GEODÉSICAS:SUAS EQUAÇÕES E ALGUMAS APLICAÇÕES

❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ ❉❊ ❋❊■❘❆ ❉❊ ❙❆◆❚❆◆❆

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

❉✐ss❡rt❛❝ã♦ ❞❡ ▼❡str❛❞♦

●❊❖❉➱❙■❈❆❙✿ ❙❯❆❙ ❊◗❯❆➬Õ❊❙ ❊ ❆▲●❯▼❆❙

❆P▲■❈❆➬Õ❊❙

❆r✐❛♥❛ ❈♦r❞❡✐r♦ ❞❡ ❆♠♦r✐♠ ▼❛t♦s

❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ ❉❊ ❋❊■❘❆ ❉❊ ❙❆◆❚❆◆❆

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

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

●❊❖❉➱❙■❈❆❙✿❙❯❆❙ ❊◗❯❆➬Õ❊❙ ❊ ❆▲●❯▼❆❙

❆P▲■❈❆➬Õ❊❙

❆r✐❛♥❛ ❈♦r❞❡✐r♦ ❞❡ ❆♠♦r✐♠ ▼❛t♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡s✲ tr❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛✲ ❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥✲ ❝✐❛s ❊①❛t❛s✱ ❯❊❋❙✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ▼s❝✳ ❈r✐st✐❛♥♦ ❍❡♥r✐q✉❡ ▼❛s❝❛r❡♥❤❛s

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s✱ ♣♦r ❡st❛r s❡♠♣r❡ ♠❡ ❣✉✐❛♥❞♦✱ ❛❜❡♥ç♦❛♥❞♦✱ ❞❡ ♦♥❞❡ ♥✉♥❝❛ ♠❡ ❢❛❧t❛ ❢♦rç❛s ♥❡✲ ❝❡ssár✐❛s ♣❛r❛ ✈❡♥❝❡r ❛s ❞✐✜❝✉❧❞❛❞❡s ❞❛ ✈✐❞❛✱ ♣♦✐s ♠❡ ❞á ❛ s❛❜❡❞♦r✐❛ ♣❛r❛ ❡s❝♦❧❤❡r tr✐❧❤❛r ♦s ♠❡❧❤♦r❡s ❝❛♠✐♥❤♦s✳

❆♦ ♠❡✉ ❡s♣♦s♦ ▲✉❝✐❛♥♦ ❡ ♠✐♥❤❛ ✜❧❤❛ ●❧❡♥❞❛ ♣❡❧♦ ❝❛r✐♥❤♦✱ ❛♠♦r✱ ❝♦♠♣r❡❡♥sã♦✱ ♣♦r ♠❡ ❞❛r❡♠ ❢♦rç❛s ❡ s❡r ♠♦t✐✈❛çã♦ ♣❛r❛ ♣r♦ss❡❣✉✐r ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s ❧✉t❛s✱ ❡ ❝♦♥s❡❣✉✐r ✈❡♥❝ê✲❧❛s✳

❆♦s ♠❡✉s ♣❛✐s ❘❛♠♦s ❡ ▼❡r② ♣♦r ❢❛③❡r❡♠ ❞❡ t♦❞❛s ❛s ♠✐♥❤❛s ❧✉t❛s✱ s✉❛s ❧✉t❛s✳P♦r ❡st❛r❡♠ s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✱ ♣♦r s❡r❡♠ r❡s♣♦♥sá✈❡✐s ♣♦r t✉❞♦ ♦ q✉❡ s♦✉✳❊ ❛ ♠✐♥❤❛ ✐r♠ã ❆❞r✐❛♥❛ ♣♦r ❡st❛r s❡♠♣r❡ ♣r♦♥t❛ ❛ ❛❥✉❞❛r✱ ♥♦ q✉❡ ♣r❡❝✐s♦✳

❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦✱ ✐♥❝❡♥t✐✈♦✱ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡✲ ❧♦s ❡♥s✐♥❛♠❡♥t♦s tã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞♦ ❝✉rs♦ ✭P❘❖❋▼❆❚✮✱ ❛❧❣✉♥s ❡♠ ❡s♣❡❝✐❛❧ ♣♦✐s s❡ t♦r♥❛r❛♠ ♠❛✐s q✉❡ ❝♦❧❡❣❛s✱❏♦✐❧♠❛✱ ❘♦s✐♣❧é✐❛✱ ❆♥❛tá❧✐❛✱ ❘♦s✐✈❛❧❞♦✱ ❚ê♥✐✈❛♠✱ ❡ t♦❞♦s ♦s ♦✉tr♦s ♣❡❧♦ ❝❛r✐♥❤♦ ❞❡ s❡♠♣r❡✱ ♦❜r✐❣❛❞♦ ♣♦r ✈✐❜r❛r❡♠ ❝♦♠✐❣♦ ❛ ❝❛❞❛ ✈✐tór✐❛✱ ♣♦rq✉❡ s❛❜❡♠ ❝♦♠♦ ♥✐♥❣✉é♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❝❛❞❛ ♣❛ss♦ ❞❛❞♦ ♥❡ss❛ ❥♦r♥❛❞❛✳

