❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛
Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚
●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✿ ❆ ❚á①✐✲❊❧✐♣s❡
❊❞✐✈❛❧❞♦ ❖❧✐✈❡✐r❛ ❞❛ ❈r✉③
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ◆❡✇t♦♥ ▲✉ís ❙❛♥t♦s
❊❞✐✈❛❧❞♦ ❖❧✐✈❡✐r❛ ❞❛ ❈r✉③
●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✿ ❆ ❚á①✐✲❊❧✐♣s❡
❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ à ❈♦♦r❞❡♥❛çã♦ ❆❝❛❞ê✲ ♠✐❝❛ ■♥st✐t✉❝✐♦♥❛❧ ❞♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ♦❢❡r❡✲ ❝✐❞♦ ❡♠ ❛ss♦❝✐❛çã♦ ❝♦♠ ❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐✲ ❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ◆❡✇t♦♥ ▲✉ís ❙❛♥t♦s
FICHA CATALOGRÁFICA
Serviço de Processamento Técnico da Universidade Federal do Piauí Biblioteca Setorial do CCN
C955g Cruz, Edivaldo Oliveira da.
Geometria do táxi: a táxi-elipse / Edivaldo Oliveira da Cruz. – Teresina, 2015.
70 f. 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. Elipse. 4. Matemática – Estudo e Ensino. I. Título
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦
➚ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉✐✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ q✉❡ ❝♦♥tr✐❜✉✐r❛♠ ❜❛st❛♥t❡ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱
➚ ❈♦♦r❞❡♥❛çã♦ ◆❛❝✐♦♥❛❧ ❞♦ P❘❖❋▼❆❚✱ ♣♦r t♦r♥❛r ♣♦ssí✈❡❧ ♦ s♦♥❤♦ ❞❡ t❛♥t♦s ♣r♦❢❡s✲ s♦r❡s✱
➚ ❈❆P❊❙✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ✐♥t❡❧❡❝t✉❛❧ ❡ ✜♥❛♥❝❡✐r♦✱
❆♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ tr❛❜❛❧❤♦ ❞❛s ❊s❝♦❧❛ ▼✉♥✐❝✐♣❛❧ ❙✐♠õ❡s ❋✐❧❤♦✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥✲ ❝❡♥t✐✈❛r❛♠ ♥❡ss❛ ❧✉t❛ ❡ ❯♥✐❞❛❞❡ ❊s❝♦❧❛r ❊♥♦q✉❡ ▼♦✉r❛✱ ♣❡❧❛s ❢❡❧✐❝✐t❛çõ❡s ❡ t♦r❝✐❞❛✱ ❆♦s ❛♠✐❣♦s ❞❡ ❝✉rs♦✱ ♣❡❧❛s ❞✐✈❡rsõ❡s ❞♦ ❉♦♠❛t ❡ ❞♦ ❋✉t❡❜♦❧✱ ♣❡❧♦s ❛❧♠♦ç♦s ♥♦ ❆❡✲ r♦♣♦rt♦✱ ♣❡❧♦s s♦r✈❡t❡s ♥♦ ❙❤♦♣♣✐♥❣ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❡❧♦s ♣r♦❞✉t✐✈♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦✱
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t♦✉ ❡♠ ♠❡✉ ♣♦t❡♥❝✐❛❧✱
➚ ♠✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛ ❊r✐♥❡t❡✱ ❝♦♠♣❛♥❤❡✐r❛ ❞❡ ✈✐❞❛✱ q✉❡ ❛❝♦♠♣❛♥❤♦✉ ♠✐♥❤❛s ❛♥❣ús✲ t✐❛s ❛♦ ❡s❝r❡✈❡r ❡ss❡ tr❛❜❛❧❤♦✱
P♦❞❡r♦s❛ é ❛ ●❡♦♠❡tr✐❛❀ ✉♥✐❞❛ à ❛rt❡✱ ✐rr❡s✐stí✈❡❧✳ ✭❊✉rí♣✐❞❡s✮
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ sã♦ ❞✐s❝✉t✐❞♦s ❛s♣❡❝t♦s ❞❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✱ ✉♠❛ ❣❡♦♠❡tr✐❛ ♥ã♦✲ ❊✉❝❧✐❞✐❛♥❛ ❡♠ q✉❡ s❡ ♠✉♥❡ ♦ ♣❧❛♥♦ ❞❛ ❞✐stâ♥❝✐❛ ❞❛ s♦♠❛✱ t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❞✐stâ♥❝✐❛ ❞♦ tá①✐✳ ❙ã♦ ❞✐s❝✉t✐❞❛s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s r❡❧❛❝✐♦♥❛❞❛s ❛ ❡st❛ ❣❡♦♠❡tr✐❛ ❡ ❛❧❣✉♥s ❞❡ s❡✉s ❛s♣❡❝t♦s q✉❡ ♣♦❞❡♠ s❡r tr❛t❛❞♦s ♣❡❧♦ Pr♦❢❡ss♦r ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡♠ s✉❛s ❛✉❧❛s✳ ➱ ❢❡✐t❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❝ô♥✐❝❛ ❡❧✐♣s❡ ♥❡st❛ ❣❡♦♠❡tr✐❛✳ ❙ã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ♣r♦♣r✐❡✲ ❞❛❞❡s ❣❡♦♠étr✐❝❛s ❞❛s ❝ô♥✐❝❛s ❡ ❝♦♥❞✐çõ❡s ❞❡ s✐♠❡tr✐❛ q✉❡ ❛✉①✐❧✐❛♠ ♥♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠❡s♠❛✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ ❞✐s❝✉ss ❛s♣❡❝ts ♦❢ ❚❛①✐❝❛❜ ●❡♦♠❡tr②✱ ❛ ♥♦♥✲❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ✐♥ ✇❤✐❝❤ ♦♥❡ ❡q✉✐♣s t❤❡ ♣❧❛♥❡ ✇✐t❤ t❤❡ s✉♠ ♠❡tr✐❝✱ ❛❧s♦ ❦♥♦✇♥ ❛s t❛①✐❝❛❜ ❞✐st❛♥❝❡✳ ■t ✐s ❞✐s❝✉ss❡❞ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s r❡❧❛t❡❞ t♦ t❤✐s ❣❡♦♠❡tr② ❛♥❞ s♦♠❡ ♦❢ ✐ts ❛s♣❡❝ts t❤❛t ❝❛♥ ❜❡ ❤❛♥❞❧❡❞ ❜② ❛ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥ ❚❡❛❝❤❡r ✐♥ t❤❡✐r ❝❧❛ss❡s✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❝♦♥✐❝s ❡❧❧✐♣s❡ ✐s ♠❛❞❡ ✐♥ t❤✐s ❣❡♦♠❡tr②✳ ■t ✐s ♣r❡s❡♥t❡❞ s♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥✐❝s ❛♥❞ s②♠♠❡tr② ❝♦♥❞✐t✐♦♥s t❤❛t ❤❡❧♣s ✐♥ t❤❡ r❡❝♦❣♥✐t✐♦♥ ♦❢ ✐ts ♣r♦♣❡rt✐❡s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷ ❘❡t❛ ❞♦ ❚á①✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸ ❘❡t❛s ❞♦ ❚á①✐ P❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹ ❍❡r♠❛♥♥ ▼✐♥❦♦✇s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✺ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✻ ❘❡♥é ❉❡s❝❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✼ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✽ ❙❡❣♠❡♥t♦ ❞❡ ❘❡t❛ ❆❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✾ ❚r✐â♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✶✵ ❚á①✐✲❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✶✶ P♦♥t♦s A ❡ B ❆❧✐♥❤❛❞♦s ❍♦r✐③♦♥t❛❧♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✶✳✶✷ P♦♥t♦s A ❡ B ❆❧✐♥❤❛❞♦s ❱❡rt✐❝❛❧♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✶ P❛rt❡ ❞❡ ✉♠❛ ❈✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ P❛rt❡ ❞❡ ✉♠❛ ❈✐❞❛❞❡ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ P❛rt❡s ❞❡ ✉♠ ❇❛✐rr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✹ ❚r❛❜❛❧❤♦ ✲ ❊s❝♦❧❛ ✲ ❙❛❧ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✶ ❊❧✐♣s❡ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷ ❚á①✐✲❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✸ ❚á①✐✲❊❧✐♣s❡ ❞❡ ❋♦❝♦sF1 ❡F2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹✳✹ ■♥t❡r✈❛❧♦ ■♥❛❞❡q✉❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✺ ❉✐stâ♥❝✐❛ ❋♦❝❛❧ ❞❛ ❚á①✐✲❊❧✐♣s❡ ❞❡ ❋♦❝♦sF1 ❡F2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✻ ❘❡t❛ ❋♦❝❛❧ ❞❡ ❈❡♥tr♦(m, k) ❡ ❋♦❝♦s F1 ❡ F2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✹✳✼ ❚r❛♥s❧❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✽ ❚á①✐✲❊❧✐♣s❡s ❚r❛♥s❧❛❞❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✾ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✶✵ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧ ❣r❛✉0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹✳✶✶ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧π/16 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✹✳✶✷ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧π/8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✹✳✶✸ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧π/6 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✶✹ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧π/4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✶✻ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧3π/8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✶✼ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧7π/16 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✶✽ ➶♥❣✉❧♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧π/2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✶✾ ❉✐stâ♥❝✐❛✿ ❈❛s♦ ✭✐✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✷✵ ❉✐stâ♥❝✐❛✿ ❈❛s♦ ✭✐✐✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷✶ ❉✐stâ♥❝✐❛✿ ❈❛s♦ ✭✐✐✐✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷✷ ❚á①✐✲❊❧✐♣s❡1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✹✳✷✸ ❚á①✐✲❊❧✐♣s❡2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✹✳✷✹ ❚á①✐✲❊❧✐♣s❡3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✹✳✷✺ ❚á①✐✲❊❧✐♣s❡4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✹✳✶ ❆♥á❧✐s❡ ❞❡ ❙✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷ ❆♥á❧✐s❡ ❞❡ ❙✐♥❛✐s ❞❛ ❊q✉❛çã♦ 4.10 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✸ ❆♥á❧✐s❡ ❞❡ ❙✐♥❛✐s ❞❛ ❊q✉❛çã♦ 4.16 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹✳✹ ❆♥á❧✐s❡ ❞❡ ❙✐♥❛✐s ❞❛ ❊q✉❛çã♦ 4.17 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✹✳✺ ❆♥á❧✐s❡ ❞❡ ❙✐♥❛✐s ❞❛ ❊q✉❛çã♦ 4.18 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶✸
✶ ❆s ❉✐❢❡r❡♥t❡s ●❡♦♠❡tr✐❛s ✶✻
✶✳✶ ❇r❡✈❡ ❍✐stór✐❝♦ ❞❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷ ❆♥❛❧✐s❛♥❞♦ ❞✐stâ♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❆ ▼étr✐❝❛ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ✷✸
✷✳✶ ❖ ❈♦♥❝❡✐t♦ ❞❡ ▼étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❆ ▼étr✐❝❛ ❞♦ ❚á①✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸ ❘❡❧❛❝✐♦♥❛♥❞♦ ❉✐stâ♥❝✐❛s ✷✼
✸✳✶ ▼étr✐❝❛ ❞♦ ❚á①✐ ❡ ❈♦♠❜✐♥❛tór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ❈♦♥✜❛♥❞♦ ♥❛ ▼étr✐❝❛ ❞♦ ❚á①✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❘❡❧❛çã♦ ❡♥tr❡ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ❡ ❛ ❉✐stâ♥❝✐❛ ♥❛ ▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✸✷
✹ ❆ ❚á①✐✲❊❧✐♣s❡ ✸✹
✹✳✶ ❊❧✐♣s❡ ❊✉❝❧✐❞✐❛♥❛ ❡ ❚á①✐✲❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷ ❆♥á❧✐s❡ ❡ ❈♦♥str✉çã♦ ❞❡ ❯♠❛ ❚á①✐✲❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹✳✷✳✶ ▼♦✈✐♠❡♥t♦s ❞❡ ❚r❛♥s❧❛çã♦ ❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧ ✭❡✉❝❧✐❞✐❛♥❛✮ ❞❛ ❚á①✐✲❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✶✳✶ ▼♦✈✐♠❡♥t♦ ❞❡ ❚r❛♥s❧❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✶✳✷ ▼♦✈✐♠❡♥t♦ ❞❡ ❘♦t❛çã♦ ❞❛ ❘❡t❛ ❋♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷✳✷ ❆♥á❧✐s❡ ❞❛ ❚á①✐✲❊❧✐♣s❡ ❝♦♠ ❘❡t❛ ❋♦❝❛❧ P❛r❛❧❡❧❛ ❛♦ ❊✐①♦ Ox ✳ ✳ ✺✼
✹✳✷✳✸ ❆♥á❧✐s❡ ❞❛ ❚á①✐✲❊❧✐♣s❡ ❝♦♠ ❘❡t❛ ❋♦❝❛❧ P❛r❛❧❡❧❛ ❛♦ ❊✐①♦ Oy ✳ ✳ ✺✾
✹✳✷✳✹ ❆♥á❧✐s❡ ❞❛ ❚á①✐✲❊❧✐♣s❡ ❝♦♠ ❘❡t❛ ❋♦❝❛❧ ❖❜❧íq✉❛ ❛♦s ❊✐①♦s ❈♦♦r✲ ❞❡♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✷✳✹✳✶ ❘❡t❛ ❋♦❝❛❧ ❝♦♠ ❈♦❡✜❝✐❡♥t❡ ❆♥❣✉❧❛r P♦s✐t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✷✳✹✳✷ ❘❡t❛ ❋♦❝❛❧ ❝♦♠ ❈♦❡✜❝✐❡♥t❡ ❆♥❣✉❧❛r ◆❡❣❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✻✸
✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✻
✶✸
■♥tr♦❞✉çã♦
●❡♦♠❡tr✐❛ é ✉♠❛ ♣❛❧❛✈r❛ ❞❡ ♦r✐❣❡♠ ❣r❡❣❛✿ ❣❡♦ ❂ t❡rr❛ ❡ ♠❡tr✐❛ ❂ ♠❡❞✐❞❛✳ ●❡♦✲ ♠❡tr✐❛ s✐❣♥✐✜❝❛✱ ❡♥tã♦✱ ♠❡❞✐❞❛ ❞❛ t❡rr❛✳ ❚❛❧ ♥♦♠❡♥❝❧❛t✉r❛ ✈❡♠ ❞♦ ❊❣✐t♦ ❛♥t✐❣♦✱ ♦♥❞❡ ♦s ❡❣í♣❝✐♦s ❝✉❧t✐✈❛✈❛♠ t❡rr❛s q✉❡ ❡r❛♠ ❞✐✈✐❞✐❞❛s ❡♠ ❧♦t❡s✱ ♥❛s ♠❛r❣❡♥s ❞♦ r✐♦ ◆✐❧♦✳ ❉❡✈✐❞♦ ❛♦ tr❛♥s❜♦r❞❛♠❡♥t♦ ❞♦ r✐♦✱ ❛ á❣✉❛ ❛❝❛❜❛✈❛ ❛♣❛❣❛♥❞♦ ❛s ❞✐✈✐sór✐❛s ❞❡ss❡s ❧♦t❡s✳ ❊♥tã♦✱✈✐♥❤❛♠ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❢❛r❛ó ❡ r❡❢❛③✐❛♠ ❛s ❞✐✈✐sõ❡s ❞❛ t❡rr❛✳ P❛r❛ ✐ss♦✱ ♠❡❞✐❛♠ ❝♦♠♣r✐♠❡♥t♦s✱ ❧❛r❣✉r❛s✱ â♥❣✉❧♦s✱ ❡ ♦✉tr❛s ♠❡❞✐❞❛s ♥❡❝❡ssár✐❛s ❛ ✉♠❛ ❜♦❛ ❞✐✈✐sã♦✳ ❆ ♣❛rt✐r ❞✐ss♦✱ ♦s ❣r❡❣♦s ❛♣r❡♥❞❡r❛♠ ❡ ❛♣r✐♠♦r❛r❛♠ ❡ss❡s ❝♦♥❝❡✐t♦s✱ ❡ ♦s ♥♦♠❡❛r❛♠ ✏●❡✲ ♦♠❡tr✐❛✑✳
❆s ♣r♦♣r✐❡❞❛❞❡s ❡ ❛s r❡❧❛çõ❡s ❡♥tr❡ ♦s ♥ú♠❡r♦s sã♦ ❡st✉❞❛❞❛s ♥❛ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❞❡♥♦♠✐♥❛❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❖ ❡st✉❞♦ ❞❡st❡s ❝♦♥❝❡✐t♦s é tr❛t❛❞♦ ❝♦♠ ♠❛✐♦r ê♥❢❛s❡ ♥♦s ❝✉rs♦s ❞❡ ❣r❛❞✉❛çã♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✳ ❬✽❪
❆♣❡s❛r ❞❡ tr❛③❡r ♥♦ ♣ró♣r✐♦ ♥♦♠❡ ✏♠❡❞✐❞❛✑✱ ❛ ●❡♦♠❡tr✐❛ t❡♠ ♠✉✐t♦ ♠❛✐s ❛ ✈❡r ❝♦♠ ❢♦r♠❛s ❣❡♦♠étr✐❝❛s✳ ❚❛✐s ❢♦r♠❛s ✐♥st✐❣❛♠ ♦ ✐♥t❡r❡ss❡ ❞♦s ❤♦♠❡♥s ❞❡s❞❡ ♦ ❝♦♠❡ç♦ ❞❛ ❤✐stór✐❛✳
❯♠ ❣r❛♥❞❡ ♣r♦❜❧❡♠❛ ❡♥❝♦♥tr❛❞♦ ❡♠ s❛❧❛s ❞❡ ❛✉❧❛ ❛❝❡r❝❛ ❞♦ ❡♥s✐♥♦ ❞❛ ●❡♦♠❡tr✐❛ é ♦ ♣♦✉❝♦ ❞❡st❛q✉❡ q✉❡ s❡ ❞á ❛ s❡✉ ❡♥s✐♥♦✳ ■ss♦ é ♦❜s❡r✈❛❞♦ ♥♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆✬s✮❬✸❪✿
