ❞✐❢❡✲ r✐♥❞♦ ❛♣❡♥❛s ♣♦r ✉♠❛ q✉❡stã♦ ❞❡ ❡s❝❛❧❛✱ ♦✉ s❡❥❛✱ ❛ ❡q✉❛çã♦ ❞❡ q✉❛❧q✉❡r ♣❛rá❜♦❧❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ ❢♦r♠❛y2
=x✱ ✉t✐❧✐③❛♥❞♦ ♦s ♠ét♦❞♦s ❞❡ r♦t❛çã♦✱ tr❛♥s❧❛çã♦ ♦✉ ❤♦♠♦t❡t✐❛✳
❊①❡♠♣❧♦ ✸✳✶✳ ❙✐♠♣❧✐✜q✉❡ ❛ ❡q✉❛çã♦ x2
+ 8x−3y+ 10 = 0 ♣♦r ✉♠❛ tr❛♥s❧❛çã♦ ❞♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✳
❈♦♠♣❧❡t❛♥❞♦ ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❝♦♠ r❡❧❛çã♦ ❛x ❡♥❝♦♥tr❛r❡♠♦s✿
x2
+ 8x−3y+ 10 = 0
x2
+ 8x+ 16−16−3y+ 10 = 0
(x+ 4)2
−3y−6 = 0 (x+ 4)2
−3(y−2) = 0
❢❛③❡♥❞♦x′ =x+ 4 ❡y′ =y−2✱ t❡♠♦s✿
x′2
−3y′ = 0
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✸✵
❊①❡♠♣❧♦ ✸✳✷✳ ❯s❛♥❞♦ ❛s té❝♥✐❝❛s ❡st✉❞❛❞❛s ♥❛ s❡çã♦ ✸✳✸✱ tr❛♥s❢♦r♠❡ ❛ ❡q✉❛çã♦4x2
+ y2
+ 4xy+x−2y = 0 ❡♠ ✉♠❛ ❞♦ t✐♣♦ y2
=x✳ ❋❛r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ✉♠❛ r♦t❛çã♦ ✉s❛♥❞♦✿
x=x′cosθ−y′senθ ❡y =x′senθ+y′cosθ✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ❞❛❞❛✱ ♦❜t❡♠♦s✿
4(x′cosθ−y′senθ)2
+ (x′senθ+y′cosθ)2
+
+4(x′cosθ−y′senθ)(x′senθ+y′cosθ) +
+(x′cosθ−y′senθ)−2(x′senθ+y′cosθ) = 0
❞❡s❡♥✈♦❧✈❡♥❞♦ ❡♥❝♦♥tr❛♠♦s✿
(4cos2
θ+sen2
θ+ 4cosθsenθ)x′2
+
+(4sen2
θ+cos2
θ−4senθcosθ)y′2
+
+(−6cosθsenθ+ 4cos2
θ−4sen2
θ)x′y′+
+(cosθ−2senθ)x′+ (−senθ−2cosθ)y′ = 0(1)
❈♦♠♦ ❛ ✐♥t❡♥çã♦ é ❡❧✐♠✐♥❛r ♦ ♣r♦❞✉t♦x′y′✱ ❢❛r❡♠♦s s❡✉ ❝♦❡✜❝✐❡♥t❡ ✐❣✉❛❧ à ③❡r♦✱ ♦✉ s❡❥❛✿
−6cosθsenθ+ 4cos2
θ−4sen2
θ= 0 ✭✷✮
◆♦t❡ q✉❡✿ sen(2θ) = 2senθcosθ✱ ♦✉ s❡❥❛✱ cosθsenθ = 1
2sen2θ ❡ ❛✐♥❞❛ cos2θ = cos2
θ−sen2
θ ✱ ♦❜t❡♠♦s s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✷✮
−3sen2θ+ 4cos2θ = 0 3sen2θ = 4cos2θ
sen2θ cos2θ =
4 3
tg2θ = 4 3.
s✉❜st✐t✉✐♥❞♦ ♥❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ tr✐❣♦♥♦♠❡tr✐❛✱ sen2
θ+cos2
θ = 1✱ t❡♠♦s q✉❡✿
cosθ = √2
5 ❡ senθ = 1
√
5 ❡ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✮✱ ♦❜t❡♠♦s✿
y′ =√5x′2
❖ ❣rá✜❝♦ ❞❡ss❛ ❡q✉❛çã♦ é ✉♠❛ ♣❛rá❜♦❧❛ ❡ ❡st❛ r❡♣r❡s❡♥t❛❞❛ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦s ❞♦✐s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r✳
❖❜s❡r✈❡ q✉❡ ❡st❛ ♣❛rá❜♦❧❛ é t❛♠❜é♠ ♦ ❣rá✜❝♦ ❞❛ ❡q✉❛çã♦4x2
+y2
❘♦t❛çã♦ ❡ ❚r❛♥s❧❛çã♦ ❞❡ ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ✸✶
❋✐❣✉r❛ ✸✳✽✿ ❘♦t❛çã♦ ❞❡ ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s
❋♦♥t❡✿ ▲❡❤♠❛♥♥
❆❣♦r❛ t❡♥❞♦ y′ =√5x′2✱ ✉s❛r❡♠♦s ❤♦♠♦t❡t✐❛✿
❙❡❥❛y′ =λy′′ ❡ x′ =λx′′✱ t❡r❡♠♦s✿
y′ =√5x′2
⇒λy′′ =√5(λx′′)2
⇒ y′′ =x′′2 ✭✸✮ ✱ ♦♥❞❡
λ= √1
5
◆♦t❡ q✉❡ ✭✸✮ ❛✐♥❞❛ ♥ã♦ é ❛ ❡q✉❛çã♦ ❞❡s❡❥❛❞❛✳ P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❡ss❛ ❡q✉❛çã♦ ❜❛st❛ ❢❛③❡r♠♦s ✉♠❛ r♦t❛çã♦✱ ❝♦♠ θ = π
2✳
❆ss✐♠✱ ❢❛③❡♥❞♦ x′′=xcosθ−ysenθ ❡ y′′ =xsenθ+ycosθ ✱ ♦❜t❡r❡♠♦s✿
xsenθ+ycosθ = (xcosθ−ysenθ)2✱ ❝♦♠♦
θ = π
2✱ ❡♥tã♦senθ= 1 ❡cosθ = 0 ⇒ y
2
=x
◆♦t❡ ♣❡❧❛ ✜❣✉r❛ ❛❜❛✐①♦✱ q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧ ❡ ❞❡y2
=x✱ é ♦ ♠❡s♠♦✳
❋✐❣✉r❛ ✸✳✾✿ ●rá✜❝♦ ❛♣ós r♦t❛çã♦
