Open Curvas de Bézier

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❈✉r✈❛s ❞❡ ❇é③✐❡r

❊❱❊❘❚ ❊▲❱■❙ ❇❆❚■❙❚❆ ❉❊ ❆▲▼❊■❉❆

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ▲❡♥✐♠❛r ◆✉♥❡s ❞❡ ❆♥❞r❛❞❡

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

❈✉r✈❛s ❞❡ ❇é③✐❡r

♣♦r

❊❱❊❘❚ ❊▲❱■❙ ❇❆❚■❙❚❆ ❉❊ ❆▲▼❊■❉❆

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

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ▲❊◆■▼❆❘ ◆❯◆❊❙ ❉❊ ❆◆❉❘❆❉❊ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❋❘❆◆❑ ❙■◆❆❚❘❆ ●❖▼❊❙ ❉❆ ❙■▲❱❆ ✲ ❯❋❘P❊

Pr♦❢✳ ❉r✳ ❊❉❙❖◆ ❉❊ ❋■●❯❊■❘✃❉❖ ▲■▼❆ ❏❯◆■❖❘ ✲ ❯❋P❇

❆❣r❛❞❡❝✐♠❡♥t♦s

◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❛s t✉r♠❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ❣❛❧❡❣♦ ✭❊r✐✲ ♥❛❧❞♦✮✱ ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡r❛♠ t♦❞❛ ♣❛❝✐ê♥❝✐❛ ♣❛r❛ tr❛♥s♠✐t✐r ♦ ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ❛♦ ❧♦♥❣♦ ❞♦ ❝✉rs♦ ♥❛ ❯❋P❇✳ ❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ s✉♣♦rt❛r❛♠ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❞✉r❛♥t❡ ♦s sá❜❛❞♦s ❡ t♦❞❛s ❛s ❤♦r❛s ❞❡ ❡st✉❞♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❛❞♦✲ rá✈❡❧ ❝♦♠♣❛♥❤❡✐r❛ ❊r✐❦❛ ▼❛r✐❛ ▲✐♥s ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❞✉r❛♥t❡ ❡st❛ ❥♦r♥❛❞❛✳

❉❡❞✐❝❛tór✐❛

❆♦ ♠❡✉ ❛✈ô ❊✉❢rás✐♦✱ q✉❡ ❞❡✉s t❡♥❤❛ ❡♠ ❜♦♠ ❧✉❣❛r✳ ■♥ ♠❡♠♦r✐✉♠

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❢❛③❡♠♦s ✉♠❛ ✐♥tr♦❞✉çã♦ às ❈✉r✈❛s ❞❡ ❇é③✐❡r✱ ✐♠♣♦rt❛♥t❡ ✐t❡♠ ❞❛ ❛♣❧✐❝❛çã♦ ♠❛t❡♠át✐❝❛ q✉❡ ♦r✐❣✐♥♦✉✲s❡ ♥❛ ✐♥❞ústr✐❛ ❛✉t♦♠♦❜✐❧íst✐❝❛ ❢r❛♥❝❡s❛ ❡ q✉❡ tê♠ ❛♣❧✐❝❛çõ❡s ❡♠ ✈ár✐❛s ár❡❛s ❝✐❡♥tí✜❝❛s✳ ❉✐✈❡rs♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s sã♦ r❡✈✐s✐t❛❞♦s t❛✐s ❝♦♠♦ ❝✉r✈❛s ❞❡✜♥✐❞❛s ♣❛r❛♠❡tr✐❝❛♠❡♥t❡✱ ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦✳ ❆♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ é ❛❜♦r❞❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❈❛st❡❧❥❛✉ ♣❛r❛ ❝♦♥str✉çã♦ ❞❡ ❝✉r✈❛ ❡ s✉❛s ❞❡r✐✈❛❞❛s✳ ❙ã♦ ✐♠♣❧❡♠❡♥t❛❞♦s ❡①❡♠♣❧♦s ❞❡ ❝♦♥str✉çã♦ ✉s❛♥❞♦ ♦ ●❡♦●❡❜r❛ ❡ ▲❆_❚❊❳✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❈✉r✈❛s ❞❡ ❇é③✐❡r✱ P♦❧✐♥ô♠✐♦ ❞❡ ❇❡r♥st❡✐♥✱ ❙♦❢t✇❛r❡ ●❡♦❣❡❜r❛✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ ♠❛❦❡ ❛♥ ✐♥tr♦❞✉❝t✐♦♥ ❛♥ ✐♠♣♦rt❛♥t ❛♣♣❧✐❝❛t✐♦♥ ♠❛t❤❡♠❛t✐❝s ❝❛❧❧❡❞ ❇é③✐❡r ❝✉r✈❡s✳ ❚❤❡ ❤✐st♦r② ♦❢ t❤✐s ❝✉r✈❡ ♦r✐❣✐♥❛t❡❞ ✐♥ ✐♥❞✉str② ❛✉t♦♠♦❜✐❧❡ ❋r❡♥❝❤ ✱ ❛♥❞ ❢♦✉♥❞ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✈❛r✐♦✉s ✜❡❧❞s ♦❢ s❝✐❡♥❝❡✳ ❘❡✈✐s✐t s♦♠❡ ❝♦♥❝❡♣ts s✉❝❤ ❛s ♣❛r❛♠❡tr✐❝ ❢✉♥❝t✐♦♥s✱ ♣♦❧②♥♦♠✐❛❧s ❇❡r♥st❡✐♥ ❛♥❞ ✐♥t❡r♣♦❧❛t✐♦♥ ❢♦r ❞❡✜♥✐t✐♦♥ t❤❡ ❝✉r✈❡s ❇é③✐❡r✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss t❤❡ ❛❧❣♦r✐t❤♠ ❈❛st❡❧❥❛✉ ✇❤✐❝❤ ❢❛❝✐❧✐✲ t❛t❡s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡ ❛♥❞ ❞❡t❡r♠✐♥❡ ❞❡r✐✈❛t✐✈❡✳ ❚❤r♦✉❣❤♦✉t t❤❡ t❡①t ✇❡ ✇✐❧❧ ✐♠♣❧❡♠❡♥t s♦♠❡ ❡①❛♠♣❧❡s ✇✐t❤ ●❡♦❣❡❜r❛ s♦❢t✇❛r❡ ❛♥❞ ▲❆_{❚❊❳ ✐♥ ❛❞❞✐t✐♦♥ t♦}

❞✐s❝✉ss r❡❧❡✈❛♥t ✐ss✉❡s t❤❛t ❛r♦✉s❡ ♣✉❜❧✐❝ ✐♥t❡r❡st✳

❑❡②✇♦r❞s✿ ❇é③✐❡r ❈✉r✈❡s✱ ❇❡r♥st❡✐♥ P♦❧②♥♦♠✐❛❧s✱ ❙♦❢t✇❛r❡ ●❡♦❣❡❜r❛✳

❙✉♠ár✐♦

❆❜str❛❝t ✈✐✐

✶ ❘❡✈✐sã♦ ♠❛t❡♠át✐❝❛ ✷

✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❈✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ P♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ■♥t❡r♣♦❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✶ ■♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✷ ■♥t❡r♣♦❧❛❞♦r ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷ ❈✉r✈❛s ❞❡ ❇é③✐❡r ✶✺

✷✳✶ ❯♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r ❡ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❉❡✜♥✐çã♦ ❞❛s ❈✉r✈❛s ❞❡

❇é③✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❈✉r✈❛s ❞❡ ❇é③✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳✷ ❆❧❣♦r✐t♠♦ ❞❡ ❈❛st❡❧❥❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✸ ❈á❧❝✉❧♦ ❞❛ ❉❡r✐✈❛❞❛ ❡♠ r❡❧❛çã♦ ❛ ❈✉r✈❛ ❞❡ ❇é③✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✹ ❈á❧❝✉❧♦ ❞❛ ✐♥t❡❣r❛❧ ❞❛ ❈✉r✈❛ ❞❡ ❇é③✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✸ ❊①❡♠♣❧♦s ✷✽

✸✳✶ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✳✶ ❊①❡♠♣❧♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✳✷ ❊①❡♠♣❧♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✳✸ ❊①❡♠♣❧♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✳✹ ❊①❡♠♣❧♦ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✳✺ ❊①❡♠♣❧♦ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✶✳✻ ❊①❡♠♣❧♦ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✶✳✼ ❊①❡♠♣❧♦ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹ ❘❡s✉❧t❛❞♦s ❡ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✸✽ ✹✳✶ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✷ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

❆ ❈♦♥str✉✐♥❞♦ ❝✉r✈❛s ❞❡ ❇é③✐❡r ✉s❛♥❞♦ ♦

●❡♦❣❡❜r❛ ❡ ❝♦♠ ♦ ▲❆_{❚❊❳ ✹✵}

❆✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ❆✳✷ ❈♦♥str✉çã♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ❆✳✸ ❯s❛♥❞♦ ♦ P❙❚r✐❝❦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ❆✳✹ ❯s❛♥❞♦ ♦ ❛♠❜✐❡♥t❡ ✜❣✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ❆✳✺ ❯s❛♥❞♦ ♦ P●❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ❆✳✻ ❯s❛♥❞♦ ♦ ❚❊❳❞r❛✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ❆✳✼ ❯s❛♥❞♦ ♦ ♣❛❝♦t❡ ❜❡③✶✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

❇ ❚❡♦r❡♠❛ ❞❛ ❆♣r♦①✐♠❛çã♦ ❞❡ ❲❡✐❡rstr❛ss ✺✻

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❘❡t❛ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛P ✔t✈ ❝♦♥str✉í❞❛ ♥♦ ●❡♦❣❡❜r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✶✳✷ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛ x2✔

y2 ₄ ❝♦♥str✉í❞❛ ♥♦ ●❡♦❣❡❜r❛✳ ✳ ✳ ✺

✶✳✸ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛ x 2 cos❼2_πt➁ ❡ _y 2 sen❼2_πt➁✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✹ ❊❧✐♣s❡ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛x Rcos❼2_πt➁ ❡ _{y r}sen❼2_πt➁✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✺ ❆ tr❛❥❡tór✐❛ ❡❧í♣t✐❝❛ ♦✉ ❝✐r❝✉❧❛r ❞❡s❝r✐t❛ ♣♦r ✉♠ ♣♦♥t♦ ✜①♦ ♥❛ ❜❛rr❛ ❞❡♣❡♥❞❡♥❞♦ ❞❛ r❛③ã♦ ❡♥tr❡ ♦s ♣♦♥t♦s ❆ ❡ ❇✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✻ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✼ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✽ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✾ ❯s❛♥❞♦ ♦ ●❡♦❣❡❜r❛ ♣❛r❛ ✈❡r✐✜❝❛r ♦ ♣♦❧✐♥ô♠✐♦ ✐♥t❡r♣♦❧❛❞♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✵ ■♠❛❣❡♠ ❞❡ ✉♠ ♣❛t♦ ✉s❛❞❛ ♥❛ ✐♥t❡r♣♦❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✶ P♦♥t♦s ✐♥t❡r♣♦❧❛❞♦s s♦❜r❡ ❛ ✐♠❛❣❡♠ ❞♦ ♣❛t♦ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ▲❛✲