◆♦ ❡♥t❛♥t♦✱ ❛ ●❡♦♠❡tr✐❛ t❡♠ t✐❞♦ ♣♦✉❝♦ ❞❡st❛q✉❡ ♥❛s ❛✉❧❛s ❞❡ ▼❛t❡♠át✐❝❛ ❡✱ ♠✉✐t❛s ✈❡③❡s✱ ❝♦♥❢✉♥❞❡✲s❡ s❡✉ ❡♥s✐♥♦ ❝♦♠ ♦ ❞❛s ♠❡❞✐❞❛s✳ ❊♠ q✉❡ ♣❡s❡ s❡✉ ❛❜❛♥❞♦♥♦✱ ❡❧❛ ❞❡s❡♠♣❡♥❤❛ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❝✉rrí❝✉❧♦✱ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ♣♦ss✐❜✐❧✐t❛ ❛♦ ❛❧✉♥♦ ❞❡s❡♥✈♦❧✈❡r ✉♠ t✐♣♦ ❞❡ ♣❡♥s❛♠❡♥t♦ ♣❛rt✐❝✉❧❛r ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r✱ ❞❡s❝r❡✈❡r ❡ r❡♣r❡s❡♥t❛r✱ ❞❡ ❢♦r♠❛ ♦r❣❛♥✐③❛❞❛✱ ♦ ♠✉♥❞♦ ❡♠ q✉❡ ✈✐✈❡✳ ❚❛♠❜é♠ é ❢❛t♦ q✉❡ ❛s q✉❡stõ❡s ❣❡♦♠étr✐❝❛s ❝♦st✉♠❛♠ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ❞♦s ❛❞♦❧❡s❝❡♥t❡s ❡ ❥♦✈❡♥s ❞❡ ♠♦❞♦ ♥❛t✉r❛❧ ❡ ❡s♣♦♥tâ♥❡♦✳ ❆❧é♠ ❞✐ss♦✱ é ✉♠ ❝❛♠♣♦ ❢ért✐❧ ❞❡ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ q✉❡ ❢❛✈♦r❡❝❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ♣❛r❛ ❛r❣✉♠❡♥t❛r ❡ ❝♦♥str✉✐r ❞❡♠♦♥str❛çõ❡s✳ ✭❇❘❆❙■▲✱ ✶✾✾✽✮
❖ ❡♥s✐♥♦ ❞❛ ●❡♦♠❡tr✐❛ ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ às ✈❡③❡s é ❞❡✐①❛❞♦ ❞❡ ❧❛❞♦ ♣♦rq✉❡ ❛❧❣✉♥s ❧✐✈r♦s tr❛③❡♠ ♦s ❝♦♥t❡ú❞♦s ❞❛ ♠❡s♠❛ ♥❛ ♣❛rt❡ ✜♥❛❧ ❞♦ ❧✐✈r♦✱ ❡ ♦✉tr❛s ✈❡③❡s ♣♦rq✉❡ ♦s ♣ró♣r✐♦s ♣r♦❢❡ss♦r❡s ❥✉❧❣❛♠ ♦s ❝♦♥t❡ú❞♦s ❞❡ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✱ ❝♦♠♦ ❈á❧❝✉❧♦ ❡ ➪❧❣❡❜r❛✱ ♠❛✐s ✐♠♣♦rt❛♥t❡s q✉❡ ❛ ●❡♦♠❡tr✐❛✳ ❈♦♠ ✐ss♦ ♦ ❛❧✉♥♦ ♣❡r❞❡ ❛❧❣✉♠❛s ❝❛♣❛❝✐❞❛❞❡s ❞❡ r❛❝✐♦❝í♥✐♦ q✉❡ só ♦ ❡♥s✐♥♦ ❞❡st❛ ♦❢❡r❡❝❡✳
✶✹
❞♦ ❚á①✐ é ✉♠❛ ❣❡♦♠❡tr✐❛ ❛♣❧✐❝❛❞❛ ❡♠ ✉♠❛ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛✱ ❛s ❧✐♥❤❛s ❤♦r✐③♦♥t❛✐s ❡ ❛s ✈❡rt✐❝❛✐s s❡ ❝♦♥❢✉♥❞❡♠ ❝♦♠ r✉❛s ❞❡ ✉♠❛ ✏❝✐❞❛❞❡ ✐❞❡❛❧✑ ♦✉ ✏❝✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛✑✳ ❬✽❪ P♦r s❡r ✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ❢á❝✐❧ ❝♦♠♣r❡❡♥sã♦✱ ♣♦✐s ♣♦❞❡ s❡r ❡♠♣r❡❣❛❞❛ ♥♦ ❝♦t✐❞✐❛♥♦ ❞❛s ♣❡ss♦❛s✱ ♥♦s s❡✉s ❞❡s❧♦❝❛♠❡♥t♦s ❛ ♣é ♦✉ ❡♠ ✉♠ tr❛♥s♣♦rt❡✱ ❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ♣♦❞❡ tr❛③❡r ✉♠❛ ót✐♠❛ ❝♦♥tr✐❜✉✐çã♦ ❛♦ ❛♣r❡♥❞✐③❛❞♦ ❞❡ ✉♠❛ ❢♦r♠❛ ❜❡♠ ♠❛✐s ✢❡①í✈❡❧ ❡ ❝r✐❛t✐✈❛ q✉❡ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳ ▼❡s♠♦ s❡♥❞♦ ♣♦✉❝♦ ❝♦♥❤❡❝✐❞❛✱ ❢❛③ ♣❛rt❡ ❞♦ ❞✐❛ ❛ ❞✐❛ ❞❡ t♦❞♦s ♥ós ❡♠ q✉❛s❡ t♦❞♦ ♦ t❡♠♣♦✳ ❬✽❪
P♦r ❡ss❛ q✉❛s❡ ♦❜r✐❣❛t♦r✐❡❞❛❞❡ ❞❛s ♣❡ss♦❛s ✈✐❛❥❛r❡♠ ♣♦r r✉❛s ❡ ❝❛❧ç❛❞❛s✱ t❛❧ ❣❡♦✲ ♠❡tr✐❛ é ✉♠ út✐❧ ♠♦❞❡❧♦ ♥❛ ❣❡♦❣r❛✜❛ ✉r❜❛♥❛✳
❱❡❥❛ ♦ q✉❡ ❛✜r♠❛ ❑r❛✉③❡ ❬✼❪✿
❆ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ é q✉❛s❡ ♦ ♠❡s♠♦ q✉❡ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳ ❖s ♣♦♥t♦s sã♦ ♦s ♠❡s♠♦s✱ ❛s ❧✐♥❤❛s sã♦ ❛s ♠❡s♠❛s✱ ❡ ♦s â♥❣✉❧♦s sã♦ ♠❡❞✐❞♦s ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛✳ ❙♦♠❡♥t❡ ❛ ❢✉♥çã♦ ❞❡ ❞✐stâ♥❝✐❛ é ❞✐❢❡r❡♥t❡✳✭❑❘❆❯❙❊✱ ✶✾✽✻✮
❯♠❛ ●❡♦♠❡tr✐❛ s❡♠❡❧❤❛♥t❡✱ ♠❛✐s ❛❝❡ssí✈❡❧✱ ❝♦♠ ♣♦✉❝❛ ❞✐❢❡r❡♥ç❛ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ❜❛s❡ ❞♦s ❡st✉❞♦s ❣❡♦♠étr✐❝♦s ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ s❡♥❞♦ ✐♥tr♦❞✉③✐❞❛ ❛♥t❡s ❞❛ ❊✉❝❧✐❞✐❛♥❛✱ t♦r♥❛r✐❛ ❢á❝✐❧ ❛ tr❛♥s✐çã♦ ♣❛r❛ ❡st❛ ❡ ❛✐♥❞❛ ❛❥✉❞❛r✐❛ ❡♠ s✉❛s ✐♥t❡r♣r❡t❛çõ❡s ♠❛✐s ❝♦♠♣❧❡①❛s✳
❙❛❜❡✲s❡ q✉❡ ✉♠❛ ❡❧✐♣s❡ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❡♠ q✉❡ ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❛ ❞♦✐s ♦✉tr♦s ♣♦♥t♦s ✜①♦s✱ ❝❤❛♠❛❞♦s ❢♦❝♦s é ❝♦♥st❛♥t❡ ❡ ♠❛✐♦r q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ t❛✐s ❢♦❝♦s✳ ❖ ♦❜❥❡t♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ ❡st✉❞♦ é ❛ ❚á①✐✲❊❧✐♣s❡✱ ❡❧✐♣s❡ ✐♥s❡r✐❞❛ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✱ q✉❡ é ✉♠ t✐♣♦ ❞❡ ❣❡♦♠❡tr✐❛ ✏♥ã♦✲❊✉❝❧✐❞✐❛♥❛✑✳ ❚❛❧ ❡①♣r❡ssã♦ s✉r❣❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ ❡ss❛ ❡stá ❡♠ ❞❡s❛❝♦r❞♦ ❝♦♠ ❛❧❣✉♥s ♣r✐♥❝í♣✐♦s ✭q✉❡ ✉♠❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ❞❡✈❡ s❛t✐s❢❛③❡r✮ ❡st❛❜❡❧❡❝✐❞♦s ♣♦r ❊✉❝❧✐❞❡s✱ ❡♠ s❡✉ ▲✐✈r♦ ✏❊❧❡♠❡♥t♦s✑✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❛①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s✳ ❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❞✐s❝✉t✐r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐stâ♥❝✐❛ q✉❛♥❞♦ s❡ ❞✐s❝✉t❡ ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡❧✐♣s❡✳ ❈♦♠♦ ✉♠❛ ❛♥á❧✐s❡ ❝r✐t❡r✐♦s❛ ❞❡ ♠ó❞✉❧♦s é ✐♠♣r❡s❝✐♥❞í✈❡❧✱ ❡st❛ é ✉♠❛ ✐♠♣♦rt❛♥t❡ ❛♣❧✐❝❛çã♦ ❞❛q✉❡❧❡ ❝♦♥❝❡✐t♦✳
P♦r s❡r ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ❣❡♦♠❡tr✐❛ ♥ã♦✲❊✉❝❧✐❞✐❛♥❛✱ ♣♦❞❡♠ s✉r❣✐r ❛❧❣✉♠❛s ✐♥❞❛❣❛✲ çõ❡s✿ ❊ ❛ ❚á①✐✲❊❧✐♣s❡ t❡♠ ❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ❊❧✐♣s❡ ❞❡✜♥✐❞❛ ♥❛ ●❡♦♠❡tr✐❛ ❊✉✲ ❝❧✐❞✐❛♥❛❄ ❙❡ ♥ã♦✱ ❡♥tã♦ q✉❛❧ é ❛ ❢♦r♠❛❄ ❆♦ ✜♠ ❞❡ss❡ tr❛❜❛❧❤♦ ❡ss❛s r❡s♣♦st❛s s❡rã♦ r❡s♣♦♥❞✐❞❛s✳
❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ ❡st✉❞♦ s❡ ❞❛rá ❡♠4❝❛♣ít✉❧♦s✱ ❞✐✈✐❞✐❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❈❛♣ít✉❧♦ ✶✿ ➱ ❢❡✐t♦ ✉♠ ❤✐stór✐❝♦ ❞❛s ❣❡♦♠❡tr✐❛s ♥ã♦✲❊✉❝❧✐❞✐❛♥❛s✱ ❞❛♥❞♦ ê♥❢❛s❡ à ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✱ tr❛ç❛♥❞♦ ✉♠ ♣❛r❛❧❡❧♦ ❡♥tr❡ ❡st❛ ❡ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳
✶✺
❈❛♣ít✉❧♦ ✸✿ ❙❡rã♦ ❝♦♠♣❛r❛❞❛s✱ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ❡ ❛ ❞♦ ❚á①✐✱ ❡st❛❜❡❧❡❝❡♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞✐stâ♥❝✐❛ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✳
❈❛♣ít✉❧♦ ✹✿ ❈♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❡ ❛♥❛❧ít✐❝❛ ❞❛ ❚á①✐✲❊❧✐♣s❡✱ ♠♦str❛♥❞♦ ♦ ❝♦♠✲ ♣♦rt❛♠❡♥t♦ ❣rá✜❝♦ ❡ ❛♥❛❧ít✐❝♦ ❞❛ ♠❡s♠❛✱ ❞❛♥❞♦ ❡ss❡♥❝✐❛❧ ✐♠♣♦rtâ♥❝✐❛ ❛♦ ♠♦✈✐♠❡♥t♦ ❞❡ r♦t❛çã♦ ❞❛ r❡t❛ ❢♦❝❛❧ ❞❛ ❚á①✐✲❊❧✐♣s❡✱ ❜❡♠ ❝♦♠♦ s✉❛ ✐♥t❡r❢❡rê♥❝✐❛ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❣rá✜❝♦ ❞❡st❛✳
✶✻
✶ ❆s ❉✐❢❡r❡♥t❡s ●❡♦♠❡tr✐❛s
❆té ♠❡❛❞♦s ❞♦ sé❝✉❧♦XIX❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛ ❛❝r❡❞✐t❛✈❛ q✉❡ ♥ã♦ ❡①✐st✐❛ ♦✉tr❛