❆♦ ♠❡✉ ❝♦❧❡❣❛ ❞❡ ❣r❛❞✉❛çã♦✱ ❛♠✐❣♦✱ ❝✉♠♣❛❞r❡✱ ✐r♠ã♦ ❉✐❧❝❡s❛r ❉❛♥t❛s ♣❡❧♦ ❛♣♦✐♦✱ ✐♥✲ ❝❡♥t✐✈♦ ❡ ❛❥✉❞❛ ♣❛r❛ ❝♦♥❝r❡t✐③❛çã♦ ❞❡s❞❡ tr❛❜❛❧❤♦✳

➚ ❝❛♣❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ✭❜♦❧s❛ ❞❡ ❡st✉❞♦s✮✳

❊♠ ❡s❡♣❝✐❛❧ ❛♦ ♠❡✉ ♣r♦❢❡ss♦r ♦r✐❡♥t❛❞♦r ❈r✐st✐❛♥♦ ▼❛s❝❛r❡♥❤❛s ♣❡❧♦ ❛♣♦✐♦✱ ❡♥s✐♥❛♠❡♥✲ t♦s✱ ♣❛❝✐ê♥❝✐❛✱ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ s❡♠♣r❡ ❜✉s❝❛♥❞♦ ♦ ♠❡❧❤♦r ♣❛r❛ ❝♦♥str✉çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❙❡r❡✐ ❡t❡r♥❛♠❡♥t❡ ❣r❛t❛✱ ❛ ✈♦❝ê t♦❞♦ ♦ ♠❡✉ r❡s♣❡✐t♦ ❡ ❛❞♠✐r❛çã♦✳❉❡✉s ♦ ❛❜❡♥ç♦❡✳

❘❡s✉♠♦

❆ ✜♥❛❧✐❞❛❞❡ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❡st✉❞❛r ❞❡ ♠♦❞♦ ❞❡t❛❧❤❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❣❡♦❞és✐❝❛ s♦❜r❡ s✉♣❡r❢í❝✐❡s r❡❣✉❧❛r❡s S _⊂R3✱ ❛♥❛❧✐s❛♥❞♦ s✉❛s ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s ❛♥❛❧ít✐❝❛s✳ ❱✐♠♦s

q✉❡ ❛s ❣❡♦❞és✐❝❛s sã♦ ✐♥trí♥s❡❝❛s✱ ✐st♦ é✱ ❞❡♣❡♥❞❡♠ ❛♣❡♥❛s ❞❛ ✶❛ _{❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧✱ ♦✉}

❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ sã♦ ♣r❡s❡r✈❛❞❛s ♣♦r ✐s♦♠❡tr✐❛✳ ❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡st❛❝á✈❡❧ ❞❛s ❣❡♦❞és✐❝❛s é q✉❡ ❡❧❛s sã♦ ❝✉r✈❛s q✉❡ ♠✐♥✐♠✐③❛♠ ❧♦❝❛❧♠❡♥t❡ ❞✐stâ♥❝✐❛s ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿❈✉r✈❛s✱ s✉♣❡r❢í❝✐❡s✱ ❝✉r✈❛t✉r❛✱ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛✱ ♠❡♥♦r ❝❛♠✐✲ ♥❤♦✳

❆❜str❛❝t

❚❤❡ ❛✐♠ ♦❢ t❤✐s ✇♦r❦ ✇❛s st✉❞② ✐♥ ❣r❡❛t ❞✐t❛✐❧s t❤❡ ❝♦♥❝❡♣t ♦❢ ❣❡♦❞❡s✐❝s ♦♥ s✉r❢❛❝❡s

S _⊂R3✱ ❛♥❛❧②③✐♥❣ ✐ts ♠❛✐♥❧② ❛♥❛❧②t✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❲❡ s❛✇ t❤❛t ❣❡♦❞❡s✐❝s ❛r❡ ✐♥tr✐✲

s✐❝s✱ t❤❛t✬s✱ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ✜rst ❢✉♥❞❛♠❡♥t❛❧ ❢♦r♠✱ ♦r t❤❡ s❛♠❡ ✇❛②✱ ❛r❡ ♣r❡s❡r✈❡❞ ❜② ✐s♦♠❡tr②✳ ❆♥♦t❤❡r ♦✉tst✉❞✐♥❣ ♣r♦♣❡rt② ♦❢ ❣❡♦❞❡s✐❝s ✐s t❤❛t ✐ts ❛r❡ ❝✉r✈❡s ✇❤✐❝❤ ♠✐♥✐♠✐③❡ ❧♦❝❛❧❧② ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦♥ ❛ s✉r❢❛❝❡✳

❑❡②✇♦r❞s✿❈✉r✈❡s✱ s✉r❢❛❝❡s✱ ❝✉r✈❛t✉r❡s✱ ❝✉r✈❛t✉r❡ ❣❡♦❞és✐❝✱ s❤♦rt❡st ♣❛t❤✳

❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✐

❘❡s✉♠♦ ✐✐✐

❆❜str❛❝t ✈

❙✉♠ár✐♦ ✶

■♥tr♦❞✉çã♦ ✸

✶ ❈✉r✈❛s ♥♦ ❊s♣❛ç♦ ✺

✶✳✶ ❈✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❈✉r✈❛t✉r❛ ❡ ❋ór♠✉❧❛s ❞❡ ❋r❡♥❡t✲❙❡rr❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷ ❙✉♣❡r❢í❝✐❡s r❡❣✉❧❛r❡s ✶✼