❣r❛♥❣❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✷ ❊①❡♠♣❧♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r ✉s❛♥❞♦ ♦ s♦❢t✇❛r❡ ✇①▼❛①✐♠❛✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶ Pr♦❥❡t♦ ❞❡ ✈❡í❝✉❧♦ ✉s❛♥❞♦ ❝✉r✈❛s ❞❡ ❇é③✐❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❋♦r♠❛s ❣❡r❛❞❛s ♣♦r ❛❧❣✉♥s s♦❢t✇❛r❡ ✉s❛♥❞♦ ❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❯s❛♥❞♦ ❝✉r✈❛s ❡♠ ❝✐r✉r❣✐❛s ♣❧ást✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✹ ■♥✈ó❧✉❝r♦s ❝♦♥✈❡①♦ ♥❛s ❝✉r✈❛s ❈✉r✈❛s ❞❡ ❇é③✐❡r ✈❛r✐❛♥❞♦ ♦s ♣♦♥t♦s

❞❡ ❝♦♥tr♦❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✺ P♦♥t♦s ❞❛ ❝✉r✈❛ ❝♦♠ três P♦♥t♦s ❞❡ ❝♦♥tr♦❧❡ ❣❡r❛❞❛ s❡❣✉♥❞♦ ♦ ❛❧❣♦✲

r✐t♠♦ ❞❡ ❈❛st❡❧❥❛✉ ❡ ❤❛❜✐❧✐t❛♥❞♦ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✷✺ ✷✳✻ ❈✉r✈❛s ❞❡ ❇❡③✐❡r ❝♦♠ três ✉s❛♥❞♦ ♦ r❛str♦ ♥♦ ❣❡♦❣❡❜r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✼ ❈✉r✈❛s ❞❡ ❇é③✐❡r ❝♦♠ q✉❛tr♦ ♣♦♥t♦s ❞❡ ❝♦♥tr♦❧❡ ✉s❛♥❞♦ ♦ r❛str♦✳ ✳ ✳ ✳ ✷✻ ✷✳✽ P♦♥t♦s ❣❡r❛❞♦s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❉❡ ❈❛st❡❧❥❛✉ ❝♦♠ t 0_.5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✷✳✾ ❉❡r✐✈❛❞❛ ❞❛ ❈✉r✈❛ ❣❡r❛❞❛ ♣❡❧♦s ♣♦♥t♦s ❼0_,0➁_,❼0_,2➁_,❼8_,2➁ ❡ ❼4_,0➁ ✳ ✳ ✳ ✷✼

✸✳✶ P♦❧í❣♦♥♦ ❞❡ ❝♦♥tr♦❧❡ ❢♦r♠❛❞♦ ♣♦r ✶✼ ♣♦♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ P♦❧í❣♦♥♦ ❞❡ ❝♦♥tr♦❧❡ ❢♦r♠❛❞♦ ♣♦r ✶✼ ♣♦♥t♦s ✈❛r✐❛♥❞♦ ♦ t ❝♦♠ ✐♥❝r❡✲

♠❡♥t♦s ❞❡0_.2_. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸✳✸ P♦í❣♦♥♦ ❞❡ ❝♦♥tr♦❧❡ ❢♦r♠❛❞♦ ♣♦r ✶✼ ♣♦♥t♦st 0_.5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✹ ❈✉r✈❛ ❞❡ ❇é③✐❡r ✉s❛♥❞♦ três ❝♦♦r❞❡♥❛❞❛s ♥♦ ❡s♣❛ç♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✺ ●❡r❛♥❞♦ ♦ P♦❧í❣♦♥♦ ❞❡ ❝♦♥tr♦❧❡ ❝♦♠ ♦s ❞♦✐s ♣❡♥tá❣♦♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✻ ■♠♣❧❡♠❡♥t❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❈❛st❡❧❥❛✉ ❡ ✈❛r✐❛♥❞♦t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✼ ❯s❛♥❞♦ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ♣❛r❛ ❣❡r❛r ❛ ❈✉r✈❛ ❞❡ ❇é③✐❡r ❡♠ ❊s♣✐r❛❧✳ ✳ ✸✷ ✸✳✽ ❊s♣✐r❛❧ ❢♦r♠❛❞❛ ♣♦r ❛❧❣✉♥s ♣♦♥t♦s ❞❡ ❝♦♥tr♦❧❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡

tr✐â♥❣✉❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✾ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❈❛st❡❥❛✉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✶✵ ❈✉r✈❛ ❣❡r❛❞❛ ♣❡❧♦s ♣♦♥t♦s ❼✏1_,0➁✱ ❼0_,1➁✱ ❼1_,1➁✱ ❼0_,2➁✱ ❼0_,1➁✱ ❼✏1_,1➁✱

❼0_,2➁✱ ❼0_,1➁ ❡ ❼1_,0➁✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✸✳✶✶ ❊s❝♦❧❤❛ ❞♦s ♣♦♥t♦s s♦❜r❡ ❛ ✐♠❛❣❡♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✶✷ ❱❛r✐❛♥❞♦ ♦ ♣❛râ♠❡tr♦ t ♦❜s❡r✈❛♠♦s ♦s ♣♦♥t♦s ✐♥t❡r♠❡❞✐ár✐♦s ❣❡r❛❞♦s✳ ✸✺

✸✳✶✸ ❈✉r✈❛ ❣❡r❛❞❛ ✉s❛♥❞♦ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✶✹ ❈✉r✈❛ ❣❡r❛❞❛ ♣❛r❛ ♦s ♣♦♥t♦s ❼1_,0➁_,❼✏1_,1➁_,❼1_,1➁ ❡ ❼✏1_,0➁ ❡♠ _t 0_.5

❛ ❞❡r✐✈❛❞❛ ♥ã♦ ❡①✐st❡ ♣❛r❛ ❡st❡s ✈❛❧♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✶✺ ❈✉r✈❛ ❣❡r❛❞❛ ♣❛r❛ ♦s ♣♦♥t♦s ❼0_,1➁_,❼0_,0➁_,❼1_,1➁ ❡ ❼✏1_,0➁ ❡♠_t 0_.5 ❛

❞❡r✐✈❛❞❛ ♥ã♦ ❡①✐st❡ ♣❛r❛ ❡st❡s ✈❛❧♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✶✻ ❈✉r✈❛ ❣❡r❛❞❛ ♣❛r❛ ♦s ♣♦♥t♦s ❼0_,✏1➁_,❼0_,2➁_,❼1_,1➁ ❡ ❼✏1_,0➁ ❡♠ _t 0_.5

❛ ❞❡r✐✈❛❞❛ ♥ã♦ ❡①✐st❡ ♣❛r❛ ❡st❡s ✈❛❧♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ❆✳✶ ❙❡❧❡çã♦ ❞❛ ♦♣çã♦ P♦♥t♦ ♥❛ ❜❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❆✳✷ ❊s❝♦❧❤❛ ❞♦s P♦♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❆✳✸ ❈♦♥str✉çã♦ ❞♦s s❡❣♠❡♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ❆✳✹ ❙❡❣♠❡♥t♦ ❣❡r❛❞♦ ♣❡❧♦s ♣♦♥t♦s ❆ ❡ ❇✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ❆✳✺ ❈♦♥str✉çã♦ ❞♦s s❡❣♠❡♥t♦s ❞♦s ♣♦♥t♦s ❇✱ ❈ ❡ ❉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ❆✳✻ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ❆✳✼ ❉❡✜♥✐♥❞♦ ♦ ✐♥t❡r❧❛❧♦ ❞♦ ♣❛râ♠❡tr♦t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

❆✳✽ ❈♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

❆✳✾ ❈♦♥str✉çã♦ ❞♦ ♣♦♥t♦ s♦❜r❡ ♦ s❡❣♠❡♥t♦ ✉s❛♥❞♦ ❛ ❝❛✐①❛ ❞❡ ❡♥tr❛❞❛✳ ✳ ✳ ✹✺ ❆✳✶✵ P♦♥t♦ s♦❜r❡ ♦ s❡❣♠❡♥t♦AB✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

❆✳✶✶ P♦♥t♦s ❋ ❡ ● s♦❜r❡ ♦s s❡❣♠❡♥t♦s BC ❡ CD✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

❆✳✶✷ ❈♦♥str✉çã♦ ❞♦ s❡❣♠❡♥t♦ EF✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

❆✳✶✸ ❙❡❣♠❡♥t♦sEF ❡ F G✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

❆✳✶✹ ❈♦♥str✉çã♦ ❞♦s ♣♦♥t♦s s♦❜r❡ ♦s s❡❣♠❡♥t♦s EF ❡F G✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

❆✳✶✺ P♦♥t♦s ❍ ❡ ●✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ❆✳✶✻ ●❡r❛♥❞♦ ♦ ♣♦♥t♦ ❏ ✉s❛♥❞♦ ❛ ❝❛✐①❛ ❞❡ ❡♥tr❛❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ❆✳✶✼ P♦♥t♦ ❏✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ❆✳✶✽ ❙❡❧❡çã♦ ❞♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ♥❛ ❜❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ❆✳✶✾ ●❡r❛♥❞♦ ❛ ❝✉r✈❛ ❞❡ ❇é③✐❡r ❡♠ r❡❧❛çã♦ ❛♦ ♣♦♥t♦ ❏ ❡ ♦ ♣❛râ♠❡tr♦t✳ ✳ ✳ ✺✵

❆✳✷✵ ❱❛r✐❛çã♦ ❞♦ ♣♦♥t♦ ❏ ✉s❛♥❞♦ ♦ ❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ❆✳✷✶ ❈✉r✈❛ ❣❡r❛❞❛ ❝♦♠ ♦ ❝♦♠❛♥❞♦ ♣s❜❡③✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ❆✳✷✷ ❈✉r✈❛ ❣❡r❛❞❛ ❝♦♠ ♦ ❝♦♠❛♥❞♦ ❝❧✈❡❝ ❞♦ ❚❊❳❞r❛✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