❢♦r♠❛ ❞❡ ✐♥t❡r♣r❡t❛r ♦ ❡s♣❛ç♦ ❡♠ q✉❡ ✈✐✈❡♠♦s s❡ ♥ã♦ ♣❡❧❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ♦✉ ♣❡❧❛ ●❡♦♠❡tr✐❛ ❊s❢ér✐❝❛✳ ❍♦✉✈❡ ❡♥tã♦ ✉♠❛ ❣r❛♥❞❡ r❡✈♦❧✉çã♦ ♥♦ ♠✉♥❞♦ ♠❛t❡♠át✐❝♦ q✉❛♥❞♦ ❢♦✐ ❞❡s❝♦❜❡rt❛ ❛ ♣r✐♠❡✐r❛ ●❡♦♠❡tr✐❛ ♥ã♦✲❊✉❝❧✐❞✐❛♥❛ ❡ ❞✐❢❡r❡♥t❡ ❞❛ ●❡♦♠❡tr✐❛ ❊s❢ér✐❝❛✱ ❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✳
❆ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ❡①❡r❝❡✉✱ ❞✉r❛♥t❡ sé❝✉❧♦s✱ ♦ ❞♦♠í♥✐♦ ❞❛ ✈✐sã♦ ❞♦ ♠✉♥❞♦ r❡❛❧ ❡ ❞❛ ♥♦çã♦ ❞❡ ❞✐stâ♥❝✐❛ ♣♦r ✉♠❛ ❧✐♥❤❛ r❡t❛✳ ❊❧❛ ❛✐♥❞❛ é ❛ ❣❡♦♠❡tr✐❛ ♠❛✐s ❡♥s✐♥❛❞❛ ♥❛s ❡s❝♦❧❛s ❡ s❡✉ ♥♦♠❡ é ❤♦♠❡♥❛❣❡♠ ❛ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ q✉❡ ✈✐✈❡✉ ❡♥tr❡ ♦s sé❝✉❧♦sIII ❡IV ❛✳❈✱ ❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ✏P❛✐ ❞❛ ●❡♦♠❡tr✐❛✑✳ ➱ ❞❡
❊✉❝❧✐❞❡s ♦ ♠❛✐s ❢❛♠♦s♦ tr❛❜❛❧❤♦ ❞❡ ❣❡♦♠❡tr✐❛ ❥á ♣✉❜❧✐❝❛❞♦✿ ✏❖s ❊❧❡♠❡♥t♦s✑✳ ❉❡s❞❡ s✉❛ ♣✉❜❧✐❝❛çã♦✱ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❛❝r❡❞✐t❛✈❛♠ q✉❡ ♦ ❱ P♦st✉❧❛❞♦ ♣♦❞❡r✐❛ s❡r ❞❡♠♦♥s✲ tr❛❞♦ ✉t✐❧✐③❛♥❞♦ ❝♦♠♦ s✉❜sí❞✐♦ ♦s q✉❛tr♦ ♣♦st✉❧❛❞♦s ❛♥t❡r✐♦r❡s✱ ❡ ♠✉✐t♦s t❡♥t❛r❛♠✱ s❡♠ s✉❝❡ss♦✱ r❡❛❧✐③❛r t❛❧ ❞❡♠♦♥str❛çã♦✱ ♦ q✉❡✱ ♥♦ sé❝✳ XIX ❧❡✈♦✉ ❛ s✉s♣❡✐t❛s ❞❡ q✉❡ t❛❧✈❡③
♣✉❞❡ss❡♠ s❡r ❞❡s❡♥✈♦❧✈✐❞❛s ♦✉tr❛s ❣❡♦♠❡tr✐❛s ❝♦♥s✐st❡♥t❡s ❡ q✉❡ ❝♦♥tr❛❞✐ss❡ss❡♠ ♦ ❱ P♦st✉❧❛❞♦ ✭❞❛s ♣❛r❛❧❡❧❛s✮ s❡♥❞♦ ❡♥tã♦✱ ✐♠♣♦ssí✈❡❧ ❞❡♠♦♥strá✲❧♦ ✉t✐❧✐③❛♥❞♦ ♦s q✉❛tr♦ ❛♥t❡r✐♦r❡s✱ ❧❡✈❛♥❞♦ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ q✉❡ t❛❧ ♣♦st✉❧❛❞♦ ♥ã♦ ❡r❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦s q✉❛✲ tr♦ ♣r✐♠❡✐r♦s✳ P♦❞❡r✐❛✱ ❝♦♠ ✐ss♦✱ ❝r✐❛r✲s❡ ♦✉tr❛s ❣❡♦♠❡tr✐❛s tã♦ ❝♦♥s✐st❡♥t❡s q✉❛♥t♦ ❛ ❊✉❝❧✐❞✐❛♥❛✱ ❝♦♥tr❛❞✐③❡♥❞♦ ♦ ❱ P♦st✉❧❛❞♦✳ ▼❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛✱ ❝♦♠♦ ◆✳ ■✳ ▲♦❜❛✲ ❝❤❡✈s❦② ✭✶✼✾✷✕✶✽✺✻✮✱ ❏✳ ❇♦❧②❛✐ ✭✶✽✵✷✕✶✽✻✵✮ ❡ ❈✳ ❋✳ ●❛✉ss ✭✶✼✼✼✕✶✽✺✺✮ ❛①✐♦♠❛t✐③❛r❛♠ q✉❡ ♦✉ ♥ã♦ ❡①✐st✐❛♠ r❡t❛s ♣❛r❛❧❡❧❛s ❛ ✉♠❛ r❡t❛ ❞❛❞❛✱ ♣❛ss❛♥❞♦ ♣♦r ✉♠ ♣♦♥t♦ ❢♦r❛ ❞❡❧❛✱ ♦✉ q✉❡ ♣♦r ❡ss❡ ♣♦♥t♦ ♣♦❞❡r✐❛♠ ♣❛ss❛r ♠❛✐s ❞❡ ✉♠❛ ♣❛r❛❧❡❧❛ ❛ t❛❧ r❡t❛✳ ■♥✐❝✐❛✈❛✲s❡ ❛ss✐♠ ♦ ❡st✉❞♦ s✐st❡♠át✐❝♦ ❞❛s ●❡♦♠❡tr✐❛s ♥ã♦✲❊✉❝❧✐❞✐❛♥❛s✳
❆✜♠ ❞❡ ❜❡♠ ❡♥t❡♥❞❡r ♦ q✉❡ s❡rá ❡①♣❧❛♥❛❞♦ ❧❡♠❜r❡♠♦s q✉❛✐s ♦s ❝✐♥❝♦ P♦st✉❧❛❞♦s ❞❡ ❊✉❝❧✐❞❡s ❬✷❪✱ ❡ q✉❡ s❡r✈❡♠ ❞❡ ❜❛s❡ ♣❛r❛ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳
✶✳ ❯♠❛ ú♥✐❝❛ r❡t❛ ♣♦❞❡ s❡r tr❛ç❛❞❛ ♣❛ss❛♥❞♦ ♣♦r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s✱ q✉❛✐sq✉❡r✳ ✷✳ ❯♠❛ ❧✐♥❤❛ r❡t❛ ♣♦❞❡ s❡r ♣r♦❧♦♥❣❛❞❛ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳
✶✼
✺✳ P♦r ✉♠ ♣♦♥t♦ P✱ ❡①t❡r✐♦r ❛ ✉♠❛ r❡t❛ r✱ ❝♦♥s✐❞❡r❛❞❛s ❡♠ ✉♠ ♠❡s♠♦ ♣❧❛♥♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛ à r❡t❛ r✱ ♣❛ss❛♥❞♦ ♣♦r P✳
❋✐❣✉r❛ ✶✳✶✿ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ❋♦♥t❡✿ ✇✇✇✳❞❡❝✳✉❢❝❣✳❡❞✉✳❜r✴❜✐♦❣r❛✜❛s✴❊✉❝❧❆❧❡①
❆ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ❝♦♥tr❛❞✐③ ♦ ❱ P♦st✉❧❛❞♦✱ ❡ ❡♠ s❡✉ ❧✉❣❛r ❛✜r♠❛ q✉❡✱ ♣♦r ✉♠ ♣♦♥t♦ ❞❛❞♦ P✱ ❡①t❡r✐♦r à r❡t❛r✱ ❛♠❜♦s ❡♠ ✉♠ ♠❡s♠♦ ♣❧❛♥♦✱ ❡①✐st❡♠ ♠❛✐s ❞❡ ✉♠❛ r❡t❛
♣❛r❛❧❡❧❛ ❛ ❡st❛ r❡t❛ r✳
❱❡❥❛ ♥❛ ✜❣✉r❛1.2✱ ❛ ✐❞❡✐❛ ❞❡ r❡t❛ s❡❣✉♥❞♦ ❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✿
❋✐❣✉r❛ ✶✳✷✿ ❘❡t❛ ❞♦ ❚á①✐ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
P❡r❝❡❜❛ q✉❡ ❛ ✐❞❡✐❛ ❞❡ r❡t❛ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ é ❛ ♠❡s♠❛ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐✲ ❞✐❛♥❛✱ ✐st♦ é✱ ✉♠❛ ❧✐♥❤❛ q✉❡ ♣❛ss❛ ♣♦r ❞♦✐s ♣♦♥t♦s s❡♠ q✉❡ ❤❛❥❛ ❝✉r✈❛s✳ ❆ q✉❡❜r❛ q✉❡ ❤á ♥❡ss❛ r❡t❛ é ❜❡♠ ❛❝❡✐t❛ ♣♦r ❡ss❛ ❣❡♦♠❡tr✐❛✳
❖❜s❡r✈❡ ♥❛ ✜❣✉r❛ 1.3 ❛ ❝♦♥tr❛❞✐çã♦ ❛♦ ❱ P♦st✉❧❛❞♦✱ ❛♣♦♥t❛❞❛ ♣❡❧❛ ●❡♦♠❡tr✐❛ ❞♦
❚á①✐✿
❖❜s❡r✈❡ q✉❡ ♣❡❧♦ ♣♦♥t♦K ❢♦r❛ ❞❛ r❡t❛ r ♣❛ss❛♠ ❛s r❡t❛s s ❡t✱ ♣❛r❛❧❡❧❛s ❛r✳
❆ ✜❣✉r❛ 1.3 ♥♦s ❞✐③ t❛♠❜é♠ q✉❡ ❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ♥ã♦ é ❞❡ ■♥❝✐❞ê♥❝✐❛✱ ❧❡♠✲
❇r❡✈❡ ❍✐stór✐❝♦ ❞❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ✶✽
❋✐❣✉r❛ ✶✳✸✿ ❘❡t❛s ❞♦ ❚á①✐ P❛r❛❧❡❧❛s ❋♦♥t❡✿ ●❡♦❣❡❜r❛
❆①✐♦♠❛ I1 ✿ ◗✉❛❧q✉❡r q✉❡ s❡❥❛ ❛ r❡t❛ ✱ ❡①✐st❡♠ ♣♦♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ ❡ ♣♦♥t♦s q✉❡
♥ã♦ ♣❡rt❡♥❝❡♠ ❛ ❡❧❛✳
❆①✐♦♠❛ I2 ✿ ❉❛❞♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛ q✉❡ ♦s ❝♦♥té♠✳
◆♦t❡ q✉❡ ❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ❝♦♥tr❛❞✐③ ♦ ❆①✐♦♠❛ I2✱ ♣♦✐s ❛s r❡t❛s s ❡ t✱ ❞✐st✐♥t❛s✱
❝♦♥té♠ ♦s ♣♦♥t♦s K ❡L✳
❈♦♥✈é♠ ❧❡♠❜r❛r q✉❡ ❤♦❥❡ ❡①✐st❡♠ ✐♥ú♠❡r❛s ●❡♦♠❡tr✐❛s ♥ã♦✲❊✉❝❧✐❞✐❛♥❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✱ ❛ ●❡♦♠❡tr✐❛ ❊❧í♣t✐❝❛✱ ❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱✳✳✳ ❚❛✐s ❣❡♦♠❡tr✐❛s ❢♦r❛♠ ❝r✐❛❞❛s ❡♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳■❳ ✭❡①❝❡t♦ ♣❡❧❛ ●❡♦♠❡tr✐❛ ❊s✲ ❢ér✐❝❛✱ q✉❡ t❡♠ s✉❛s ♦r✐❣❡♥s ♥❛ ●ré❝✐❛ ❆♥t✐❣❛✮ ❡ ❞❡s❞❡ ❡♥tã♦ ✈ê♠ ❛❜r✐♥❞♦ ❣r❛♥❞❡s ♣❡rs♣❡❝t✐✈❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛✳
✶✳✶ ❇r❡✈❡ ❍✐stór✐❝♦ ❞❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐
❆ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ♣❡❧♦ ♠❛t❡♠át✐❝♦ r✉ss♦ ❍❡r♠❛♥♥ ▼✐♥❦♦✇s❦✐ ✭✶✽✻✹✕✶✾✵✾✮✳ ➱ ✉♠❛ ❣❡♦♠❡tr✐❛ ❛♣❧✐❝❛❞❛ ❡♠ ✉♠❛ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛✱ ♥❛ q✉❛❧ ❧✐♥❤❛s ❤♦r✐③♦♥t❛✐s ❡ ✈❡rt✐❝❛✐s ❝♦rr❡s♣♦♥❞❡♠ às r✉❛s ❞❡ ✉♠❛ ✏❝✐❞❛❞❡ ✐❞❡❛❧✑ ♦✉ ✏❝✐❞❛❞❡ ✐♠❛❣✐✲ ♥ár✐❛✑✳
❆♥❛❧✐s❛♥❞♦ ❞✐stâ♥❝✐❛s ✶✾
❋✐❣✉r❛ ✶✳✹✿ ❍❡r♠❛♥♥ ▼✐♥❦♦✇s❦✐
❋♦♥t❡✿ ✇✇✇✲❤✐st♦r②✳♠❝s✳st✲❛♥❞r❡✇s✳❛❝✳✉❦✴P✐❝t❉✐s♣❧❛②✴▼✐♥❦♦✇s❦✐
✶✳✷ ❆♥❛❧✐s❛♥❞♦ ❞✐stâ♥❝✐❛s
❆ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ♠♦❞❡r♥❛ ❢♦✐ ❞❡s❝♦❜❡rt❛ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❡ q✉❛s❡ s✐✲ ♠✉❧tâ♥❡❛ ♣♦r P✐❡rr❡ ❞❡ ❋❡r♠❛t ✭✶✻✵✶ ✕ ✶✻✻✺✮ ❡♠ ✶✻✷✾ ✭♥✉♠ tr❛❜❛❧❤♦ ♣✉❜❧✐❝❛❞♦ ❛♣❡♥❛s ❡♠ ✶✻✼✾✮ ❡ ♣♦r ❘❡♥è ❉❡s❝❛rt❡s ✭✶✺✾✻ ✕ ✶✻✺✵✮ ❡♠ ✶✾✸✼ ♥✉♠ tr❛❜❛❧❤♦ ❞❡♥♦♠✐✲ ♥❛❞♦ ▲❛ ●❡♦♠étr✐❡ ♣✉❜❧✐❝❛❞♦ ♥♦ ♠❡s♠♦ ❛♥♦ ❝♦♠♦ ❛♣ê♥❞✐❝❡ ❞❛ s✉❛ ♦❜r❛ ❉✐s❝✉rs♦ ❞♦ ♠ét♦❞♦ ♣❛r❛ ❜❡♠ ❝♦♥❞✉③✐r ❛ r❛③ã♦ ❡ ♣r♦❝✉r❛r ❛ ✈❡r❞❛❞❡ ♥❛s ❝✐ê♥❝✐❛s✮✳ ❬✺❪