✷✳✶ P❛r❛♠❡tr✐③❛çã♦✱ ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ❡ ✈❡t♦r t❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✶ ❈✉r✈❛s ❈♦♦r❞❡♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✷ P❧❛♥♦ ❚❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ Pr✐♠❡✐r❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸ ❙❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹ ❈✉r✈❛t✉r❛s ♥❛ s✉♣❡r❢í❝✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✹✳✶ ❈✉r✈❛t✉r❛ ♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✸ ●❡♦❞és✐❝❛s ✸✾

✸✳✶ ❊q✉❛çõ❡s ●❡♦❞és✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✷ Pr❡s❡r✈❛çã♦ ❞❛ ●❡♦❞és✐❝❛ ♣♦r ■s♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✸ ●❡♦❞és✐❝❛s ❡ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✹ ❖ ♠❡♥♦r ❝❛♠✐♥❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✶ ❊①❡r❝í❝♦s ♣r♦♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✹✳✷ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✸

■♥tr♦❞✉çã♦

▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❛ ♥♦çã♦ ❞❡ ❣❡♦❞és✐❝❛ ♠♦❞❡❧❛ ❡♠ ✉♠ ❝❡rt♦ s❡♥t✐❞♦ ❛ r❡s♣♦st❛ ❛ ✉♠❛ ♣❡r❣✉♥t❛ ♥❛t✉r❛❧✿ q✉❛❧ é ❛ ❝✉r✈❛ q✉❡ ❧✐❣❛ ❞♦✐s ♣♦♥t♦s ❡♠ ✉♠ ❡s♣❛ç♦ ♣♦r ❛r❝♦s ❝✉❥♦ ❝♦♠♣r✐♠❡♥t♦ é ♦ ♠❡♥♦r ♣♦ssí✈❡❧❄ ❊ss❛ ♣❡r❣✉♥t❛ ❥á ❡r❛ ❢♦r♠✉❧❛❞❛ ♣❡❧♦s ❣r❡❣♦s ❛♥t✐❣♦s q✉❡ ❞❡❞✉③✐r❛♠ ❛ ♣❛rt✐r ❞❡ ♦❜s❡r✈❛çõ❡s ❛str♦♥ô♠✐❝❛s q✉❡ ❛ t❡rr❛ ❞❡✈✐❛ s❡r r❡❞♦♥❞❛✳ ❆ ❣❡♦♠❡tr✐❛ ❝❧áss✐❝❛✱ ♥♦ ♣❧❛♥♦✱ ❡st✉❞❛ ❝♦♠ ❣r❛♥❞❡s ❞❡t❛❧❤❡s r❡❧❛çõ❡s ❡♥tr❡ ♣♦♥t♦s✱ r❡t❛s ❡ ❝ír❝✉❧♦s✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ♥♦çã♦ ❞❡ ✧r❡t✐❞ã♦✧✭✉♠❛ ♥♦çã♦ ❣❧♦❜❛❧✮ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝ír❝✉❧♦ ✭✉♠❛ ❝✉r✈❛ ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❣❧♦❜❛❧✮ ♥ã♦ ❢❛③ s❡♥t✐❞♦ ♥❛ t❡♦r✐❛ ❧♦❝❛❧ ❞❛s s✉♣❡r❢í❝✐❡s✳

❊✉❝❧✐❞❡s ❞❡✜♥✐✉ ✉♠❛ r❡t❛ ❝♦♠♦ ✉♠ ♦❜❥❡t♦ s❡♠ ❧❛r❣✉r❛ ❛❧❣✉♠❛ ❡ ✉♠❛ ❧✐♥❤❛ r❡t❛ ❝♦♠♦ ✧✉♠❛ ❧✐♥❤❛ ❛ q✉❛❧ ❡♥❝♦♥tr❛✈❛✲s❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♠ ♦s ♣♦♥t♦s s♦❜r❡ s✐ ♠❡s♠❛✧✳

❊st❛s ❞❡✜♥✐çõ❡s ♥ã♦ ♥♦s ❛❥✉❞❛♠ ❛ ❣❡♥❡r❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ✉♠❛ ❧✐♥❤❛ r❡t❛ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❣❡r❛❧✳ ❈♦♥t✉❞♦✱ é s❛❜✐❞♦ q✉❡ ❞❛❞♦s ❞♦✐s ♣♦♥t♦sP ❡Q❡♠ Rn_{✱ ✉♠ s❡❣♠❡♥t♦ ❞❡} r❡t❛ ❝♦♥❡❝t❛♥❞♦ P ❛Q ❢♦r♥❡❝❡ ♦ ❝❛♠✐♥❤♦ ❞❡ ♠❡♥♦r ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡st❡s ❞♦✐s ♣♦♥t♦s✳

❙♦❜ ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛rS _⊂ Rn _{q✉❡ ♥ã♦ é ♣❧❛♥❛✱ ♠❡s♠♦ ❛ ♥♦çã♦ ❞❡ ❞✐stâ♥❝✐❛ ❡♠}

S ❡♥tr❡ ❞♦✐s ♣♦♥t♦s P ❡Q ♣♦ss✉❡ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡✱ ❞❡s❞❡ q✉❡ ♥ã♦ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡

✉♠❛ ❧✐♥❤❛ r❡t❛ ❝♦♥❡❝t❛♥❞♦ P ❛ Q ❡st❛rá ❝♦♥t✐❞❛ ❡♠ S✳ ◆♦ ❡♥t❛♥t♦✱ ❝♦♠♦ ❞✐s♣♦♠♦s ❞♦

❝♦♥❝❡✐t♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ é ♣♦ssí✈❡❧ ❢❛❧❛r♠♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝✉r✈❛s❀ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❞✐stâ♥❝✐❛ s♦❜r❡ S ❡♥tr❡ P ❡ Q ❝♦♠♦✿

✐♥❢ ④❝♦♠♣r✐♠❡♥t♦ ❞❡ α:α é ✉♠❛ ❝✉r✈❛ s♦❜r❡ S ❝♦♥❡❝t❛♥❞♦P ❡ Q⑥✳

❆ss✐♠ ♣♦❞❡♠♦s t♦♠❛r ✉♠❛ ♣r✐♠❡✐r❛ ❢♦r♠✉❧❛çã♦ ✐♥t✉✐t✐✈❛ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✧r❡t✐❞ã♦✧s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r S ❛ s❡❣✉✐♥t❡✿ ❯♠❛ ❝✉r✈❛ α s♦❜r❡ S é ✧r❡t❛✧s❡ ♣❛r❛ t♦❞♦s ♦s ♣❛r❡s

❞❡ ♣♦♥t♦s s♦❜r❡α✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s P ❡Qé ✐❣✉❛❧ ❛ ❞✐stâ♥❝✐❛ P Q⌢

❡♥tr❡ ❡❧❡s✳

◆❡st❡ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈❡♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ s♦❜r❡ ❣❡♦❞és✐❝❛s✱ q✉❡ sã♦ ❝✉r✈❛s ❡s♣❡❝✐❛✐s s♦❜r❡ ❛s s✉♣❡r❢í❝✐❡s✱ ❡ q✉❡ ❣❡♥❡r❛❧✐③❛♠ ♥♦ ❛♠❜✐❡♥t❡ R3 ♦ ❝♦♥❝❡✐t♦ ❞❡ r❡t✐❞ã♦✱

❛❧é♠ ❞❡ t❛♠❜é♠ ❡st❛r ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ àq✉❡❧❛s ❝✉r✈❛s q✉❡ ♠✐♥✐♠✐③❛♠ ✭❡♠ ✉♠ ❝❡rt♦ s❡♥t✐❞♦✮ ❞✐stâ♥❝✐❛s ❡♥tr❡ ♣♦♥t♦s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡♥tr❡ q✉❛✐sq✉❡r ♦✉tr❛s ❝✉r✈❛s s✐t✉❛❞❛s ♥❛ s✉♣❡r❢í❝✐❡ ❡♠ ❡st✉❞♦✳

❈❛♣ít✉❧♦ ✶

❈✉r✈❛s ♥♦ ❊s♣❛ç♦

■♥✐❝✐❛r❡♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠ ♣❡♥s❛♠❡♥t♦ ♣r✐♠ár✐♦ ❞♦ q✉❡ s❡❥❛ ✉♠❛ ❝✉r✈❛ ♥♦s ❞✐❢❡✲ r❡♥t❡s r❛♠♦s ❞❛ ❣❡♦♠❡tr✐❛✱ ♣❛r❛ q✉❡ ❛ ♣❛rt✐r ❞❡ss❡ ♣♦ss❛♠♦s ❡♥t❡♥❞❡r ♠❡❧❤♦r ❛ ❞❡✜♥✐çã♦ q✉❡ s❡ ❡♥❝❛✐①❛ ♥♦ q✉❡ ♣r❡t❡♥❞❡♠♦s tr❛❜❛❧❤❛r✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♠❡❞✐❞❛ q✉❡ ♦ t❡①t♦ s❡ ❞❡✲ s❡♥✈♦❧✈❡ ♦ ❧❡✐t♦r ♣♦❞❡rá ♥♦t❛r ❛ ♣r❡❞✐❧❡çã♦ ♣❡❧❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞❛ ❜❛s❡ ❞♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧ q✉❡ é ♦ ❢♦❝♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ P❛r❛ t❛♥t♦✱ ❡st✉❞❛r❡♠♦s ❛s ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s r❡❣✉❧❛r❡s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✱ ❝♦♥✲ ❝❡✐t✉❛♥❞♦ ❡❧❡♠❡♥t♦s ❝♦♠♦ ✈❡t♦r t❛♥❣❡♥t❡✱ ❝✉r✈❛t✉r❛ ❡ tr✐❡❞r♦ ❞❡ ❋r❡♥❡t✲❙❡rr❡t✳ ❈❛s♦ ♦ ❧❡✐t♦r q✉❡✐r❛ ❛♣r♦❢✉♥❞❛r✲s❡ ♥❛s ❞❡✜♥✐çõ❡s ✉t✐❧✐③❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦✱ ♣♦❞❡rá ❝♦♥s✉❧t❛r ❛s r❡❢❡rê♥❝✐❛s ❬✶❪✱ ❬✸❪✱ ❬✺❪✱ ❬✼❪✱ ❬✶✵❪ ❡ ❬✶✶❪✳