❈❛♣ít✉❧♦ ✶

❘❡✈✐sã♦ ♠❛t❡♠át✐❝❛

✶✳✶ ■♥tr♦❞✉çã♦

❆ ❡❞✉❝❛çã♦ ❜r❛s✐❧❡✐r❛ t❡♠ ♣❛ss❛❞♦ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s ♣♦r ✉♠ ♣❡rí♦❞♦ ❞❡ ❣r❛♥✲ ❞❡s tr❛♥s❢♦r♠❛çõ❡s✱ t❛♥t♦ ♥♦ â♠❜✐t♦ ❞❛ ❝♦♥❞✉çã♦ ❞❛s ♣♦❧ít✐❝❛s ❡❞✉❝❛❝✐♦♥❛✐s q✉❛♥t♦ ♥♦ ♣r♦❝❡ss♦ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠✱ ❡st❛❜❡❧❡❝✐❞♦ ❛ ♣❛rt✐r ❞❛ ❡✈♦❧✉çã♦ ✭às ✈❡③❡s ✐♥✲ ✈♦❧✉çã♦✮ ❞❛s t❡♦r✐❛s ❞❡ ❡♥s✐♥♦✳ ❚❛❧ r❡✈♦❧✉çã♦ ✈❡♠ ♣r♦❞✉③✐♥❞♦ ✉♠❛ ♥♦✈❛ ❝✉❧t✉r❛✱ ❝♦❧♦❝❛♥❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦✲t❡❝♥♦❧ó❣✐❝♦ ♥❛ ❝♦♥❞✐çã♦ ❞❡ ✉♠ ❡❢❡t✐✈♦ ❡ ✐♥❞✐s✲ ♣❡♥sá✈❡❧ ❢❛t♦r ❞❡ ♣r♦❞✉çã♦ ♥♦ ♠✉♥❞♦ ❝❛♣✐t❛❧✐st❛✳

❖ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ ❡✐①♦s ♥♦rt❡❛❞♦r❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❛çõ❡s q✉❡ ♣r♦♠♦✈❛♠ ❛ ❡❢❡t✐✈✐❞❛❞❡ ❡ ❛ ❡✜❝á❝✐❛ ❞♦ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❞❡✈❡ s❡r ✉♠❛ ♣rát✐❝❛ ♥❛s ✐♥st✐t✉✐çõ❡s ❞❡ ❡♥s✐♥♦✱ ♦❜❥❡t✐✈❛♥❞♦ ♦ ❛t❡♥❞✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛r❝❡❧❛ ❝❛❞❛ ✈❡③ ♠❛✐♦r ❞❛ ♣♦♣✉❧❛çã♦✱ ❡♠ ❜✉s❝❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❧♦❝❛❧✱ r❡❣✐♦♥❛❧ ❡ ♥❛❝✐♦♥❛❧✳

❖ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ ✈❡♠ ♣❛ss❛♥❞♦ ♣♦r ✉♠❛ ❢❛s❡ ❞❡ r❡✈✐sã♦ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❝♦♥t❡ú❞♦ ♠✐♥✐str❛❞♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ r❡✈✐s✐t❛r ❛❧❣✉♥s t❡♠❛s✱ às ✈❡③❡s s♦✜st✐❝❛❞♦s ♠❛s ❞❡ ❢á❝✐❧ ❡♥t❡♥❞✐♠❡♥t♦✱ ♣♦❞❡ s❡r ✉♠❛ ❡①❝❡❧❡♥t❡ ❡str❛té❣✐❛ ♣❛r❛ ✐♥❝❡♥t✐✈❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦s ❡st✉❞❛♥t❡s✱ ❡ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ♣❡❧❛ ♠❛t❡♠át✐❝❛✳

◆❡st❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞♦ ❝✉rs♦ ❞❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧ ❢❛③❡♠♦s ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ s♦❜r❡ ❛ ❤✐stór✐❛ ❞❛ ❈✉r✈❛ ❞❡ ❇é③✐❡r ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❡♠ ✐♥ú♠❡r❛s ár❡❛s ❞❛ ❝✐ê♥❝✐❛✳

❯♠❛ r❡✈✐sã♦ ♠❛t❡♠át✐❝❛ ❝♦♠ ❛❧❣✉♥s tó♣✐❝♦s q✉❡ ❢✉♥❞❛♠❡♥t❛♠ ❛ ❝♦♥str✉çã♦ ❞❛s ❈✉r✈❛s ❞❡ ❇é③✐❡r s❡rá r❡❛❧✐③❛❞❛ ♥❡st❡ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✳ ❱❛♠♦s r❡✈✐s✐t❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ♣❛r❛ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❈✉r✈❛ ❞❡ ❇é③✐❡r✳ ❈♦♠❡ç❛♠♦s ❞❡✜♥✐♥❞♦ ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥✳ P♦r ✜♠ ❞❡✜♥✐♠♦s ✐♥✲ t❡r♣♦❧❛çã♦✱ ❡ ✐♥tr♦❞✉③✐♠♦s ❡①❡♠♣❧♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ▲❛❣r❛♥❣❡ ❡ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r✳

◆♦ ❡st✉❞♦ ❞❛ ❝✉r✈❛ ❞❡ ❇é③✐❡r ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ❢♦r❛♠ ✐♠♣❧❡♠❡♥t❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛ r❡✈✐sã♦ ♠❛t❡♠át✐❝❛ ♣❛r❛ ✐❧✉str❛r ♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❝✉r✈❛ ❞❡ ❇é③✐❡r ❡ s✉❛ ❢✉♥❞❛♠❡♥t❛çã♦ ♠❛t❡♠át✐❝❛✱ ❝♦♠ ❛❧❣✉♥s

✶✳✷✳ ❈❯❘❱❆❙ P❆❘❆▼❊❚❘■❩❆❉❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❡①❡♠♣❧♦s ✐♠♣❧❡♠❡♥t❛❞♦s ♥♦ ●❡♦❣❡❜r❛✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❛❜♦r❞❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣r♦♣♦st♦s ❡ ♦✉tr♦s r❡t✐r❛❞♦s ❞♦ ❧✐✈r♦ ❞❡ r❡❢❡rê♥❝✐❛ ❋❛r✐♥ ❬✷❪✳

❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❢♦r❛♠ ❞✐s❝✉t✐❞♦s ♥❛s ❝♦♥❝❧✉sõ❡s ❡ ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s✱ ♥♦ q✉❛rt♦ ❝❛♣ít✉❧♦✳

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r✲ ❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✳

✶✳✷ ❈✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s

❊♠ ❛♣❧✐❝❛çõ❡s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡①✐st❡♠ ❞✐✈❡rs❛s ♠❛♥❡✐r❛s ❞❡ r❡♣r❡s❡♥t❛r ✉♠ ♦❜✲ ❥❡t♦ ❣rá✜❝♦✱ ♠❛s ❞❡st❛❝❛♠♦s ❞✉❛s q✉❡ sã♦ ❝♦♠✉♠❡♥t❡ ✉s❛❞❛s ❛ ❞❡s❝r✐çã♦ ♣❛r❛♠é✲ tr✐❝❛ ♦✉ ✐♠♣❧í❝✐t❛✳

◆❛ ❞❡s❝r✐çã♦ ♣❛r❛♠étr✐❝❛ ❛ ❝✉r✈❛ é ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦f ✂I ❵_{R R}2

, f❼t➁

❼x❼t➁, y❼t➁➁✱ ♦♥❞❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ♥❛ r❡t❛✳ ◆❛ ❢ís✐❝❛ ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s

♣♦ss✉❡♠ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ✐♥t❡r❡ss❛♥t❡✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✈❛r✐á✈❡❧ t❃I ❝♦♠♦ s❡♥❞♦ ♦

t❡♠♣♦✱ ❛ ❝✉r✈❛ ❞❛❞❛ ♣♦r f❼t➁ r❡♣r❡s❡♥t❛ ✉♠❛ ♣❛rtí❝✉❧❛ ♥♦ ♣❧❛♥♦ ❬✶✵❪✳

❯♠❛ r❡t❛✱ ♣♦r ❡①❡♠♣❧♦✱ ✐♠♣❧✐❝✐t❛♠❡♥t❡ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❡q✉❛çã♦r✂ax✔by✔c 0✱ ❡st❛ r❡t❛ _r ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♥♦ ♣❧❛♥♦ ♣♦r ✉♠ ♣♦♥t♦ _P ❃_r ❬✾❪✱ ❡ ♦ ✈❡t♦r ✈

q✉❡ ❞❡t❡r♠✐♥❛ ❛ s✉❛ ❞✐r❡çã♦✱ ✜❣✉r❛ ✶✳✶ ✳ ❆ ❢♦r♠❛ ✈❡t♦r✐❛❧ ❞❛ ❡q✉❛çã♦ ♣❛r❛♠étr✐❝❛ ❞❛ r❡t❛ é ❞❛❞❛ ♣♦r

f❼t➁ P ✔t✈, t❃_R.

❙❡ P ❼x0, y0➁,✈ ❼v1, v2➁ ❡ f❼t➁ ❼x❼t➁, y❼t➁➁✱ ❡♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦

♣❛r❛♠étr✐❝❛ ❞❡ r ❡♠ ❝♦♦r❞❡♥❛❞❛s✿

❼x❼t➁, y❼t➁➁ ❼x0, y0➁ ✔t❼v1, v2➁

➱ ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉s❛♥❞♦ ❞✉❛s ♠❛✲ ♥❡✐r❛s ❞✐st✐♥t❛s✱ ♣♦r ❡①❡♠♣❧♦✳

x2

✔y2 ₄

♦✉

x 2 sen❼2_πt➁

y 2 cos❼2_πt➁

✳

✶✳✷✳ ❈❯❘❱❆❙ P❆❘❆▼❊❚❘■❩❆❉❆❙ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✶✿ ❘❡t❛ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛P ✔t✈ ❝♦♥str✉í❞❛ ♥♦ ●❡♦❣❡❜r❛✳

❆ ♣r✐♠❡✐r❛ ❢♦r♠❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡✱ ❛♣❡♥❛s ♦s ✈❛❧♦r❡s ♣♦s✐t✐✈♦s ❞❡ ② sã♦ ❣❡r❛❞♦s✱ ♣♦✐s ❛s ❝♦♦r❞❡♥❛❞❛s sã♦ ❞❛❞❛s ♣♦r ❼x,➸4✏_x2➁✱ ✜❣✉r❛ ✶✳✷ ✳ ❆ ❢♦r♠❛ ♣❛r❛✲

♠étr✐❝❛ ❣❡r❛ t♦❞♦s ♦s ♣♦♥t♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❞❡ ♠❛♥❡✐r❛ s✉❛✈❡ ♣❛r❛ ❛ ♠áq✉✐♥❛✱ ✜❣✉r❛ ✶✳✸ ✳