❋✐❣✉r❛ ✶✳✺✿ P✐❡rr❡ ❞❡ ❋❡r♠❛t
❋♦♥t❡✿♠❛t❤s❢♦r❡✉r♦♣❡✳❞✐❣✐❜❡❧✳❜❡✴♣✐❡rr❡❞❡❢❡r♠❛t
❋✐❣✉r❛ ✶✳✻✿ ❘❡♥é ❉❡s❝❛rt❡s ❋♦♥t❡✿❡❝❛❧❝✉❧♦✳✐❢✳✉s♣✳❜r✴❤✐st♦r✐❛✴❞❡s❝❛rt❡s
❆ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❛❣r❡❣❛ ♦s ❝♦♥❝❡✐t♦s ❞❛ ➪❧❣❡❜r❛ ❡ ❞❛ ●❡♦♠❡tr✐❛ ❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳
P❛r❛ ❡st✉❞❛r ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ✉t✐❧✐③❛✲s❡ ♦ P❧❛♥♦ ❈❛rt❡s✐❛♥♦✱ ✉♠ s✐st❡♠❛ ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ❡✐①♦s ♦rt♦❣♦♥❛✐s✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ 1.7✳
◆♦ ♣❧❛♥♦✱ Ox r❡♣r❡s❡♥t❛ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❡ Oy✱ ♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✳ ❈❛❞❛
❆♥❛❧✐s❛♥❞♦ ❞✐stâ♥❝✐❛s ✷✵
❋✐❣✉r❛ ✶✳✼✿ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
P❛r❛ s❡ ❝❛❧❝✉❧❛r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✱ s❡❣✉♥❞♦ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r ♦s ♣♦♥t♦s A(xA, yA) ❡B(xB, yB)✱ r❡♣r❡s❡♥t❛❞♦s ♥❛ ✜❣✉r❛ 1.8✿
❋✐❣✉r❛ ✶✳✽✿ ❙❡❣♠❡♥t♦ ❞❡ ❘❡t❛ ❆❇ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
❙❡♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❊✉❝❧✐❞✐❛♥❛ ❡♥tr❡ ♦s ♣♦♥t♦sA❡B❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦AB✱ ❡ q✉❡
é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ABC✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ 1.9✱ ❛♦ ✉t✐❧✐③❛r✲s❡
♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♦❜té♠✲s❡✿
(AB)2
= (AC)2
+ (BC)2 ♦✉
d2
E(A, B) = (xB−xA)
2
+ (yB−yA)
2
❆ss✐♠✱ ❛ ❞✐stâ♥❝✐❛ ❊✉❝❧✐❞✐❛♥❛ ❡♥tr❡ ♣♦♥t♦s A(xA, yA) ❡ B(xB, yB) q✉❛✐sq✉❡r ❞♦
♣❧❛♥♦✱ é ❞❛❞❛ ♣♦r
dE(A, B) =
p
❆♥❛❧✐s❛♥❞♦ ❞✐stâ♥❝✐❛s ✷✶
❋✐❣✉r❛ ✶✳✾✿ ❚r✐â♥❣✉❧♦ ❆❇❈ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
◆❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ♣♦❞❡✲s❡ ♣❡♥s❛r ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❝♦♠♦ ✉♠ ♣✐s♦ ❞❡ ❝❡râ✲ ♠✐❝❛s q✉❛❞r❛❞❛s✱ s❡♥❞♦ q✉❡ ❛ ♠❛♥❡✐r❛ ❞❡ ❧♦❝❛❧✐③❛r ♦s ♣♦♥t♦s é s❡♠❡❧❤❛♥t❡ à ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❆ ❞✐❢❡r❡♥ç❛ ❡stá ♥❛ ♦❜s❡r✈❛çã♦ ❞♦ ❝❛♠✐♥❤♦ ❞❡ ♠❡♥♦r ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✳ ❉✐❢❡r❡♥t❡♠❡♥t❡ ❞❡ ❝♦♠♦ é ❢❡✐t♦ ♥❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✱ ❡ss❛ ♠❡♥♦r ❞✐stâ♥❝✐❛ só s❡rá ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ s❡ ❛♠❜♦s ♦s ♣♦♥t♦s ❡st✐✈❡r❡♠ ♥❛ ♠❡s♠❛ ❤♦r✐③♦♥t❛❧ ♦✉ ♥❛ ♠❡s♠❛ ✈❡rt✐❝❛❧✳ ❊♠ ✉♠❛ ❝✐❞❛❞❡ ♣❧❛♥❡❥❛❞❛✱ ❞❡s❝♦♥s✐❞❡✲ r❛♥❞♦ ♦❜✈✐❛♠❡♥t❡ ❛ ❧❛r❣✉r❛ ❞❛s r✉❛s✱ ❝❛s❛s✱ ♣ré❞✐♦s✱ ❡t❝✳✱ ♣❡ss♦❛s ❡ ✈❡í❝✉❧♦s s❡ ❞❡s✲ ❧♦❝❛♠ ❤♦r✐③♦♥t❛❧♠❡♥t❡ ❡ ✈❡rt✐❝❛❧♠❡♥t❡ ♥❛s r✉❛s ❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❝❛s❛s ❡ ♣ré❞✐♦s r❡♣r❡s❡♥t❛çõ❡s ❞❡ ♣♦♥t♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ❝❛❧❝✉❧❛r ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s q✉❡ ♥ã♦ ❡stã♦ ❡♠ ✉♠❛ ♠❡s♠❛ r✉❛✱ é ♣r❡❝✐s♦ s♦♠❛r ❛s ♠❡❞✐❞❛s ❞♦s s❡❣♠❡♥t♦s ❤♦r✐③♦♥t❛✐s ❡ ✈❡rt✐❝❛✐s ♣❡r❝♦rr✐❞♦s✳
❉❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❡t♦❞♦❧ó❣✐❝♦ ♣❛r❛ ✉♠❛ ♣r✐♠❡✐r❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ss❡s ❝♦♥❝❡✐✲ t♦s ❛♦s ❛❧✉♥♦s✱ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r ❛♣❡♥❛s ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s ✐♥t❡✐r❛s✱ t♦r♥❛♥❞♦ q✉❛❧q✉❡r ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s s❡♠♣r❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♣♦✐s t❛❧ ❞✐stâ♥❝✐❛ é ♠❡✲ ❞✐❞❛ ❡♠ ♥ú♠❡r♦ ❞❡ ❜❧♦❝♦s q✉❡ ♦ tá①✐ ✉❧tr❛♣❛ss❛ ♣❛r❛ s❡ ❞❡s❧♦❝❛r ❞❡ ✉♠ ♣♦♥t♦ ❛ ♦✉tr♦ ❞❛ ❝✐❞❛❞❡✱ ♥ú♠❡r♦ ❡ss❡ q✉❡ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳
❉❡ss❡ ♠♦❞♦✱ ❛ss✐♠ ❝♦♠♦ ♥❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❛ ❞✐stâ♥❝✐❛ ❤♦r✐③♦♥t❛❧ x ❡♥tr❡
❞♦✐s ♣♦♥t♦s é ❞❛❞❛ ♣❡❧♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❛s ❛❜s❝✐ss❛s ❞❡ss❡s ♣♦♥t♦s✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❛ ❞✐stâ♥❝✐❛ ✈❡rt✐❝❛❧ y ❡♥tr❡ ❡ss❡s ♣♦♥t♦s é ♦ ♠ó❞✉❧♦ ❞❛
❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❛s ♦r❞❡♥❛❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s✳
❈♦♥s✐❞❡r❛♥❞♦ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ♥ã♦✲❛❧✐♥❤❛❞♦s A(xA, yA) ❡ B(xB, yB)✱ ✈❡❥❛ ❛
✜❣✉r❛ 1.10✳
▲♦❣♦✱ ✜❝❛ ❝❧❛r♦ q✉❡ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ✭❝♦♠♦ é ❝❤❛♠❛❞❛ ❛ ❞✐stâ♥❝✐❛ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✮ ❡♥tr❡ ♦s ♣♦♥t♦s A ❡B s❡rá✿
dT(A, B) = |xB−xA|+|yB−yA|,
❆♥❛❧✐s❛♥❞♦ ❞✐stâ♥❝✐❛s ✷✷
❋✐❣✉r❛ ✶✳✶✵✿ ❚á①✐✲❉✐stâ♥❝✐❛ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
❖❜s❡r✈❡ q✉❡ ♠❡s♠♦ ♥♦ ❝❛s♦ ❡♠ q✉❡ A ❡ B ❡stã♦ ♥❛ ♠❡s♠❛ ❧✐♥❤❛ ❤♦r✐③♦♥t❛❧ ♦✉
✈❡rt✐❝❛❧ ✈❛❧❡ ❛ ❢ór♠✉❧❛ ❛❝✐♠❛✳ ❱❡❥❛ ♥❛s ✜❣✉r❛s 1.11❡ 1.12✳
❋✐❣✉r❛ ✶✳✶✶✿ P♦♥t♦s A ❡ B ❆❧✐♥❤❛❞♦s
❍♦r✐③♦♥t❛❧♠❡♥t❡
❋♦♥t❡✿●❡♦❣❡❜r❛
❋✐❣✉r❛ ✶✳✶✷✿ P♦♥t♦s A ❡ B ❆❧✐♥❤❛❞♦s
❱❡rt✐❝❛❧♠❡♥t❡
❋♦♥t❡✿●❡♦❣❡❜r❛
✷✸
✷ ❆ ▼étr✐❝❛ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐
❯♠ ❝♦♥❥✉♥t♦ ♣❛r❛ s❡r ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♣r❡❝✐s❛ ♦❜❡❞❡❝❡r às ♣r♦♣r✐❡✲ ❞❛❞❡s ♣ró♣r✐❛s ❞❡ ✉♠❛ ♠étr✐❝❛✱ q✉❡ s❡rã♦ ❧❡♠❜r❛❞❛s ♥❛ s❡çã♦ s❡❣✉✐♥t❡✳ ▲♦❣♦ ❛♣ós✱ ♥❛ s❡çã♦ 2.2 s❡rá ♠♦str❛❞♦ q✉❡ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ♦❜❡❞❡❝❡ ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❡ ♣♦r ✐ss♦✱
♣♦❞❡ s❡r ❝❤❛♠❛❞❛ ♠étr✐❝❛✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ▼étr✐❝❛ ❞♦ ❚á①✐✳
✷✳✶ ❖ ❈♦♥❝❡✐t♦ ❞❡ ▼étr✐❝❛
❙❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ M✳ ❯♠❛ ♠étr✐❝❛ ❡♠ M é ✉♠❛ ❢✉♥çã♦❬✻❪
d :M×M→R
q✉❡ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✐✳ ➱ s❡♠♣r❡ ♥ã♦✲♥❡❣❛t✐✈❛✱ ♦✉ s❡❥❛✱
d(x, y)≥0,∀x, y ∈M.