✶✳✶ ❈✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s r❡❣✉❧❛r❡s

■♥t✉✐t✐✈❛♠❡♥t❡✱ ♣❡♥s❛r ♥✉♠❛ ❝✉r✈❛ ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❞✐♠❡♥sã♦ 1 é ♦ q✉❡

❣♦st❛rí❛♠♦s ❞❡ ❢❛③❡r✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❞❡s❡♥❤❛r ✜❣✉r❛s ❝♦♠ ✉♠ ú♥✐❝♦ tr❛ç♦✱ s❡♠ t✐r❛r ♦ ❧á♣✐s ❞♦ ♣❛♣❡❧ ✭❋✐❣✉r❛✿ ✶✳✶✮✳

❋✐❣✉r❛ ✶✳✶✿ ❈✉r✈❛ ❛ ♠ã♦✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

P♦❞❡♠♦s ❞✐③❡r ❞❡ ❢♦r♠❛ ♠❛✐s ❛①✐♦♠át✐❝❛ q✉❡ ❝✉r✈❛ é ✉♠❛ ❞❡❢♦r♠❛çã♦ ❝♦♥tí♥✉❛ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦✱ ♦✉ ❛té ♠❡s♠♦✱ ❛ tr❛❥❡tór✐❛ ❞❡s❝r✐t❛ ♣♦r ✉♠❛ ♣❛rtí❝✉❧❛ ♥♦ ♣❧❛♥♦✳

❋✐❣✉r❛ ✶✳✷✿ ❊①❡♠♣❧♦s ❞❡ ❝✉r✈❛s✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

❆ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ tr❛❞✉③ ❡♠ ❡q✉❛çõ❡s ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s q✉❡ ✈❡r✐✜❝❛♠ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s(x, y, z)_∈R3✱ q✉❡ s❛t✐s❢❛③❡♠ ✉♠❛ ❡q✉❛çã♦

❞♦ t✐♣♦ F(x, y, z) = 0✳ ❉❛r❡♠♦s ❡♥tã♦ ✉♠❛ ❞❡✜♥✐çã♦ ❞❡ ❝✉r✈❛✱ q✉❡ ♥ã♦ ❞❡s♣r❡③❡ ❛

r❡♣r❡s❡♥t❛çã♦ ♣♦r ❢✉♥çõ❡s✱ ♠❛s t❛♠❜é♠✱ q✉❡ ❛ ❝♦♥s✐❞❡r❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ R3✭q✉❡ ♥♦s

✐♥t❡r❡ss❛ ♥♦ ♠♦♠❡♥t♦✮✳ ◆♦ ❡♥t❛♥t♦✱ ❛❧❣✉♠❛s r❡str✐çõ❡s s❡rã♦ ❢❡✐t❛s✱ ❡♠ r❡❧❛çã♦ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ❝✉r✈❛s✱ q✉❛♥t♦ à ❝❧❛ss❡ ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❡ ❛ ♥ã♦ ♥✉❧✐❞❛❞❡ ❞❛ ❞❡r✐✈❛❞❛ ♣r✐♠❡✐r❛ ♣♦r s❡r❡♠ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ♦❜❥❡t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ∁_⊂R3 é ✉♠❛ ❝✉r✈❛ s❡ ❡①✐st✐r ✉♠❛ ❛♣❧✐❝❛çã♦ α:I _⊂

R_−→R3✱ ❝♦♠ α(I) =∁_✳

❆ ❞❡✜♥✐çã♦ ♣❡r♠✐t❡ ♣❡♥s❛r ♥✉♠❛ ❝✉r✈❛∁ _{❝♦♠♦ s❡♥❞♦ ✉♠ ♣❡❞❛ç♦ ❞❡ r❡t❛ ❞❡❢♦r♠❛❞♦} ♣♦r ✉♠❛ ❛♣❧✐❝❛çã♦ α✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ♦✉ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❛r❛ ∁_✳ ❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❯♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐❛✈❡❧ ❞♦ ♣❧❛♥♦ é ✉♠❛ ❛♣❧✐❝❛çã♦α✱ ❞❡

❝❧❛ss❡ C∞✱ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦

I _⊂R_→R3✳

❆ ✈❛r✐❛✈❡❧t_∈Ié ❞✐t❛ ♣❛râ♠❡tr♦ ❞❛ ❝✉r✈❛ ❡ ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡R3 ❞♦s ♣♦♥t♦sα(t), t _∈I✱

é ❝❤❛♠❛❞♦ tr❛ç♦ ❞❛ ❝✉r✈❛ ✭❋✐❣✉r❛✿ ✶✳✸✮✳

❋✐❣✉r❛ ✶✳✸✿ P❛r❛♠❡tr✐③❛çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ♥♦ ❡s♣❛ç♦✳ ❋♦♥t❡✿ ❬✶✵❪

❉❡✈❡♠ ♦❜s❡r✈❛r✱ ♣❡❧❛ ❞❡✜♥✐çã♦✱ q✉❡ ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ♥♦ ❡s♣❛ç♦ é ✉♠❛ ❛♣❧✐✲ ❝❛çã♦ α(t) = (x(t), y(t), z(t)), t_∈I✱ ♦♥❞❡ ❛s ❢✉♥çõ❡s x, y, z sâ♦ ❞❡ ❝❧❛ss❡ C∞✳

❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❆ ❛♣❧✐❝❛çã♦

α(t) = (x0 +at, y0+bt, z0+ct)✱ t∈R✱ ❝♦♠ a2+b2+c2 6= 0

é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❝✉❥♦ tr❛ç♦ é ✉♠❛ ❧✐♥❤❛ r❡t❛ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦

(x0, y0, z0) ❡ ♣❛r❛❧❡❧❛ ❛♦ ✈❡t♦r~v = (a, b, c) ✭ ❋✐❣✉r❛✿ ✶✳✹✮✳

❋✐❣✉r❛ ✶✳✹✿ r❡t❛✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❙❡❥❛ a > 0 ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧✱ ❛ ❝✉r✈❛ ❝✉❥❛ ❡q✉❛çã♦ ✈❡t♦r✐❛❧ é r❡♣r❡✲

s❡♥t❛❞❛ ♣♦r

α(t) = (rcost, rsint, at) ❝♦♠ t_∈R

é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❝✉❥❛s ❢♦r♠❛s ❝♦♦r❞❡♥❛❞❛s sã♦x=rcost, y=

rsint ❡ z =at✳ ❙❡❣✉❡ ❝❧❛r❛♠❡♥t❡ q✉❡ α é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❆❧é♠

❞✐ss♦✱ t❡♠✲s❡ q✉❡ x2 _+y2 _=r2_(cos2_{t+ sin}2_{t) = r}2_{✱ ❧♦❣♦ ♦ tr❛ç♦ ❞❛ ❝✉r✈❛ ❡stá ❝♦♥t✐❞♦ ♥♦} ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r x2_+y2 _=r2_{✳ ❈♦♠♦z =at❝♦♠ a >0✱ ❛ ❝✉r✈❛ ❢❛③ ✉♠❛ ❡s♣✐r❛❧ ♣❛r❛ ❝✐♠❛} ❛♦ r❡❞♦r ❞♦ ❝✐❧✐♥❞r♦ ❛ ♠❡❞✐❞❛ q✉❡ t ❛✉♠❡♥t❛✳ ❊ss❛ ❝✉r✈❛ é ❝❤❛♠❛❞❛ ❞❡ ❤é❧✐❝❡ ❝✐r❝✉❧❛r

✭❋✐❣✉r❛✿ ✶✳✺✮✳

❋✐❣✉r❛ ✶✳✺✿ ❤é❧✐❝❡ ❝✐r❝✉❧❛r✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

❊①❡♠♣❧♦ ✶✳✶✳✺✳ ❆ ❝✉r✈❛ ❞❡ ❱✐✈✐❛♥✐ ❢♦r♠❛❞❛ ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❞♦ ❝✐❧✐♥❞r♦(x₋a)2_+y2 _=a2 ❝♦♠ ❛ ❡s❢❡r❛ x2_+y2_+z2 _{= 4a}2 _{❡ q✉❡ s❡ ♣♦❞❡ ♣❛r❛♠❡tr✐③❛r ♣♦r}

α(t) = a(1 + cost,sint,2 sin(t 2))

❚❛♠❜é♠ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ✭❋✐❣✉r❛✿ ✶✳✻✮

❋✐❣✉r❛ ✶✳✻✿ ❈✉r✈❛ ❞❡ ❱✐✈✐❛♥✐✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❙❡❥❛ α : I _⊂ R _−→ R3 ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧ q✉❡✱ ❛

❝❛❞❛ t _∈ I✱ ❛ss♦❝✐❛ α(t) = (x(t), y(t), z(t))✱ ♦ ✈❡t♦r α′

(t) = (x′

(t), y′

(t), z′

(t)) é ❝❤❛♠❛❞♦

✈❡t♦r t❛♥❣❡♥t❡ ❛ α ❡♠ t✳

❊st❛ ❞❡✜♥✐çã♦ ♥♦s trás ✉♠❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ✉♠ ✈❡t♦r t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛ q✉❡ ♦❝✉♣❛ ❛ ♣♦s✐çã♦ ❧✐♠✐t❡ q✉❛♥❞♦ p_−→p0 ❞❛ r❡t❛ ♣♦r ❡❧❡s ❞❡t❡r♠✐♥❛❞❛✳

❆ss✐♠ s❡ α é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛✱ t♦♠❛♥❞♦ α(t) =p0 ❡ α(t+h) =p ♣❛r❛ t∈I ✜①❛❞♦✱ t❛❧ q✉❡ t+h _∈I✱ t❡♠♦s q✉❡ α(t+h)−α(t)

h é ♦ ✈❡t♦r ❞✐❢❡r❡♥ç❛ ∆α ♠✉❧t✐♣❧✐❝❛❞♦

♣❡❧♦ ❡s❝❛❧❛r 1

h ✱ ❛ss✐♠ ❝❤❡❣❛♠♦s ❡①❛t❛♠❡♥t❡ ❛ ❞❡✜♥✐çã♦ ❞❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦α ❡♠ t✳

lim

h→0

α(t+h)₋α(t)

h

◗✉❛♥❞♦ s❡ ♣❡♥s❛ ♥✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❝♦♠♦ ❛ ❞❡s❝r✐çã♦ ❞❛ tr❛❥❡t♦r✐❛ ❈✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s α(t0) ❡α(∆t+t0)❝♦♠♦ s❡♥❞♦ ♦

lim

∆t→₀

α(t0+ ∆t)−α(t0)