◆❛ ❡q✉❛çã♦ ❞❛ ❡❧✐♣s❡ ♣♦❞❡♠♦s s✐♠♣❧✐✜❝❛r ❜❛st❛♥t❡ ❛ ❡①♣r❡ssã♦✿

x2

R2 ✔

y2

r2 1

♦♥❞❡ R é ♦ r❛✐♦ ♠❛✐♦r ❡ r é ♦ r❛✐♦ ♠❡♥♦r✱ ✜❣✉r❛ ✶✳✹ ✳ ◆❛ ❢♦r♠❛ ♣❛r❛♠étr✐❝❛ t❡♠♦s✿

x Rcos❼2_πt➁

y rsen❼2_πt➁

❯♠❛ s✐t✉❛çã♦ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✶✳✺✱ ♦♥❞❡ ✉♠❛ ❜❛rr❛ t❡♠ s✉❛s ❡①tr❡♠✐❞❛❞❡s ♥♦s ❡✐①♦s ❳ ❡ ❨✱ ♣♦❞❡♥❞♦ ❞❡s❧✐③❛r ❛♣❡♥❛s ❛♦ ❧♦♥❣♦ ❞♦s ❡✐①♦s✱ ✉♠ ♣♦♥t♦ ✜①♦ ❞❡s❝r❡✈❡ ✉♠❛ tr❛❥❡tór✐❛ q✉❡ ✈❛r✐❛ ❡♥tr❡ ❊❧✐♣s❡s ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✈❛r✐❛♥❞♦ ❛ ♣♦s✐çã♦ ❞♦ ♣♦♥t♦✱ ✜❣✉r❛ ✶✳✺✳

✶✳✸✳ P❖▲■◆Ô▼■❖❙ ❉❊ ❇❊❘◆❙❚❊■◆❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✷✿ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛ x2✔

y2 ₄❝♦♥str✉í❞❛ ♥♦ ●❡♦❣❡❜r❛✳

✶✳✸ P♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥

❇❛s✐❝❛♠❡♥t❡✱ ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❇❡r♥st❡✐♥ é ❝❛♣❛③ ❞❡ ❛♣r♦①✐♠❛r ✉♠❛ ❢✉♥çã♦ ❝♦♥✲ tí♥✉❛f ❞❡✜♥✐❞❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦ ❬❛✱ ❜❪✱ ❜❡♠ ❝♦♠♦ ❛❥✉st❛r ❝✉r✈❛s

♦✉ s✉♣❡r❢í❝✐❡s✳

❊st❡s ♣♦❧✐♥ô♠✐♦s t❛♠❜é♠ ♣❡r♠✐t❡♠ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳ ▼❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ r❡♣r♦❞✉③ ❡①❛t❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳ ■st♦ é✱ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♥ã♦ ♣❛ss❛ ❛tr❛✈és ❞♦s ❞❛❞♦s✱ ♠❛s ♣❡rt♦ ❞❡❧❡s✳ ❈♦♠♦ ❢❡rr❛✲ ♠❡♥t❛ ❞❡ ❛❥✉st❡✱ ❡st❡s ♣♦❧✐♥ô♠✐♦s tê♠ ✈ár✐❛s ✈❛♥t❛❣❡♥s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❞❡s❡♥❤♦ ❛ss✐st✐❞♦ ♣♦r ❝♦♠♣✉t❛❞♦r✭❈❆❉✮ ❬✺❪✳

❍á ❝✐r❝✉♥stâ♥❝✐❛s ♦✉ ❡①♣❡r✐ê♥❝✐❛s q✉❡ ❞❡✈❡♠ ❡♥❝♦♥tr❛r ♦✉ ♠❛♥t❡r ✉♠❛ ❜♦❛ ❛♣r♦✲ ①✐♠❛çã♦ ❡♠ ✈❡③ ❞❡ ♣r♦❝✉r❛r ✐♥t❡r♣♦❧❛çã♦ ❞♦s ❞❛❞♦s✳❖s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ sã♦ ❝❤❛♠❛❞❛s ❛ss✐♠ ♣♦rq✉❡ sã♦ ❛ ❜❛s❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ♣r♦✈❛ ❞❡ ❇❡r♥st❡✐♥ q✉❡ ❝♦♥❞✉✲ ③✐✉ ❛♦ t❡♦r❡♠❛ ❞❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❲❡✐❡rstr❛ss ✭✈❡r ❛♣ê♥❞✐❝❡ ❇✮ ✳ ❊st❛ ❞❡♠♦♥str❛çã♦ t❡♠ ✉♠ ♣r♦❢✉♥❞♦ ✐♠♣❛❝t♦ ❡♠ ♠✉✐t❛s ár❡❛s✳ ❖s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ❡stã♦ ❧✐❣❛✲ ❞♦s ❝♦♠ ❛ t❡♦r✐❛ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡✱ ❝♦♠ ♣r♦❜❧❡♠❛s ❞❡ ♠♦♠❡♥t♦s ❡ ❝♦♠ ❛ t❡♦r✐❛ ❞❡ s♦♠❛s ❞❡ sér✐❡s ❞✐✈❡r❣❡♥t❡s✳

❖ ❝á❧❝✉❧♦ ❞❛ ❡①♣❛♥sã♦ ❜✐♥♦♠✐❛❧

✶✳✸✳ P❖▲■◆Ô▼■❖❙ ❉❊ ❇❊❘◆❙❚❊■◆❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✸✿ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛ x 2 cos❼2_πt➁ ❡ _y 2 sen❼2_πt➁✳

1 ❼_t✔ ❼1✏_t➁➁n

n

◗

i 0

➀ n_i ➅.❼1✏_t➁n✏i_.ti

❝♦♠ t ❃ 0_,1✆✱ _i 0_{, ..., n}❀ ✐st♦ ♥♦s ♣❡r♠✐t❡ ♦❜t❡r ♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ❇❡r♥st❡✐♥ ❡♠

❞✐❢❡r❡♥t❡s ❣r❛✉s n✳

B_in❼t➁ ➀ n

i ➅.❼1✏t➁

n✏i_.ti

❝♦♠ t❃ 0_,1✆✱_i 0_{, ..., n}

❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✶✱ ✜❣✉r❛ ✶✳✻✿

B1

0❼t➁ ➀

1

0 ➅.❼1✏t➁

1✏0

.t0 ₁

✏t

B1

1❼t➁ ➀

1

1 ➅.❼1✏t➁

1✏1

.t1

t

❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✷✳ ✜❣✉r❛ ✶✳✼✿

✶✳✸✳ P❖▲■◆Ô▼■❖❙ ❉❊ ❇❊❘◆❙❚❊■◆❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✹✿ ❊❧✐♣s❡ ❣❡r❛❞❛ ❞❛ ❢♦r♠❛ x Rcos❼2_πt➁ ❡ _{y r}sen❼2_πt➁✳

B02❼t➁ ➀

2

0 ➅.❼1✏t➁

2

.t0 ❼1✏_t➁2 B12❼t➁ ➀

2

1 ➅.❼1✏t➁

2✏1

.t1 2❼1✏_t➁_t B22❼t➁ ➀

2

2 ➅.❼1✏t➁

2✏2

.t2 t2

❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✸✱ ✜❣✉r❛ ✶✳✽✿

B3

0❼t➁ ➀

3

0 ➅.❼1✏t➁

3

.t0

❼1✏_t➁3

B3

1❼t➁ ➀

2

1 ➅.❼1✏t➁

3✏1

.t1 ₃

❼1✏_t➁2

t

B2

3❼t➁ ➀

3

2 ➅.❼1✏t➁

3✏2

.t2 ₃

❼1✏_t➁_t2

✶✳✸✳ P❖▲■◆Ô▼■❖❙ ❉❊ ❇❊❘◆❙❚❊■◆❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✺✿ ❆ tr❛❥❡tór✐❛ ❡❧í♣t✐❝❛ ♦✉ ❝✐r❝✉❧❛r ❞❡s❝r✐t❛ ♣♦r ✉♠ ♣♦♥t♦ ✜①♦ ♥❛ ❜❛rr❛ ❞❡♣❡♥❞❡♥❞♦ ❞❛ r❛③ã♦ ❡♥tr❡ ♦s ♣♦♥t♦s ❆ ❡ ❇✳

❋✐❣✉r❛ ✶✳✻✿ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✶✳

B33❼t➁ ➀

3

3 ➅.❼1✏t➁

3✏3

.t3 t3

✶✳✸✳ P❖▲■◆Ô▼■❖❙ ❉❊ ❇❊❘◆❙❚❊■◆❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✼✿ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✷✳

❋✐❣✉r❛ ✶✳✽✿ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ ✸✳

✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥

P♦❞❡♠♦s ❝✐t❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ❬✺❪✿

❼ ❙ã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❞❡ ❢❛t♦✱ ❞✐✈✐❞✐♥❞♦ Pn

i 0citi❼1✏t➁n✏i 0 ♣♦r

❼1✏_t➁n ❡ ✉s❛♥❞♦ _s t

1✏_t ♦❜t❡♠♦s P

n

i 0cisi 0 ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ c0 c1 c2

... cn 0✳

❼ ❙ã♦ ❙✐♠étr✐❝♦s✱

B_in❼t➁ B_nn_✏i❼1✏_t➁ ✟ ➀ n i ➅.t

i_.❼1✏t➁n✏i _➀ n

n✏i ➅.❼1✏t➁

n✏i_.ti

✳

❈♦♠♦ ❛ ❡①♣r❡ssã♦ 1 ❼_t✔ ❼1✏_t➁➁n ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

1 ❼❼1✏_t➁ ✔_t➁n✱ ❡①♣❧✐❝❛ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ♥❛ ❡①♣❛♥sã♦ ❜✐♥♦♠✐❛❧✳

❼ ❆s r❛í③❡s sã♦ ✵ ❡ ✶✱

✶✳✸✳ P❖▲■◆Ô▼■❖❙ ❉❊ ❇❊❘◆❙❚❊■◆❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

B_in❼0➁ _Bnn_✏i❼1➁ ➐ 1 0 ♣❛r❛

i 0

i❆0 →

❱❡r✐✜q✉❡ q✉❡ ♣❛r❛ t 0 ♦s t❡r♠♦s _t ❛♣r❡s❡♥t❛rã♦ ✈❛❧♦r❡s ♥✉❧♦s✳ P❛r❛ _t 1 ♦s

t❡r♠♦s 1✏_t ❛♣r❡s❡♥t❛rã♦ ✈❛❧♦r❡s ♥✉❧♦s✱ ✐st♦ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r♦✈❛❞♦