✐✐✳ ➱ ♥✉❧❛✱ ❛♣❡♥❛s q✉❛♥❞♦ ♦s ♣♦♥t♦s ❝♦✐♥❝✐❞❡♠✱ ♦✉ s❡❥❛✱
d(x, y) = 0⇔x=y.
✐✐✐✳ ➱ s✐♠étr✐❝❛✱ ♦✉ s❡❥❛✱ é t❛❧ q✉❡
d(x, y) =d(y, x),∀x, y ∈M.
✐✈✳ ❖❜❡❞❡❝❡ ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r✱ ♦✉ s❡❥❛✱
d(x, z)≤d(x, y) +d(y, z),∀x, y, z ∈M.
❆ ▼étr✐❝❛ ❞♦ ❚á①✐ ✷✹
✷✳✷ ❆ ▼étr✐❝❛ ❞♦ ❚á①✐
❆ ❞✐stâ♥❝✐❛ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✱ ♣❛r❛ s❡r ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ♠étr✐❝❛✱ ❞❡✈❡rá ♦❜❡✲ ❞❡❝❡r às q✉❛tr♦ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❧✐st❛❞❛s ❛❝✐♠❛✳ ▼♦str❡♠♦s q✉❡ ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s ♣♦r t❛❧ ❣❡♦♠❡tr✐❛✳
P❛r❛ ❞❡♠♦♥str❛r ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s✱ ✉s❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ♠ó❞✉❧♦ ♦✉ ✈❛❧♦r ❛❜s♦❧✉t♦✱ q✉❡ s❡❣✉❡✿❬✶✶❪
❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ x∈R✳ ❖ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ x✱ ❞❡♥♦t❛❞♦ ♣♦r |x| é t❛❧ q✉❡
|x|=
(
−x, s❡ x <0, x, s❡ x≥0.
❉❡✜♥✐❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ✈❡❥❛♠♦s✿
❙❡♥❞♦ três ♣♦♥t♦s A(xA, yA), B(xB, yB) ❡ C(xC, yC) ❡♠ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❛
❞✐stâ♥❝✐❛✲❚á①✐ dT ❞❡A ❛té B✱ ♣♦r ❡①❡♠♣❧♦✱ é ❛ss✐♠ ❞❡✜♥✐❞❛
dT(A, B) = |xB−xA|+|yB−yA|.
Pr♦✈❛r❡s♠♦s q✉❡ ❡ss❛ ❡q✉❛çã♦ ♦❜❡❞❡❝❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠étr✐❝❛✳ ✐✳ dT(A, B)≥0 ❉❡ ❢❛t♦✱
❆ ❞✐stâ♥❝✐❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r
dT(A, B) =|xB−xA|+|yB−yA|
s❡rá s❡♠♣r❡ ♥ã♦✲♥❡❣❛t✐✈❛✱ ♣♦✐s
|xB−xA| ≥0
❡
|yB−yA| ≥0.
▲♦❣♦✱ dT ≥0✳
✐✐✳ dT(A, B) = 0⇔A =B
(⇒) ❙❡♥❞♦dT(A, B) = 0✱ t❡♠ s❡
|xB−xA|+|yB−yA|= 0 ⇒ |xB−xA|=−|yB−yA|.
❈♦♠♦ |xB−xA| ❡ |yB −yA| sã♦ ❛♠❜♦s ♥ã♦ ♥❡❣❛t✐✈♦s✱ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ só s❡
✈❡r✐✜❝❛ s❡
❆ ▼étr✐❝❛ ❞♦ ❚á①✐ ✷✺
❉❛í✱
xB =xA ❡yB =yA✳
▲♦❣♦✱ A=B✳
(⇐) ❙❡ A=B✱ ❡♥tã♦✱ (xA, yA) = (xB, yB)⇒ xA=xB ❡yA =yB✳ ❉❡ss❛ ❢♦r♠❛✱
s❡♥❞♦
dT(A, B) = |xB−xA|+|yB−yA|=|xA−xA|+|yA−yA|=|0|+|0|= 0.
▲♦❣♦✱ dT(A, B) = 0✳
P♦rt❛♥t♦✱ dT(A, B) = 0⇔A=B✳
✐✐✐✳ dT(A, B) = dT(B, A)
❈♦♠ ❡❢❡✐t♦✱
dT(A, B) = |xB−xA|+|yB−yA|
= | −(xA−xB)|+| −(yA−yB)|
= | −1|.|xA−xB|+| −1|.|yA−yB|
= 1.|xA−xB|+ 1.|yA−yB|
= |xA−xB|+|yA−yB|=dT(B, A)
✐✈✳ dT(A, C)≤dT(A, B) +dT(B, C)
❆♥t❡s ❞❡ ❞❡♠♦♥str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ r❡❝♦r❞❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿ Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡❥❛♠ ❛ ❡ ❜ ♥ú♠❡r♦s r❡❛✐s q✉❛✐sq✉❡r✱ ❡♥tã♦ ✭❝❢ ❬✶✶❪✮
|a+b| ≤ |a|+|b|.
❉❡♠✳✿ ❈♦♠♦ x≤ |x|✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ x∈R✱ s❡ a+b≥0✱ ❡♥tã♦
|a+b|=a+b≤ |a|+|b|.
❈❛s♦ ❝♦♥trár✐♦✱ s❡ a+b <0✱ ❡♥tã♦
❆ ▼étr✐❝❛ ❞♦ ❚á①✐ ✷✻
❱❛♠♦s à ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡iv ❞❡ ♠étr✐❝❛ ♣❛r❛ ❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐✿
❙❡❥❛
dT(A, C) = |xC −xA|+|yC −yA|
= |(xC −xB) + (xB−xA)|+|(yC −yB) + (yB−yA)|.
❯s❛♥❞♦ ❛ ♣r♦♣♦s✐çã♦ 3.1✱ t❡♠♦s q✉❡✱
dT(A, C) = |(xC −xB) + (xB−xA)|+|(yC−yB) + (yB−yA)|
≤ |xC −xB|+|xB−xA|+|yC −yB|+|yB−yA|
= (|xB−xA|+|yB−yA|) + (|xC −xB|+|yC−yB|)
= dT(A, B) +dT(B, C)
▲♦❣♦✱
dT(A, C)≤dT(A, B) +dT(B, C).