∆t ❂∆limt→₀

∆s

∆t ❂v(t0)✳

❆ss✐♠ q✉❛♥❞♦ ♦ t❡♠♣♦ t❡♥❞❡ ❛ ③❡r♦✱ ♣♦❞❡✲s❡ ♣❡♥s❛r ♥❛ ✈❡❧♦❝✐❞❛❞❡ ♥♦ ✐♥st❛♥t❡ t0✱ ❞❛❞♦ ♣❡❧♦ ❧✐♠✐t❡

lim

t→₀

α(t+t0)−α(t0)

t ❂ α

′

(t0)❂ v(t0) ❖✉ s❡❥❛✱ α′

(t0) é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡s❝r✐t♦ ♣♦r α ♥♦ ✐♥st❛♥t❡ t0✳ ◆❡st❡ ❝❛s♦✱ ❞❡s✐❣♥❛r❡♠♦s | α′

(t0) | ❝♦♠♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❡s❝❛❧❛r ♥♦ ✐♥st❛♥t❡ t0❀ ✜s✐❝❛♠❡♥t❡ ♣♦❞❡♠♦s ♣❡♥s❛r ❡♠ α′′ ❝♦♠♦ ❛ ❛❝❡❧❡r❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡s❝r❡✈❡♥❞♦

α(t)✱ ♣♦✐s

♠❡❞❡ ❛ ✈❛r✐❛çã♦ ❞❡ α′

✳

❉❡✜♥✐çã♦ ✶✳✶✳✼✳ ❯♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧ α : I _⊂⊂ R _−→ R3 é ❞✐t❛

r❡❣✉❧❛r s❡ ♣❛r❛ t♦❞♦ t _∈I✱ t❡♠✲s❡ α′

(t)₆= 0

P♦❞❡♠♦s ♦❜t❡r ✈ár✐❛s ❝✉r✈❛s r❡❣✉❧❛r❡s q✉❡ t❡♠ ♦ ♠❡s♠♦ tr❛ç♦ q✉❡ ✉♠❛ ❞❛❞❛ ❝✉r✈❛α✳

▼❛t❡♠❛t✐❝❛♠❡♥t❡ ✐st♦ é ❢❡✐t♦ ✉s❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ r❡♣❛r❛♠❡tr✐③❛çã♦ h✳

❙❡❥❛♠ ■ ❡ ❏ ✐♥t❡r✈❛❧♦s ❛❜❡rt♦s ❞❡ R ✱ α : I _−→ R3 ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r ❡ h : J _−→ I

✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧C∞

(I)✱ ❝✉❥❛ ❞❡r✐✈❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ♥ã♦ ♥✉❧❛ ❡♠ t♦❞♦s ♦s

♣♦♥t♦s t❛❧ q✉❡ h(J) = I✳ ❊♥tã♦ ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛

β =α_◦h:J _−→R3

é ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r✱ q✉❡ t❡♠ ♦ ♠❡s♠♦ tr❛ç♦ q✉❡ α✳ ❉❡ ❢❛t♦✱ β(t) =α(h(t))✱ ❧♦❣♦

β′

(t) = α′

(h(t))(h′

(t))₆= 0✱ _∀t_∈J

❡

β(J) =α(h(J)) =α(I)

❆ ❛♣❧✐❝❛çã♦ β é ❝❤❛♠❛❞❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞❡ α ♣♦r h✳ ❆ ❢✉♥ç❛õ h é ❛ ♠✉❞❛♥ç❛ ❞❡

♣❛râ♠❡tr♦✳

❊①❡♠♣❧♦ ✶✳✶✳✽✳ ❆ ❝✉r✈❛

α(s) = (cos_√s 2,sin

s

√

2,

s

√

2)✱ s ∈R✱

é ❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❤é❧✐❝❡

β(t) = (cost,sint, t)

♣❡❧❛ ♠✉❞❛♥ç❛ ❞❡ ♣❛râ♠❡tr♦

h(s) = _√s

2 ✱ s∈R

❉❛❞❛ ✉♠❛ ❝✉r✈❛ α(t)✱ s❛❜❡r ❞❡t❡r♠✐♥❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡♥tr❡ ♦s ♣♦♥t♦sα(t0) ❡ α(t) é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♥♦ss♦s ❢✉t✉r♦s ❝á❧❝✉❧♦s✳ ❙❡❣✉✐♥❞♦ ♦ ♣❡♥s❛♠❡♥t♦

❞❡ q✉❡ α ❞❡s❝r❡✈❡ ✉♠ ♠♦✈✐♠❡♥t♦✱ t❡♠♦s q✉❡ ♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡ é ❞❛❞♦ ♣♦r α′

(t)✳ ▲♦❣♦✱

s❡ | α′

(t) _|= 1✱ ❛ q✉❡stã♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ é ❢❛❝✐❧♠❡♥t❡ r❡s♦❧✈✐❞❛✱ ♣♦✐s✱ ❝♦♠♦ ♦ ♠ó❞✉❧♦