♥♦s ❣rá✜❝♦s ❛♣r❡s❡♥t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✳

❼ ❆ s♦♠❛ ❞❡ t♦❞❛s ♣❛rt✐çõ❡s ❞♦ ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ♣♦ss✉✐ ✈❛❧♦r ✶✱ Pn

i 0B_in❼t➁ 1, ♣❛r❛ t♦❞♦t❃R✳ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ é ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛❞❛ ♣❛r❛

❡①♣r❡ssã♦✿

1 ❼_t✔ ❼1✏_t➁➁n

n

◗

i 0

➀ n_i ➅.❼1✏_t➁n✏i_.ti

▲♦❣♦ ❛♦ s♦♠❛r ❛ ❡①♣r❡ssã♦ t❡r❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ♦ ✈❛❧♦r1✳

❼ ❙ã♦ ♣♦s✐t✐✈♦s ♣❛r❛ ❼0_,1➁✱

Pn

i 0B_in❼t➁ ❆0✱ ♣❛r❛ t♦❞♦ t❃ ❼0,1➁

❈♦♠♦t❈0✱ ❼1✏_t➁ ❈0❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛❧ sã♦ ♣♦s✐t✐✈♦s✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r

q✉❡ Pn

i 0B_in❼t➁ ❆0✳

❼ ❙❛t✐s❢❛③ ❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛

B_in✔1❼t➁ tB_in_✏1❼t➁ ✔ ❼1✏t➁B

n i ❼t➁,

♦♥❞❡ Bn

✏1✁B

n

n✔1✁0 ❡ B 0 0 ✁1

❊st❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ✈❡♠ ❞❛ ✐❞❡♥t✐❞❛❞❡✿

➀ n✔_i 1 ➅ ➀ _i✏n₁ ➅ ✔ ➀ n_i ➅

❖s ♣♦❧✐♥ô♠✐♦s ❇❡r♥st❡✐♥✱ é ❣❡r❛❞♦s ♣❡❧❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ✱ ❡ ♣♦❞❡ s❡r ❝❛❧✲ ❝✉❧❛❞♦ ✉t✐❧✐③❛♥❞♦ ✉♠ ♠ét♦❞♦ s❡♠❡❧❤❛♥t❡ ❛♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ❜✐♥♦♠✐❛❧ ♦♥❞❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❡stã♦ ♥♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✳ ❈❛❞❛ ✈❛❧♦r é ❛ ♠é❞✐❛ ♣♦♥❞❡r❛❞❛ ❞♦s ❞♦✐s ✈❛❧♦r❡s ♠❛✐s ♣ró①✐♠♦s ❛❝✐♠❛ ❞❡❧❡✱ ♦♥❞❡ ♦s ♣❡s♦s sã♦ t ❡ 1✏_t✳

❖ ♣♦❧✐♥ô♠✐♦ ❞❡ ❇❡r♥st❡✐♥ s❡❣✉❡ ♦ ❡sq✉❡♠❛ ❞❡ ❞✐str✐❜✉✐çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❛❜❛✐①♦✿

B0 0 B1 0 B 1 1 B2 0 B 2 1 B 2 2 ✞ ✝ ✝ ✟ Bn 0 B n 1 B n

2 . . . Bnn

✶✳✹✳ ■◆❚❊❘P❖▲❆➬➹❖ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

P♦❞❡♠♦s ✈❡r ❛ ❡①♣❛♥sã♦ ❛té ♦ q✉✐♥t♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦✿

1 t✝ ✞_❼1✏_t➁

t ❼1✏_t➁

t✝ ✞_❼1✏_t➁ t✝ ✞❼1✏_t➁

t2 ✷t✭✶✲t✮ ❼₁✏

t➁2

t✝ ✞_❼1✏_t➁ t✝ ✞❼1✏_t➁ t✝ ✞❼1✏_t➁

t 3_t2❼₁✏

t➁ 3_t❼1✏_t➁2 ❼₁✏

t➁3

t✝ ✞_❼1✏_t➁ t✝ ✞❼1✏_t➁ t✝ ✞❼1✏_t➁ t✝ ✞❼1✏_t➁

t4 ₄

t3❼₁✏

t➁ 6_t2❼₁✏

t➁2 ₄

t❼1✏_t➁3 ❼₁✏

t➁4

✶✳✹ ■♥t❡r♣♦❧❛çã♦

❊♥tr❡ ♦s ♣r♦❜❧❡♠❛s q✉❡ ♥♦s ❞❡♣❛r❛♠♦s ♥♦ ❞✐❛ ❛ ❞✐❛ ❝♦♥s✐st❡ ❡♠ ❞❛❞❛ ✉♠❛ t❛❜❡❧❛ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❞❡ f❼x➁ ♣❛r❛ ✉♠ ❞❛❞♦ x✳ ❆ ❛♣r♦①✐♠❛çã♦ ❞❡ ❢✉♥çõ❡s

♣♦r ♣♦❧✐♥ô♠✐♦s é ✉♠❛ ❞❛s ✐❞❡✐❛s ♠❛✐s ❛♥t✐❣❛s ❞❛ ❛♥á❧✐s❡ ♥✉♠ér✐❝❛✱ ❡ ❛✐♥❞❛ ✉♠❛ ❞❛s ♠❛✐s ✉s❛❞❛s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ s✉♣r❛❝✐t❛❞♦✳ ➱ ❜❛st❛♥t❡ ❢á❝✐❧ ❡♥t❡♥❞❡r ♣♦r q✉❡ r❛③ã♦ ✐ss♦ ❛❝♦♥t❡❝❡✳ ❖s ♣♦❧✐♥ô♠✐♦s sã♦ ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣✉tá✈❡✐s✱ s✉❛s ❞❡r✐✈❛❞❛s ❡ ✐♥t❡❣r❛✐s sã♦ ♥♦✈❛♠❡♥t❡ ♣♦❧✐♥ô♠✐♦s ❡ s✉❛s r❛í③❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❝♦♠ r❡❧❛t✐✈❛ ❢❛❝✐❧✐❞❛❞❡ ❬✶❪✳

❆ s✐♠♣❧✐❝✐❞❛❞❡ ❞♦s ♣♦❧✐♥ô♠✐♦s ♣❡r♠✐t❡ q✉❡ ❛ ❛♣r♦①✐♠❛çã♦ ♣♦❧✐♥♦♠✐❛❧ s❡❥❛ ♦❜t✐❞❛ ❞❡ ✈ár✐♦s ♠♦❞♦s✱ ❡♥tr❡ ♦s q✉❛✐s ♣♦❞❡♠♦s ❝✐t❛r✿ ■♥t❡r♣♦❧❛çã♦✱ ▼ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s✱ ▼✐♥✐✲▼❛①✱ ❡t❝✱ ♣♦rt❛♥t♦ é ✈❛♥t❛❥♦s♦ s✉❜st✐t✉✐r ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧✐❝❛❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ q✉❡ ❛ r❡♣r❡s❡♥t❡✳ ❆❧é♠ ❞✐ss♦ t❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❲❡✐rstr❛ss q✉❡ ❛✜r♠❛ q✉❡✿ t♦❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♣♦❞❡ s❡r ❛r❜✐tr❛r✐❛♠❡♥t❡ ❛♣r♦①✐♠❛❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦✳

❱❡r❡♠♦s ❛q✉✐ ❝♦♠♦ ❛♣r♦①✐♠❛r ✉♠❛ ❢✉♥çã♦ ✉s❛♥❞♦ ▼ét♦❞♦s ❞❡ ■♥t❡r♣♦❧❛çã♦ P♦✲ ❧✐♥♦♠✐❛❧ ❞❡ ▲❛❣r❛♥❣❡ ❡ ▲✐♥❡❛r✳ ❊st❡s ♠ét♦❞♦s sã♦ ✉s❛❞♦s ❝♦♠♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f❼x➁ ❡♠ s✐t✉❛çõ❡s ❡♠ q✉❡ ❛ ❢✉♥çã♦ é ❡①tr❡♠❛♠❡♥t❡ ❝♦♠♣❧✐❝❛❞❛ ❡

❞❡ ❞✐❢í❝✐❧ ♠❛♥❡❥♦✳ ❊♥tã♦✱ às ✈❡③❡s✱ é ✐♥t❡r❡ss❛♥t❡ s❛❝r✐✜❝❛r ❛ ♣r❡❝✐sã♦ ❡♠ ❜❡♥❡❢í❝✐♦ ❞❛ s✐♠♣❧✐✜❝❛çã♦ ❞♦s ❝á❧❝✉❧♦s✳

✶✳✹✳✶ ■♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡

❉❛❞♦s ✉♠ ❝♦♥❥✉♥t♦ ❞❡ n✔1 ♣♦♥t♦s ♥♦ ♣❧❛♥♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ _{yi f}❼_xi➁✱ t❛✐s

q✉❡ ❼x1, y1➁,❼x2, y2➁, ...,❼xn✔1, yn✔1➁✱ ❤á ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉n✱P❼x➁✱ ❝✉❥♦ ❣rá✜❝♦ ♣❛ss❛

♣❡❧♦sn✔1♣♦♥t♦s ❞❛❞♦s✱ ♦✉ s❡❥❛✱_P❼_xi➁ _{yi, i} 1_{, ...n}✔1✳ ❆ss✐♠✱ ♣♦r ✉♠ ♣♦♥t♦ ú♥✐❝♦

✶✳✹✳ ■◆❚❊❘P❖▲❆➬➹❖ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

♣❛ss❛ ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧ ✭♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ③❡r♦ ❂ ❝♦♥st❛♥t❡✮✱ ♣♦r ❞♦✐s ♣♦♥t♦s ♣❛ss❛ ✉♠❛ r❡t❛ ✭♣♦❧✐♥ô♠✐♦ ❞❡ ♣r✐♠❡✐r♦ ❣r❛✉✮✱ ♣♦r três ♣♦♥t♦s ♣❛ss❛ ✉♠❛ ♣❛rá❜♦❧❛ ✭♣♦❧✐♥ô♠✐♦ ❞❡ s❡❣✉♥❞♦ ❣r❛✉✮✱ ❡t❝✳ ❍á ♠✉✐t❛s ♠❛♥❡✐r❛s ♣❡❧❛ q✉❛❧ s❡ ♣♦❞❡ ❡♥❝♦♥tr❛r ♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ nq✉❡ ♣❛ss❛ ♣❡❧♦s n✔1♣♦♥t♦s ❞❛❞♦s✱ ❡♠❜♦r❛ ❡❧❡ s❡❥❛ ú♥✐❝♦✱ ♦✉

s❡❥❛✱ ♦ ♣♦❧✐♥ô♠✐♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ❞✐✈❡rs❛s ❢♦r♠❛s ❡q✉✐✈❛❧❡♥t❡s✳ ❚❛❧✈❡③ ❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s s❡❥❛ ❛ q✉❡ ❝❤❛♠❛♠♦s P♦❧✐♥ô♠✐♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ❬✼❪✳

Pnx n✔1

◗

i 1

pi❼x➁yi,

♦♥❞❡ ❝❛❞❛ pi é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ n ❞❡✜♥✐❞♦ ♣♦r

pi❼x➁ ❼

x✏x1➁❼x✏x2➁...❼x✏x_i✏1➁❼x✏x_i✔1➁...❼x✏x_n✔1➁

❼xi✏x1➁❼xi✏x2➁...❼xi✏xi✏1➁❼xi✏xi✔1➁...❼xi✏xn✔1➁

.