✷✼
✸ ❘❡❧❛❝✐♦♥❛♥❞♦ ❉✐stâ♥❝✐❛s
❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ q✉❡ ❛❝♦♥t❡❝❡ ♥❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❛❞✐❛♥❛✱ ❡♠ q✉❡ ❛ ♠❡♥♦r ❞✐s✲ tâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s é ❞❡t❡r♠✐♥❛❞❛ ❛♣❡♥❛s ♣❡❧❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❡ ❡①tr❡♠✐❞❛❞❡s ✐❣✉❛✐s ❛ ❡ss❡s ♣♦♥t♦s✱ ♥❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ♠í♥✐♠❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ tr❛ç❛♥❞♦ q✉❛❧q✉❡r ✉♠ ❞❡ ✈ár✐♦s ❝❛♠✐♥❤♦s✳
✸✳✶ ▼étr✐❝❛ ❞♦ ❚á①✐ ❡ ❈♦♠❜✐♥❛tór✐❛
❖❜s❡r✈❡♠♦s ❛ ✜❣✉r❛3.1✱ r❡♣r❡s❡♥t❛♥❞♦ ✉♠❛ ♣❛rt❡ ❞❡ ✉♠❛ ❝✐❞❛❞❡✱ ❡♠ q✉❡ ❛s q✉❛✲
❞r❛s sã♦ r❡♣r❡s❡♥t❛❞❛s ♣❡❧♦s q✉❛❞r❛❞♦s✱ t♦❞♦s ❞❡ ✐❣✉❛✐s ❞✐♠❡♥sõ❡s✱ ❡ ❡♥tr❡ ❡ss❛s q✉❛✲ ❞r❛s ❡stã♦ r✉❛s ❡ ❛✈❡♥✐❞❛s✳
❊①❡♠♣❧♦ ✸✳✶✳ ❯♠ ♣❛ss❛❣❡✐r♦ ❡♥tr❛ ❡♠ ✉♠ ❚á①✐ ♥♦ ♣♦♥t♦ A ❡ ❞❡s❝❡rá ♥♦ ♣♦♥t♦ O✱
s❡❣✉♥❞♦ ❛ ✜❣✉r❛ 3.1✳ ◗✉❛♥t♦s s❡r✐❛♠ ♦s ❝❛♠✐♥❤♦s ♠í♥✐♠♦s ♣♦ssí✈❡✐s❄
❋✐❣✉r❛ ✸✳✶✿ P❛rt❡ ❞❡ ✉♠❛ ❈✐❞❛❞❡ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
❖❜s❡r✈❡ q✉❡ ♦s ♣♦♥t♦s ♠❛r❝❛❞♦s sã♦ ❝r✉③❛♠❡♥t♦s ❞❡ r✉❛s ♦✉ ❛✈❡♥✐❞❛s✳ P♦r s❡ tr❛t❛r ❞❡ ✉♠❛ ❝✐❞❛❞❡✱ ♦ ❚á①✐ ♥ã♦ ♣♦❞❡rá s❡❣✉✐r ✉♠ ❝❛♠✐♥❤♦ r❡t♦ q✉❡ ✈á ❞❡ A ❛té O✱ ♣♦✐s
▼étr✐❝❛ ❞♦ ❚á①✐ ❡ ❈♦♠❜✐♥❛tór✐❛ ✷✽
❆ss✐♠ s❡♥❞♦✱ ♦❜s❡r✈❡ ♦ ❞✐❛❣r❛♠❛ ♥❛ ♣á❣✐♥❛ s❡❣✉✐♥t❡ ❞❡t❛❧❤❛♥❞♦ ♦s ❝❛♠✐♥❤♦s ♠í♥✐♠♦s q✉❡ ❡ss❡ ❚á①✐ ♣♦❞❡ s❡❣✉✐r✿
✭◆♦t❡ q✉❡ ❛ ❧✐♥❤❛ ❞♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ A → F → K → L → M → N → O é ♦
❝❛♠✐♥❤♦ ❞❡ s♦♠❛ AF ∪F K ∪KL∪LM ∪M N ∪N O✳ ❆s ❞❡♠❛✐s ❧✐♥❤❛s s❡❣✉❡♠ ♦
♠❡s♠♦ ♣❛❞rã♦✮✳
K v //L v //M v //N v //O //hhvvvv
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E h //J h //O //vvvvhh
P♦rt❛♥t♦✱ ♣❛r❛ ♦ ❚á①✐ ♣❡r❝♦rr❡r ❞❡A❛téO ❡❧❡ t❡♠15♣♦ssí✈❡✐s ❝❛♠✐♥❤♦s ♠í♥✐♠♦s✳
❊ s❡ ❡ss❡ ♠❡s♠♦ ❚á①✐ t✐✈❡ss❡ q✉❡ ♣❡r❝♦rr❡r15 q✉❛❞r❛s ♥❛ ❤♦r✐③♦♥t❛❧ ❡ 10q✉❛❞r❛s
♥❛ ✈❡rt✐❝❛❧ ❞❡♥tr♦ ❞❡ss❛ ❝✐❞❛❞❡❄✭ ✈❡❥❛ ✜❣✉r❛ 3.2✮
❉❡♠♦r❛rí❛♠♦s ❜❛st❛♥t❡ ♣❛r❛ ❡s❝r❡✈❡r t♦❞♦s ♦s ♣♦ssí✈❡✐s tr❛❥❡t♦s✳ ❉❛í ❛ ✐♥✈✐❛❜✐✲ ❧✐❞❛❞❡ ❞❡ r❡s♦❧✈❡r ❡ss❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ s❡♠♣r❡ ✉s❛♥❞♦ ♦ r❛❝✐♦❝í❝✐♦ ❞❛ ❡s❝r✐t❛ ❞♦s ❝❛♠✐♥❤♦s ♣❛r❛ só ❞❡♣♦✐s s❛❜❡r♠♦s q✉❛♥t♦s ❡❧❡s sã♦✳
❙❡ ♦❜s❡r✈❛r♠♦s ❜❡♠ ♦ ❡①❡♠♣❧♦ 3.1✱ t♦❞♦s ♦s ❝❛♠✐♥❤♦s tê♠ ❡①❛t❛♠❡♥t❡ 2 q✉❛❞r❛s
❤♦r✐③♦♥t❛✐s (h) ❡ 4 q✉❛❞r❛s ✈❡rt✐❝❛✐s (v)✱ ♠✉❞❛♥❞♦✲s❡ ❛♣❡♥❛s ❛ ♦r❞❡♠ ❞♦s ♠❡s♠♦s✳
▼étr✐❝❛ ❞♦ ❚á①✐ ❡ ❈♦♠❜✐♥❛tór✐❛ ✷✾
❋✐❣✉r❛ ✸✳✷✿ P❛rt❡ ❞❡ ✉♠❛ ❈✐❞❛❞❡ ✷ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
❙❛❜❡✲s❡ q✉❡ ❛ ❢ór♠✉❧❛ ❞❛ P❡r♠✉t❛çã♦ ❝♦♠ r❡♣❡t✐çã♦ ❡♠ q✉❡ ❞♦✐s ❡❧❡♠❡♥t♦s s❡ r❡♣❡t❡♠ é ✭❝❢ ❬✾❪✮✿
Pa,b
n =
n!
a!b!, ✭✸✳✶✮
♦♥❞❡ n é ♦ t♦t❛❧ ❞❡ ❡❧❡♠❡♥t♦s ♣❡r♠✉t❛❞♦s ❡ a ❡b sã♦ ♦s ♥ú♠❡r♦s ❞❡ ✈❡③❡s ❡♠ q✉❡
❝❛❞❛ ❡❧❡♠❡♥t♦ ❞✐st✐♥t♦ s❡ r❡♣❡t❡✳
❊ ♥❡ss❡ ❝❛s♦ ❝♦♠♦ ❡♠ q✉❛❧q✉❡r ♦✉tr♦ ❝✐t❛❞♦✱ a ❡ b s❡rã♦ ♥ú♠❡r♦s ❞❡ tr❡❝❤♦s ❤♦✲
r✐③♦♥t❛✐s ❡ ✈❡rt✐❝❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞♦ ♣❧❛♥♦✳ ❈♦♠♦ ♦s ♠♦✈✐♠❡♥t♦s só ♣♦❞❡♠ s❡r ♥❡ss❛s ❞✐r❡çõ❡s✱ ❡♥tã♦✱ a+b=n✳
❉❡ss❛ ❢♦r♠❛ ♦ ♣r♦❜❧❡♠❛3.1s❡r✐❛ ❢❛❝✐❧♠❡♥t❡ r❡s♦❧✈✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
P❛r❛ ♣❡r❝♦rr❡r ❛ ♠❡♥♦r ❞✐stâ♥❝✐❛ ♣♦ssí✈❡❧ ♦ ❚á①✐ t❡rá 2 tr❡❝❤♦s ❤♦r✐③♦♥t❛✐s ❡ 4
tr❡❝❤♦s ✈❡rt✐❝❛✐s✱ t♦t❛❧✐③❛♥❞♦ ✻ tr❡❝❤♦s✳ ❯t✐❧✐③❛♥❞♦ ❛ ❡q✉❛çã♦ 3.1✱ ❢❛③❡♠♦s ❞❛ s❡❣✉✐♥t❡
❢♦r♠❛✿
P2,4 6 =
6! 2!4! =
6.5.4! 2!4! =
6.5 2.1 =
30 2 = 15
P♦rt❛♥t♦✱ sã♦ ✶✺ ♦s ❝❛♠✐♥❤♦s ❞❡ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ ♣♦ssí✈❡❧✱ ❝♦♥✜r♠❛♥❞♦ ❛ q✉❛♥t✐✲ ❞❛❞❡ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❡♥❝♦♥tr❛❞❛s✳
❆ss✐♠ ✜❝❛ ✏❢á❝✐❧✑ r❡s♦❧✈❡r ♦ ❝❛s♦ ❞❡15 ♠♦✈✐♠❡♥t♦s ❤♦r✐③♦♥t❛✐s ❡ 10✈❡rt✐❝❛✐s✿
P15,10 25 =
25!
15!10! = 3268760 ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s✳
▼étr✐❝❛ ❞♦ ❚á①✐ ❡ ❈♦♠❜✐♥❛tór✐❛ ✸✵
❊①❡♠♣❧♦ ✸✳✷✳ ❈♦♥s✐❞❡r❡ ❛ ✜❣✉r❛ 3.3 ❡♠ q✉❡ ❝❛❞❛ q✉❛❞r❛❞♦ tr❛❝❡❥❛❞♦ é ✉♠❛ q✉❛❞r❛
❞❡ ❝❡rt❛ ♣❛rt❡ ❞❡ ✉♠ ❜❛✐rr♦✿
❋✐❣✉r❛ ✸✳✸✿ P❛rt❡s ❞❡ ✉♠ ❇❛✐rr♦ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
◗✉❛♥t♦s sã♦ ♦s ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s ♣❛r❛ ✉♠❛ ♣❡ss♦❛ ♣❡r❝♦rr❡r ❞❛ ❡sq✉✐♥❛ C ❛té ❛
❡sq✉✐♥❛ D❄ ❊ ❞❛ ❡sq✉✐♥❛ E ❛té ❛ ❡sq✉✐♥❛F❄
❙♦❧✉çã♦ ✸✳✶✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❜❛st❛ ✉t✐❧✐③❛r♠♦s ❛ ❡q✉❛çã♦ 3.1✳
✐✳ ❉❡ A ❛ B ❡❧❛ ♣❡r❝♦rr❡rá 3 tr❡❝❤♦s ❤♦r✐③♦♥t❛✐s ❡ 1 ✈❡rt✐❝❛❧✳ ❆ss✐♠✱
P3,1 4 =
4! 1!3! =
4.3! 3!1! =
4 1 = 4.
❙❡rã♦ 4 ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s✳
✐✐✳ ❉❡ C ❛ D ❡❧❛ ♣❡r❝♦rr❡rá 6 tr❡❝❤♦s ❤♦r✐③♦♥t❛✐s ❡ 4 ✈❡rt✐❝❛✐s✳ ❆ss✐♠✱
P6,4 10 =
10! 6!4! =
10.9.8.7.6! 6!4.3.2.1 =
5040
24 = 210.