❈♦♠♦ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❛❞❛ ❛ t❛❜❡❧❛ ❡♥❝♦♥tr❛r ♦ ♣♦❧✐♥ô♠✐♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡✿

x f❼x➁

✲✶ ✶✺ ✵ ✽ ✸ ✲✶

❙♦❧✉çã♦✿

P3❼x➁ 15✕ ❼

x_✏0➁❼x_✏3➁

❼✏1✏0➁❼✏1✏3➁✔8✕

❼x_✔1➁❼x_✏3➁

❼0✔1➁❼0✏3➁✏1✕

❼x_✔1➁❼x_✏0➁

❼3✔1➁❼3✏0➁ 15✕

x2_✏3x

4 ✔8✕

❼x2_✏2x_✏3➁

✏3 ✏1✕

x2_✔x

12 .

❆❣r✉♣❛♥❞♦ ♦s t❡r♠♦s s❡♠❡❧❤❛♥t❡s✱ s❡❣✉❡ q✉❡✿

P3❼x➁ x2✏6x✔8✱ ✜❣✉r❛ ✶✳✾ ✳

❱❛♠♦s ✉s❛r ♥❛ ✜❣✉r❛ ✶✳✶✵ ♣❛r❛ ✐♥t❡r♣♦❧❛r ✉t✐❧✐③❛♥❞♦ ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡✱ ✜❣✉r❛ ✶✳✶✶ ✳

✶✳✹✳✷ ■♥t❡r♣♦❧❛❞♦r ▲✐♥❡❛r

❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❢✉♥çã♦ f❼x➁ s❡❥❛ ❝♦♥❤❡❝✐❞❛ ♥♦s ♣♦♥t♦s xi, i 1,2,3, ..., n✔1

♦♥❞❡ f❼xi➁ yi✳ ❚❡♠♦s ❡♥tã♦ n ✐♥t❡r✈❛❧♦s ❡♥tr❡ ♣♦♥t♦s ❝♦♥s❡❝✉t✐✈♦s ❼xi, xi_✔1➁✳ ❆

♠❛♥❡✐r❛ ♠❛✐s s✐♠♣❧❡s ❞❡ ✉♥✐r ❡st❡s ♣♦♥t♦s ♣♦r ✉♠❛ ❧✐♥❤❛ ❝♦♥tí♥✉❛ é tr❛ç❛r ✉♠❛ r❡t❛ ❡♥tr❡ ❝❛❞❛ ❞♦✐s ♣♦♥t♦s ❼xi, yi➁ ❡ ❼xi✔1, y_i✔1➁✳ ❱❛♠♦s ❝❤❛♠❛r ❞❡ g_i❼x➁ ❛ r❡t❛ q✉❡ ✉♥❡

♦s ♣♦♥t♦s ❼xi, yi➁ ❡ ❼xi_✔1, yi_✔1➁✳

❆ ❡q✉❛çã♦ ❞❛ gi❼x➁ é✱ ❡♥tã♦✱

gi❼x➁ yi✔ ❼x✏xi➁

yi✔1✏y_i

xi✔1✏x_i

✁yi✔ ❼x✏xi➁

∆_yi ∆_xi.

✶✳✹✳ ■◆❚❊❘P❖▲❆➬➹❖ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✾✿ ❯s❛♥❞♦ ♦ ●❡♦❣❡❜r❛ ♣❛r❛ ✈❡r✐✜❝❛r ♦ ♣♦❧✐♥ô♠✐♦ ✐♥t❡r♣♦❧❛❞♦r✳

❋✐❣✉r❛ ✶✳✶✵✿ ■♠❛❣❡♠ ❞❡ ✉♠ ♣❛t♦ ✉s❛❞❛ ♥❛ ✐♥t❡r♣♦❧❛çã♦✳

❯s❛r❡♠♦s ♦s ❞♦✐s ♠ét♦❞♦s ❡st✉❞❛❞♦s ♣❛r❛ ❢❛③❡r ✉♠❛ ✐♥t❡r♣♦❧❛çã♦ ❡♠ ✉♠❛ ✜❣✉r❛ ❞✐s♣♦st❛ ❡♠ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ♣❛r❛ r❡❛❧✐③❛r ❛ ✐♥t❡r♣♦❧❛çã♦ ❢♦✐ ✉s❛❞♦ ♦ s♦❢t✇❛r❡ ✇①▼❛①✐♠❛✳ ❖ P♦❧✐♥ô♠✐♦ ♠❛✐s s✐♥✉♦s♦ ❧✐♥❤❛ ✈❡r♠❡❧❤❛ é ♦ ✐♥t❡r♣♦❧❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ ❡♥q✉❛♥t♦ q✉❡ ❛ ❧✐♥❤❛ ❆③✉❧ é ♦ ✐♥t❡r♣♦❧❛❞♦r ❧✐♥❡❛r✱ ✜❣✉r❛ ✷✳✶✷ ❬✶❪ ✳

✶✳✹✳ ■◆❚❊❘P❖▲❆➬➹❖ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❱■❙➹❖ ▼❆❚❊▼➪❚■❈❆

❋✐❣✉r❛ ✶✳✶✶✿ P♦♥t♦s ✐♥t❡r♣♦❧❛❞♦s s♦❜r❡ ❛ ✐♠❛❣❡♠ ❞♦ ♣❛t♦ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ▲❛✲ ❣r❛♥❣❡✳

❋✐❣✉r❛ ✶✳✶✷✿ ❊①❡♠♣❧♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r ✉s❛♥❞♦ ♦ s♦❢t✇❛r❡ ✇①▼❛①✐♠❛✳

❈❛♣ít✉❧♦ ✷

❈✉r✈❛s ❞❡ ❇é③✐❡r

✷✳✶ ❯♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r ❡ s✉❛

✐♠♣♦rtâ♥❝✐❛

❆rq✉✐t❡t♦s ♥❛✈❛✐s r❡♥❛s❝❡♥t✐st❛s ♥❛ ■tá❧✐❛ ❢♦r❛♠ ♦s ♣r✐♠❡✐r♦s ❛ ✉s❛r té❝♥✐❝❛s ❡♥✲ ✈♦❧✈❡♥❞♦ s❡❝çõ❡s ❝ô♥✐❝❛s ❬✷❪✳ ❆♥t❡s ❞✐ss♦✱ ♦s ♥❛✈✐♦s ❡r❛♠ ❢❡✐t♦s ❛ ♠ã♦✱ s❡♠ q✉❛❧q✉❡r ♠❛t❡♠át✐❝❛ ❡♥✈♦❧✈✐❞❛✳ ❊ss❛s té❝♥✐❝❛s ❞❡ ❞❡s✐❣♥ ❢♦r❛♠ r❡✜♥❛❞❛s ❛tr❛✈és ❞♦s sé❝✉❧♦s✱ ❝✉❧♠✐♥❛♥❞♦ ❝♦♠ ♦ ✉s♦ ❞❡ ✈✐❣❛s ❞❡ ♠❛❞❡✐r❛s ❞♦❜r❛❞❛s ♥❛s ❢♦r♠❛s ✐❞❡❛✐s ❛tr❛✈és ❞❡ ♣❡s♦s✱ ✈❡♠ ❞❛í ❛ ♦r✐❣❡♠ ❞♦ ♥♦♠❡ ✧s♣❧✐♥❡s✧✳ ◆♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳✱ ❝♦♠ ♦ ✐♥✈❡♥t♦ ❞♦s ❛✈✐õ❡s ♦ ✉s♦ ❞❛s s❡❝çõ❡s ❝ô♥✐❝❛s ❢♦r❛♠ ♠❛✐s ❞✐❢✉♥❞✐❞❛s ❞❡✈✐❞♦ à ♥❡❝❡ss✐❞❛❞❡ ❞❡ s✉❛✈✐③❛r ❛ tr❛❥❡tór✐❛ ❞♦ ❛r ❛♦ ❧♦♥❣♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛❡r♦♥❛✈❡✳ ❖ ❛✉t♦♠ó✈❡❧ s✉r❣✐✉ ❝♦♠♦ ✉♠ ❞♦s í❝♦♥❡s ❞♦ sé❝✉❧♦ ❳❳✱ ♣r❡❝✐s❛♥❞♦ ❞❡ ♥♦✈♦s ♣r♦❥❡t♦s q✉❡ ✈✐❛❜✐❧✐③❛ss❡ ❛ ♣r♦❞✉çã♦ ❡♠ ♠❛ss❛✱ ❝♦♠ ♦ ✉s♦ ❞❡ ❈❆❉✴❈❆▼✭❞❡s❡♥❤♦ ❛✉①✐❧✐❛❞♦ ♣♦r ❝♦♠♣✉t❛❞♦r ❋❛❜r✐❝❛çã♦ ❛ss✐st✐❞❛ ♣♦r ❝♦♠♣✉t❛❞♦r✮✳

❋✐❣✉r❛ ✷✳✶✿ Pr♦❥❡t♦ ❞❡ ✈❡í❝✉❧♦ ✉s❛♥❞♦ ❝✉r✈❛s ❞❡ ❇é③✐❡r✳

◆♦ ✜♥❛❧ ❞❛ ❞é❝❛❞❛ ❞❡ ✶✾✺✵✱ t♦r♥♦✉✲s❡ ♣♦ssí✈❡❧ ❛ ✉s✐♥❛❣❡♠ ❡♠ ✸❉ ❞❡ ❜❧♦❝♦s ❞❡

✷✳✶✳ ❯▼❆ ❇❘❊❱❊ ❍■❙❚Ó❘■❆ ❉❆❙ ❈❯❘❱❆❙ ❉❊ ❇➱❩■❊❘ ❊ ❙❯❆

■▼P❖❘❚➶◆❈■❆ ❈❆P❮❚❯▲❖ ✷✳ ❈❯❘❱❆❙ ❉❊ ❇➱❩■❊❘

♠❛❞❡✐r❛ ♦✉ ❛ç♦✱ ♥❛ ❝♦♥str✉çã♦ ❞♦s ♠♦❧❞❡s✳ ❊st❡s ♠♦❧❞❡s ♣♦❞❡r✐❛♠✱ ❡♥tã♦✱ s❡r ✉s❛✲ ❞♦s ❝♦♠♦ ♠❛tr✐③❡s ♣❛r❛ ❡st❛♠♣❛❣❡♠ ❞❡ ♣❡ç❛s ❝♦♠♦ ♦ ❝❛♣ô ❡ ❛ ❧❛t❛r✐❛ ❞❡ ❝❛rr♦s✳ ❖ ❣r❛♥❞❡ ♣r♦❜❧❡♠❛ ❞❡st❡s ♠ét♦❞♦s ❞❡ ♣r♦❞✉çã♦ ❢♦✐ ❛ ❢❛❧t❛ ❞❡ s♦❢t✇❛r❡ ❛❞❡q✉❛❞♦ q✉❡ ♣❡r♠✐t✐ss❡ ❛ ❝♦♠✉♥✐❝❛çã♦ ❝♦♠ ❛s ♠áq✉✐♥❛s✳ ❆ ✜♠ r❡❛❧✐③❛r ❞❡s❡♥❤♦s ❡♠ ❝♦♠♣✉t❛✲ ❞♦r❡s✱ ❣❡r❛❧♠❡♥t❡ ❝♦♥t❛♠♦s ❝♦♠ ❧✐♥❤❛s ❡ ❝✉r✈❛s✳ ❆s ❧✐♥❤❛s sã♦ ❢á❝❡✐s ❞❡ ❞❡s❡♥❤❛r ❞❛♥❞♦ ❛♦ ❝♦♠♣✉t❛❞♦r ♦ ♣r✐♠❡✐r♦ ❡ ú❧t✐♠♦ ♣♦♥t♦ t❡♠♦s ✐♥❢♦r♠❛çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❡r❛r ♦s ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s ❛♦s ♣♦♥t♦s ❞❛❞♦s✳ ❆s ❝✉r✈❛s✱ ♥♦ ❡♥t❛♥t♦ ♣♦ss✉❡♠ ✉♠ ❣r❛✉ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ♠❛✐♦r✱ ❡♠❜♦r❛ ♣♦ss❛♠♦s ❞❡s❡♥❤❛r ❝✉r✈❛s ❝♦♠ ❢❛❝✐❧✐❞❛❞❡ à ♠ã♦ ❧✐✈r❡✱ ♥♦s ❝♦♠♣✉t❛❞♦r❡s ❡st❛s ❝✉r✈❛s ♣r❡❝✐s❛♠ s❡r ❞❡✜♥✐❞❛s ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ♠❛t❡♠át✐❝❛✱ ♠❛s ♥❡♠ t♦❞❛s ❛s ❢✉♥çõ❡s q✉❡ ❝♦♥❤❡❝❡♠♦s ♣♦❞❡♠ s❡ ❛❞❡q✉❛r ❛ ♠❛✐✲ ♦r✐❛ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♠♣✉t❛çã♦ ❣rá✜❝❛✳ ❖✉tr♦ ♣r♦❜❧❡♠❛ ❡♥❝♦♥tr❛❞♦ ❡r❛ q✉❡ ❛s ❝✉r✈❛s ♣r❡❝✐s❛✈❛♠ s❡r r❡♣r❡s❡♥t❛❞❛s ♥✉♠❡r✐❝❛♠❡♥t❡✳ ❖ ✉s♦ ❞❡ ❢♦r♠❛s ♣❛r❛♠étr✐❝❛s s✉r❣✐r❛♠ ❝♦♠♦ s♦❧✉çã♦✱ ♣♦✐s ❛té ❡♥tã♦ ♥ã♦ ❡r❛ ♣♦ssí✈❡❧ ❣❡r❛r ❝✉r✈❛s ♠✉✐t♦ ❝♦♠♣❧❡①❛ q✉❡ ♣✉❞❡ss❡♠ s❡r ✐♥t❡r♣r❡t❛❞❛s ♣♦r ♠áq✉✐♥❛s ❞❡ ❝♦♥tr♦❧❡ ♥✉♠ér✐❝♦ ♦s ❈◆❈✬s✳

❆s ❝✉r✈❛s ❞❡ ❇é③✐❡r ❢♦r❛♠ ❜❛st❛♥t❡ ♣r♦♠✐ss♦r❛s ❡ sã♦ ✉s❛❞❛s ♥❛ ♠❛✐♦r✐❛ ❞❛s s✐t✉❛çõ❡s ❡♠ q✉❡ s❡ r❡q✉❡r ♦ ❞❡s❡♥❤♦ ❞❡ ✉♠❛ ❝✉r✈❛✳ ❆ ❤♦♠❡♥❛❣❡♠ ❛♦ ❡♥❣❡♥❤❡✐r♦ ❋r❛♥❝ês P✐❡rr❡ ❇é③✐❡r✭✶✾✶✵✲✶✾✾✾✮✱ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡❧❡ s❡r ♦ ♣r✐♥❝✐♣❛❧ r❡s♣♦♥sá✈❡❧ ♣♦r ❧❡✈á✲❧❛s ❛♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ t♦❞♦s ❛tr❛✈és ❞❡ s❡✉s tr❛❜❛❧❤♦ ❞❡ ❞❡s✐❣♥ ✭tr❛❜❛✲ ❧❤❛♥❞♦ ♣❛r❛ ❛ ❘❡♥❛✉❧t ❞✉r❛♥t❡ ♠✉✐t♦s ❛♥♦s ❡ ♣✉❜❧✐❝❛♥❞♦ s✉❛s ✐♥✈❡st✐❣❛çõ❡s ❡♠ ✶✾✻✷✮✱ ❡♠❜♦r❛ ❡❧❡ ♥ã♦ t❡♥❤❛ s✐❞♦ ♦ ♣r✐♠❡✐r♦✳ P♦❞❡♠♦s s❡r t❡♥t❛❞♦s ❛ ❞✐③❡r q✉❡ ♦ ♠❛t❡♠át✐❝♦ P❛✉❧ ❉❡ ❈❛st❡❧❥❛✉ ❢♦✐ ♦ ♣r✐♠❡✐r♦✱ ✐♥✈❡st✐❣❛♥❞♦ ❛ ♥❛t✉r❡③❛ ❞❡ss❛s ❝✉r✈❛s ❡♠ ✶✾✺✾✱ ❡♥q✉❛♥t♦ tr❛❜❛❧❤❛✈❛ ♥❛ ❈✐tr♦ë♥✱ ♥♦ ❡♥t❛♥t♦✱ ❉❡ ❈❛st❡❧❥❛✉ ♥ã♦ ♣✉❜❧✐❝♦✉ s❡✉ tr❛❜❛❧❤♦✳

❋✐❣✉r❛ ✷✳✷✿ ❋♦r♠❛s ❣❡r❛❞❛s ♣♦r ❛❧❣✉♥s s♦❢t✇❛r❡ ✉s❛♥❞♦ ❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r✳

❆s ❝✉r✈❛s ❇é③✐❡r ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡ ♣♦r ❉❡ ❈❛st❡❧❥❛✉ ❞❛ ❈✐tr♦ë♥ ❡ ♣♦r ❇é③✐❡r ♥❛ ❘❡♥❛✉❧t s❡r✈✐♥❞♦ ♣❛r❛ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ ♣r♦❞✉çã♦ ❞❡ ❛✉t♦♠ó✈❡✐s✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❡st❛s ❡♠♣r❡s❛s ♣♦ss✉íss❡♠ ✉♠ ❞❡s✐❣♥ ♠❛✐s s✉❛✈❡ ❡ ❛❣r❛❞á✈❡❧ ❡♠ s❡✉s ♣r♦❞✉t♦s✳ ■♥❞❡♣❡♥❞❡♥t❡ ❞❡ q✉❡♠ ❝r✐♦✉ ♦ q✉❡ ✐♠♣♦rt❛ é q✉❡ ❡ss❛s ❝✉r✈❛s sã♦ ❜❛st❛♥t❡ út❡✐s✱ ♣♦✐s ❧✐❣❛♥❞♦ ✈ár✐❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r s✉❛ ❝♦♠❜✐♥❛çã♦

✷✳✶✳ ❯▼❆ ❇❘❊❱❊ ❍■❙❚Ó❘■❆ ❉❆❙ ❈❯❘❱❆❙ ❉❊ ❇➱❩■❊❘ ❊ ❙❯❆

■▼P❖❘❚➶◆❈■❆ ❈❆P❮❚❯▲❖ ✷✳ ❈❯❘❱❆❙ ❉❊ ❇➱❩■❊❘

❣❡r❛ ✉♠❛ ú♥✐❝❛ ❝✉r✈❛ s✉❛✈❡✳ ❆♦ ❞❡s❡♥❤❛r ❡♠ s♦❢t✇❛r❡s ❝♦♠♦ ♦ P❤♦t♦s❤♦♣✱ ❋❧❛s❤ ♦✉ ■♥❦s❝❛♣❡✱ ❛s ❝✉r✈❛s ❞❡s❡♥❤❛❞❛s sã♦ s✐♠♣❧❡s♠❡♥t❡ ❈✉r✈❛s ❞❡ ❇é③✐❡r✱ ✜❣✉r❛ ✷✳✷ ✳

❆s ❈✉r✈❛s ❞❡ ❇é③✐❡r s❡ ❞✐❢❡r❡♥❝✐❛♠ ❞❡ ♦✉tr❛s té❝♥✐❝❛s ♥❛ é♣♦❝❛ ♣♦r t❡r❡♠ ✉♠❛ ❣r❛♥❞❡ ✢❡①✐❜✐❧✐❞❛❞❡ ❡ ❛❧t❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛❞❛♣t❛çã♦✳ ❊❧❛s ♣♦❞❡♠ s❡r ❝♦♥str✉í❞❛s ❡♠ q✉❛❧q✉❡r ❝♦♠♣r✐♠❡♥t♦ ❡ ❞✐✈❡rs✐❞❛❞❡ ❞❡ ❢♦r♠❛✱ ✉♥✐♥❞♦ ♦ ♣♦♥t♦ ❞❡ ❡①tr❡♠✐❞❛❞❡ ❞❡ ✉♠❛ ❝✉r✈❛ ❛♦ ♣♦♥t♦ ❞❡ ✐♥í❝✐♦ ❞❡ ✉♠❛ ♦✉tr❛✳ ❊st❛s ❝✉r✈❛s ♣♦❞❡♠ ❡①♣❛♥❞✐r ❡ s❡ tr❛♥s❢♦r♠❛r ❡♠ s✉♣❡r❢í❝✐❡s ❞❡ ❇é③✐❡r✳