❙❡rã♦ 210 ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s✳
◆♦t❛✲s❡✱ ♣❡❧♦s ❡①❡♠♣❧♦s✱ q✉❡ s❡♥❞♦ ❞♦✐s ♣♦♥t♦s A ❡ B✱ q✉❛♥t♦ ♠❛✐s ♣♦♥t♦s ♦s
s❡♣❛r❡♠ ❤♦r✐③♦♥t❛❧ ❡ ✈❡rt✐❝❛❧♠❡♥t❡✱ ♦ ♥ú♠❡r♦ ❞❡ ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s ❞❡A ❛B t❛♠❜é♠
❈♦♥✜❛♥❞♦ ♥❛ ▼étr✐❝❛ ❞♦ ❚á①✐ ✸✶
✸✳✷ ❈♦♥✜❛♥❞♦ ♥❛ ▼étr✐❝❛ ❞♦ ❚á①✐
❊♠ s❡ tr❛t❛♥❞♦ ❞❡ ❛♣❧✐❝❛çã♦ ❝♦t✐❞✐❛♥❛✱ ❡♠ q✉❡ ❛s ♣❡ss♦❛s ♣r❡❝✐s❛♠ t❡r ✉♠❛ ♥♦çã♦ ♠❛✐s ✏❝♦rr❡t❛✑ ❞❛ ❞✐stâ♥❝✐❛ q✉❡ s❡♣❛r❛ ❞♦✐s ❧✉❣❛r❡s ❡♠ ✉♠❛ ❝✐❞❛❞❡ ♦✉ ✉♠ ❜❛✐rr♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠♦ ♦ s❡✉ ✐♥t❡r❡ss❡✱ ❛ ▼étr✐❝❛ ❞♦ ❚á①✐ é ♠❛✐s ❝♦♥✜á✈❡❧ q✉❡ ❛ ▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛✳ ❱❡❥❛ ✉♠ ❡①❡♠♣❧♦✿
❊①❡♠♣❧♦ ✸✳✸✳ ❏éss✐❝❛ ❡stá ♥❛ ❡s❝♦❧❛ ❧♦❝❛❧✐③❛❞❛ ♥❛ ❡sq✉✐♥❛ A(1,1)✳ ❙❡✉ ♣❛✐ tr❛❜❛❧❤❛
♥♦ ♣ré❞✐♦ ❧♦❝❛❧✐③❛❞♦ ♥❛ ❡sq✉✐♥❛ B(2,6) ❡ s✉❛ ♠ã❡ ❡stá ❡♠ s❡✉ ❙❛❧ã♦ ❞❡ ❇❡❧❡③❛✱ q✉❡
✜❝❛ ❧♦❝❛❧✐③❛❞♦ ♥❛ ❡sq✉✐♥❛C(5,4)✱ ❝♦♠♦ ✐❧✉str❛❞❛ ♥❛ ✜❣✉r❛3.4✳ ❙❛❜❡♥❞♦ q✉❡ ♣❛✐ ❡ ♠ã❡
s❛❡♠ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❞♦s r❡s♣❡❝t✐✈♦s ❧♦❝❛✐s✱ ❝♦♠ ❛ ♠❡s♠❛ ✈❡❧♦❝✐❞❛❞❡ ❡ ❝♦♠ ♠❡s♠♦ t✐♣♦ ❞❡ trâ♥s✐t♦✱ ♣❛r❛ q✉❛❧ ❞♦s ❞♦✐s é ♠❛✐s ✈✐á✈❡❧ ❜✉s❝❛r ❏éss✐❝❛❄
❋✐❣✉r❛ ✸✳✹✿ ❚r❛❜❛❧❤♦ ✲ ❊s❝♦❧❛ ✲ ❙❛❧ã♦ ❋♦♥t❡✿ ●❡♦❣❡❜r❛
❙♦❧✉çã♦ ✸✳✷✳ ✐✳ P❡❧❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✿ ❉✐stâ♥❝✐❛ ❞❛ ❊s❝♦❧❛ ❛♦ ❚r❛❜❛❧❤♦ ❞♦ P❛✐✿
dE(A, B) =
p
(2−1)2
+ (6−1)2
=√1 + 25 =√26∼= 5,10
❉✐stâ♥❝✐❛ ❞❛ ❊s❝♦❧❛ ❛♦ ❙❛❧ã♦ ❞❛ ▼ã❡✿
dE(A, C) =
p
(5−1)2+ (4
−1)2 =√16 + 9 =√25 = 5
❘❡❧❛çã♦ ❡♥tr❡ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ❡ ❛ ❉✐stâ♥❝✐❛ ♥❛ ▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛ ✸✷
❉✐stâ♥❝✐❛ ❞❛ ❊s❝♦❧❛ ❛♦ ❚r❛❜❛❧❤♦ ❞♦ P❛✐✿
dT(A, B) = |2−1|+|6−1|=|1|+|5|= 6
❉✐stâ♥❝✐❛ ❞❛ ❊s❝♦❧❛ ❛ ❙❛❧ã♦ ❞❛ ▼ã❡✿
dT(A, C) =|5−1|+|4−1|=|4|+|3|= 7
▲♦❣♦✱ é ♠❛✐s ✈✐á✈❡❧ q✉❡ ♦ P❛✐ ✈á ❜✉s❝❛r ❏éss✐❝❛ ♥❛ ❊s❝♦❧❛✳
❖❜s❡r✈❡ q✉❡ ♣♦r ♠❛✐s q✉❡ ❛ ♠ã❡ ❡st❡❥❛ ✏❡✉❝❧✐❞✐❛♥❛♠❡♥t❡✑ ♠❛✐s ♣❡rt♦ ❞❛ ❊s❝♦❧❛ q✉❡ ♦ ♣❛✐ s❡ ❛♠❜♦s ♣✉❞❡ss❡♠ ❞✐r✐❣✐r ❡♠ ❧✐♥❤❛ r❡t❛✱ ♦ ♣❛✐ ❝❤❡❣❛r❛ ♣r✐♠❡✐r♦✱ ♣❡❧♦ ❢❛t♦ q✉❡ ♦ trá❢❡❣♦ ❞❡ ✈❡í❝✉❧♦s s❡ ❞á ♣♦r r✉❛s ❡✴♦✉ ❛✈❡♥✐❞❛s✳
✸✳✸ ❘❡❧❛çã♦ ❡♥tr❡ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ❡ ❛ ❉✐stâ♥❝✐❛ ♥❛
▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛
❈♦♥✈é♠ ♥♦t❛r q✉❡dE(A, B)≤dT(A, B)✱ ✈❛❧❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛♣❡♥❛s q✉❛♥❞♦ ❛♠❜♦s
♦s ♣♦♥t♦s ❡st✐✈❡r❡♠ ♥❛ ♠❡s♠❛ ❤♦r✐③♦♥t❛❧ ♦✉ ♥❛ ♠❡s♠❛ ✈❡rt✐❝❛❧✳ P♦❞❡✲s❡ ❡♥tã♦ ❛✜r♠❛r q✉❡✿
❆ ❚á①✐✲❞✐stâ♥❝✐❛ é s❡♠♣r❡ ♠❛✐♦r ♦✉ ✐❣✉❛❧ q✉❡ ❛ ❞✐stâ♥❝✐❛ ♥❛ ▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛✳ ❬✶✵❪
❱❛♠♦s ✈❡r✐✜❝❛r t❛❧ ❛✜r♠❛çã♦✳
❱❡r✐✜çã♦ ✸✳✶✳ ❙❡❥❛♠ ♦s ♣♦♥t♦s A(xA, yA) ❡ B(xB, yB)✳ ❉❡ ❢❛t♦✱ é ✈❡r❞❛❞❡ q✉❡
2.|xB−xA|.|yB−yA| ≥0, ✭✸✳✷✮
♣❡❧❛ ❞❡✜♥✐çã♦ 3.1✱
❡ t❛♠❜é♠ é ✈❡r❞❛❞❡ q✉❡
(xB−xA)
2
+ (yB−yA)
2
≥0. ✭✸✳✸✮
❙♦♠❛♥❞♦ ❛ ❡①♣r❡ssã♦ 3.3 ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ 3.2✱ t❡♠✲s❡ q✉❡
(xB−xA)
2
+ (yB−yA)
2
+ 2.|xB−xA|.|yB−yA| ≥(xB−xA)
2
+ (yB−yA)
2
✭✸✳✹✮ ❖ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❛ ✐♥❡q✉❛çã♦3.4 é ✉♠ tr✐♥ô♠✐♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❝✉❥♦s t❡r♠♦s
sã♦ |xB−xA|❡ |yB−yA|✳ ❊♥tã♦✱ ❛ ❡ss❛ ✐♥❡q✉❛çã♦ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
(|xB−xA|+|yB−yA|)
2
≥(xB−xA)
2
+ (yB−yA)
2
❘❡❧❛çã♦ ❡♥tr❡ ❛ ❚á①✐✲❉✐stâ♥❝✐❛ ❡ ❛ ❉✐stâ♥❝✐❛ ♥❛ ▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛ ✸✸
❈♦♠♦ ♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ✐♥❡q✉❛çã♦ 3.5 sã♦ ♥ã♦✲♥❡❣❛t✐✈♦s✱ ❛♦ s❡ ❡①tr❛✐r ❛s r❛í③❡s
q✉❛❞r❛❞❛s ❞❡ ❛♠❜♦s✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❝♦♥t✐♥✉❛ ✈á❧✐❞❛✱ ♦✉ s❡❥❛✱
p
(|xB−xA|+|yB−yA|)2 ≥
p
(xB−xA)2+ (yB−yA)2 ✭✸✳✻✮
▲♦❣♦✱
|xB−xA|+|yB−yA| ≥
p
(xB−xA)2+ (yB−yA)2 ✭✸✳✼✮
P♦rt❛♥t♦✱
dT(A, B)≥dE(A, B).
✸✹
✹ ❆ ❚á①✐✲❊❧✐♣s❡
◆❡st❡ ❝❛♣ít✉❧♦ ❞❡✜♥❡✲s❡ ❛ ❚á①✐✲❊❧✐♣s❡✱ ♥❛ q✉❛❧ ❛ ❞✐stâ♥❝✐❛✱ ❞❡♥♦♠✐♥❛❞❛ ❚á①✐✲ ❉✐stâ♥❝✐❛✱ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ●❡♦♠❡tr✐❛ ❞♦ ❚á①✐ ♦✉ s✐♠♣❧❡s♠❡♥t❡✱ ▼étr✐❝❛ ❞♦ ❚á①✐✳ ❆q✉✐✱ ❛s r❡❣✐õ❡s ✐♥t❡r♥❛s às ❚á①✐✲❊❧✐♣s❡s sã♦ ❜❡♠ ❞✐❢❡r❡♥t❡s ❞❛q✉❡❧❛s ♦✈❛✐s ✐♥t❡r♥❛s à ❊❧✐♣s❡ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❝♦♠♦ ✈❡r❡♠♦s✳
❆ ❡q✉❛çã♦ ❞❛ ❚á①✐✲❊❧✐♣s❡ t❛♠❜é♠ t❡♠ ✉♠ ❢♦r♠❛t♦ ❞✐❢❡r❡♥t❡ ❞♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❊❧✐♣s❡ ❊✉❝❧✐❞✐❛♥❛✱ ❝♦♠♦ é ❞❡ s❡ ❡s♣❡r❛r✳ ❋❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞❡ss❛ ❡q✉❛çã♦ ❡ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❣rá✜❝♦✳
✹✳✶ ❊❧✐♣s❡ ❊✉❝❧✐❞✐❛♥❛ ❡ ❚á①✐✲❊❧✐♣s❡
❆ ❊❧✐♣s❡ ♥❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❊✉❝❧✐❞✐❛♥❛ é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❉❡✜♥✐çã♦ ✹✳✶✳ ❯♠❛ ❡❧✐♣s❡ εE
F1,F2 ❞❡ ❢♦❝♦s F1 ❡ F2 é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦
❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❛ F1 ❡ F2 é ✐❣✉❛❧ ❛ ✉♠❛ ❝♦♥st❛♥t❡ 2a > 0✱ ♠❛✐♦r q✉❡ ❛
❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s 2c≥0✳ ❖✉ s❡❥❛✱ s❡♥❞♦ 0≤c < a ❡ d(F1, F2) = 2c ✭❝❢ ❬✺❪✮✱
εE
F1,F2 ={P|d(P, F1) +d(P, F2) = 2a} ✭✹✳✶✮
❱❡❥❛ ❛ ✜❣✉r❛ 4.1 q✉❡ r❡tr❛t❛ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❊❧✐♣s❡ s❡❣✉♥❞♦ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧í✲
t✐❝❛✭❝♦♠ ❞✐stâ♥❝✐❛ ❊✉❝❧✐❞✐❛♥❛✮✳