❊♥tr❡ ❛s ❛♣❧✐❝❛çõ❡s r❡❝❡♥t❡s✱ t❡♠♦s ♦ ✉s♦ ❞❡ s♦❢t✇❛r❡ ❡♠ ❝✐r✉r❣✐❛s ♣❧ást✐❝❛s ♦♥❞❡ ♦ ♣❛❝✐❡♥t❡ ♣♦❞❡ ✈✐s✉❛❧✐③❛r ♦s ❝♦♥t♦r♥♦s ❛♥t❡s ❞♦ ♣r♦❝❡❞✐♠❡♥t♦✱ ✜❣✉r❛ ✷✳✸✳

❋✐❣✉r❛ ✷✳✸✿ ❯s❛♥❞♦ ❝✉r✈❛s ❡♠ ❝✐r✉r❣✐❛s ♣❧ást✐❝❛s✳

❖ Pr♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞❛ ❝✉r✈❛ ❞❡ ❇é③✐❡r é ❜❛st❛♥t❡ ✐♥t✉✐t✐✈♦✱ ❧❡✈❛♥❞♦ ♦ ✉s✉ár✐♦ ❛ r❡❛❧✐③❛r ✉♠❛ ❝♦♥str✉çã♦ s❡♠ ❣r❛♥❞❡s ❝♦♥❤❡❝✐♠❡♥t♦s✱ ✐st♦ ❡r❛ ❢❛✈♦rá✈❡❧✱ ♣♦✐s ♦s ❞❡s✐❣♥s ♥ã♦ ❞♦♠✐♥❛✈❛♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♠❛t❡♠át✐❝❛✳ ❆♦ ❧♦♥❣♦ ❞❛s ❞é❝❛❞❛s ❛ ❝♦♠♣✉t❛çã♦ ❣rá✜❝❛ s❡ t♦r♥♦✉ ✉♠❛ ❣r❛♥❞❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s✳ ◆♦ â♠❜✐t♦ ❞❡ ❡♥s✐♥♦ ♦ ❡st✉❞♦ ❞❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r ♣❡r♠✐t❡ r❡✈✐s✐t❛r ❛❧❣✉♥s ❝♦♥❝❡✐✲ t♦s ❝♦♠♦ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ❝✉r✈❛s✱ ♠é❞✐❛s✱ ❢✉♥çõ❡s ❡ ♣♦❧✐♥ô♠✐♦s✳ ❆♣❧✐❝❛çõ❡s ❝♦♠♦ ❝r✐❛çã♦ ❞❡ ❛♠❜✐❡♥t❡s ✈✐rt✉❛✐s✱ ❝♦♥str✉çã♦ ❞❡ ❞❡s❡♥❤♦s ❛♥✐♠❛❞♦s✱ ❛❥✉st❛♠❡♥t♦ ❞❡ ❝✉r✈❛s✱ r❡❝♦♥str✉çã♦ ❞❡ ✜❣✉r❛s ❡♠ ✸❉ ❡ ♥❛ ♠❡✲ ❞✐❝✐♥❛ ❡♠ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♠❛❣❡♥s ♣♦❞❡♠ s❡r ❝✐t❛❞❛s✱ ♦ q✉❡ é ❜❛st❛♥t❡ ♠♦t✐✈❛❞♦r ♦❜s❡r✈❛r ♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s s❡♥❞♦ ✉t✐❧✐③❛❞♦s ♥♦ ♠✉♥❞♦ r❡❛❧✳

❆♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s q✉❡ ❧❡✈❛r❛♠ ❛♦ s❡✉ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❝♦♠♦ ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ❡ ❝✉r✈❛s ♣❛r❛♠étr✐❝❛s✳ ❋❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❝✉r✈❛s ❞❡ ❇é③✐❡r✱ ❡ ❝❛s♦s ❡s♣❡❝✐❛✐s✳ ❈♦♠ ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡st✉❞❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ❞❡✜♥✐♠♦s ❛s ❈✉r✈❛s ❞❡ ❇é③✐❡r ❡ ❞❡s❡♥✈♦❧✈❡♠♦s ❛s ✐❞❡✐❛s ✐♥t✉✐t✐✈❛s ♣❛r❛ ❣❡r❛r ❡st❛s ❝✉r✈❛s✳ ❖ ❈♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❆❧✲ ❣♦r✐t♠♦ ❉❡ ❈❛st❡❧❥❛✉ ❢❛❝✐❧✐t❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ♣♦st❡r✐♦r ❡♠ s♦❢t✇❛r❡ ♠❛t❡♠át✐❝♦s ❝♦♠♦ ✇①▼❛①✐♠❛ ❡ ♦ ●❡♦❣❡❜r❛✳

P♦r ✜♠✱ ♠♦str❛♠♦s ❡①❡♠♣❧♦s ♣r♦♣♦st♦s ❡ ♦✉tr♦s s✉❣❡r✐❞♦s ♥♦ ❧✐✈r♦ ❞❡ r❡❢❡rê♥❝✐❛ ❬✷❪✳ ❆ ❝♦♠♣✉t❛çã♦ ❣rá✜❝❛ é ✉♠ ❝♦♠❜✉stí✈❡❧ ♠♦t✐✈❛❞♦r ♣❛r❛ ♦ ❡st✉❞♦ ❞❛s ❈✉r✈❛s ❞❡

✷✳✷✳ ❉❊❋■◆■➬➹❖ ❉❆❙ ❈❯❘❱❆❙ ❉❊

❇➱❩■❊❘ ❈❆P❮❚❯▲❖ ✷✳ ❈❯❘❱❆❙ ❉❊ ❇➱❩■❊❘

❇é③✐❡r✱ ❡stá ❛t✉❛❧✐③❛❞♦ ❝♦♠ ❛s ❜❛s❡s ❞❡st❡ t❡♠❛ tr❛③ ✉♠❛ s❡❣✉r❛♥ç❛ ♣❛r❛ ❛❜♦r❞á✲❧♦ ❞✉r❛♥t❡ ❛s ❛✉❧❛s✳ ❖ t❡♠❛ r❡✈✐s❛ ❛❧❣✉♥s ❝♦♥t❡ú❞♦s ♠❛t❡♠át✐❝♦s ❜ás✐❝♦s✱ ❡ s❡r✈❡ ❜❡♠ ❝♦♠♦ ❛♣❧✐❝❛çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❋❛③❡r ✉♠❛ ❝♦♠♣❛r❛çã♦ ❞❛ ❝✉r✈❛ ❞❡ ❇é③✐❡r ❝♦♠ ♦✉tr♦s ♠ét♦❞♦s ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❝♦♠♦ ♦ P♦❧✐♥ô♠✐♦ ❞❡ ▲❛❣r❛♥❣❡ ❡ ❛ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r é ❜❛st❛♥t❡ r❡❧❡✈❛♥t❡ ♣❛r❛ ♦ ❡st✉❞♦✱ ♣♦✐s tr❛③ ✉♠❛ r❡s♣♦st❛ ❝❧❛r❛ ♣❡❧❛ ♣r❡❞✐❧❡çã♦ ♣♦r ❡ss❛s ❝✉r✈❛s✳

✷✳✷ ❉❡✜♥✐çã♦ ❞❛s ❈✉r✈❛s ❞❡

❇é③✐❡r

❊ss❡♥❝✐❛❧♠❡♥t❡ ♦ q✉❡ ❛ ❝♦♥str✉çã♦ ❞❡ ❉❡ ❈❛st❡❧❥❛✉ ❢❛③ é ✉♠❛ ♠é❞✐❛ ♣♦♥❞❡r❛❞❛ ❡♥tr❡ ❞✐✈❡rs♦s ♣♦♥t♦s ❞❛❞♦s ✐♥✐❝✐❛❧♠❡♥t❡✳ ❖s ♣❡s♦s ❞❡st❛ ♠é❞✐❛ ❡♠ ❝❛❞❛ ♣♦♥t♦ sã♦ ❢✉♥çõ❡s ❞♦ ♣❛râ♠❡tr♦ t❀ ❝♦♠♦ ❡♠ ❝❛❞❛ ❡t❛♣❛ ❢❛③✲s❡ ✉♠❛ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r s♦❜r❡

♦s ♣♦♥t♦s ❞❛ ❡t❛♣❛ ❛♥t❡r✐♦r ❡ss❛s ❢✉♥çõ❡s sã♦ ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ n✱ ✈❡r✐✜q✉❡ q✉❡

t❡♠♦s n✔1 ♣♦♥t♦s ❡ ♦ ♣♦❧✐♥ô♠✐♦ é ✉♠ ❣r❛✉ ♠❡♥♦r q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ❬✾❪✳

❆s ❝✉r✈❛s ❞❡ ❇é③✐❡r ♣♦❞❡♠ s❡r ❞❡✜♥✐❞❛s ❝♦♠♦✿

C❼t➁

N

◗

i 0

PiBin❼t➁

❡♠ q✉❡ Bn

i sã♦ ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇❡r♥st❡✐♥ ❞❡ ❣r❛✉ n ❡ Pi sã♦ ♦s ♣♦♥t♦s ❞❡ ❝♦♥tr♦❧❡

❡s❝♦❧❤✐❞♦s ♣r❡✈✐❛♠❡♥t❡✳ ❚❡♠♦s ❡♥tã♦✿

x❼t➁

N

◗

i 0

xiBin❼t➁

y❼t➁

N

◗

i 0

yiBin❼t➁

Pr♦❜❧❡♠❛✿ ❆❝❤❡ ❛ ❝✉r✈❛ ❞❡ ❇❡③✐❡r q✉❡ ♣♦ss✉✐ ♦s ♣♦♥t♦s ❞❡ ❝♦♥tr♦❧❡✿ ❼x, y➁ ❼2_,2➁✱

❼1_,1_.5➁✱❼3_.5_,0➁✱❼4_,1➁

❙♦❧✉çã♦✿

x❼t➁ 2_B3

0❼t➁ ✔1B 3

1❼t➁ ✔3.5B 3

2❼t➁ ✔4B 3 3❼t➁

2_t3

✔t2

❼1✏_t➁ ✔3_.5_t❼1✏_t➁2

✔4❼1✏_t➁3

t3✔₁₂

t2✏₁₇

t✔8

2 .

y❼t➁ 2_B3

0❼t➁ ✔1.5B 3

1❼t➁ ✔0B 3

2❼t➁ ✔1B 3 3❼t➁

2❼_t3

➁ ✔1_.5❼_t2

➁❼1✏_t➁ ✔0_t❼❼1✏_t➁2

➁ ✔1❼❼1✏_t➁3

➁

t3✏_9t2✔_6t✏₂

2 .