❞♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡ ✭✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✮ é ✉♥✐tár✐♦✱ q✉❡r ❞✐③❡r q✉❡ ❡♠ ❝❛❞❛ ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ♣❡r❝♦rr❡♠♦s ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ s❡♥❞♦ ❛ss✐♠✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ tr❛❥❡tór✐❛ ❡♥tr❡ ♦s ✐♥st❛♥t❡s t0 ❡ t s❡rá |t−t0 |✳

❊♥tã♦ ♦ q✉❡ t❡♠♦s ❛ ❢❛③❡r é ❡♥❝♦♥tr❛r ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞❡α ♦♥❞❡ _|α′

(t)_|= 1✳

❉❡✜♥✐çã♦ ✶✳✶✳✾✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛rα:I _−→R3✱ é ❞✐t❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦

❞❡ ❛r❝♦ s❡✱ ♣❛r❛ t♦❞♦ t0, t∈I, t0 ≤t✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❞❛ ❝✉r✈❛ α ❞❡ t0 ❛ t é ✐❣✉❛❧ ❛ t₋t0✳

P❡❧❛ ❞❡✜♥✐çã♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡

ds

dt =v =|α

′

(t)_|⇒ds=_|α′

(t)_|dt

P♦r ✐♥t❡❣r❛çã♦ s❡ ♦❜té♠✿

s(t) =Rt t0 |α

′

(t)_|dt=t₋t0

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✵✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r α : I _⊂ R _−→ R3✱ ❡st❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦

❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ t_∈I✱ t❡♠✲s❡ _|α′

(t)_|= 1✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦sα♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡ ✜①❡♠♦st0 ∈I✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ s :I _−→R✱ q✉❡ ♣❛r❛ ❝❛❞❛ t_∈I✱ ❛ss♦❝✐❛

s(t) =Rt t0 |α

′

(t)_|dt

❙❡t0 6t ✱ ❡♥tã♦✱ ♣♦r ❤✐♣ót❡s❡✱ Rt

t0 |α

′

(t)_|dt=t₋t0

❧♦❣♦✱ ds

dt = 1✳

❞♦ ♠❡s♠♦ ♠♦❞♦✱ s❡t 6t0✱ ❡♥tã♦ Rt0

t |α

′

(t)_|dt =t0−t=−(t−t0) =−s(t)

s❡♥❞♦ ❛ss✐♠✱

−s(t) =t0−t ✭❞❡r✐✈❛♥❞♦✮ −ds_dt =₋1

❡♥tã♦ ds

dt = 1✳

P♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦t _∈I✱s(t) =t₋t0✱ ❡s′(t) = 1✳ ❈♦♠♦s′(t) =|α′(t)|✱ ❝♦♥❝❧✉✐♠♦s q✉❡ |α′

(t)_|= 1✱ ♣❛r❛ t♦❞♦ t_∈I✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡|α′

(t)_|= 1✱ ❡♥tã♦ _∀t_∈I ❝♦♠ t0 < tt❡♠♦s✿

s(t) =Rt t0 |α

′

(t)_|dt=Rt

t0dt=t−t0

❧♦❣♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦✿ ✶✳✶✳✾✱ α ❡st❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✳

❊①❡♠♣❧♦ ✶✳✶✳✶✶✳ ❆ ❛♣❧✐❝❛çã♦

α(t) = (√5 cos_√t 5,

√

5 sin_√t 5,

√

5)✱ t _∈R

é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱ ❥á q✉❡ |α′

(t)_|= 1

❱❡r❡♠♦s ❛❣♦r❛ q✉❡ t♦❞❛ ❝✉r✈❛ r❡❣✉❧❛r α ❛❞♠✐t❡ ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ β ♣❡❧♦ ❝♦♠✲

♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✷✳ ❙❡❥❛ α : I _⊂ R _−→ R3 ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r ❡ s : I _−→ s(I) _⊂ R ❛

❢✉♥çã♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❞❡ α ❛ ♣❛rt✐r ❞❡ t0✳ ❊♥tã♦ ❡①✐st❡ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ h ❞❡ s✱ ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ J = s(I)✱ ❡ β =α_◦h é ✉♠❛ r❡♣❛r❛♠❡r✐③❛çã♦ ❞❡ α✱ ♦♥❞❡ β

❡stá ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✳ ❉❡♠♦♥str❛çã♦✳ α é ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r✱ ♣♦rt❛♥t♦

s′

(t) =_|α′

(t)_|>0

✐st♦ é✱sé ✉♠❛ ❢✉♥çã♦ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✱t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ t♦❞❛ ❢✉♥çã♦ ❝♦♠ ❞❡r✐✈❛❞❛

❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛ ❡♠ ✉♠ ❝❡rt♦ ✐♥t❡r✈❛❧♦ I✱ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✳ ▲♦❣♦✱ ❡①✐st❡ ❛

❢✉♥çã♦ ✐♥✈❡rs❛ ❞❡ s, h : J _−→ I✳ ❈♦♠♦ ♣❛r❛ t♦❞♦ t _∈ I ✱ h(s(t)) = t ✱ t❡♠♦s q✉❡ dh

ds ds

dt = 1✱ ♣♦rt❛♥t♦✱

dh

ds =

1

s′

(t) = 1

|α′

(t)_| >0.

❈♦♥❝❧✉í♠♦s q✉❡✱ β(s) =α_◦h(s)✱s _∈J✱ é ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞❡α ❡

dβ ds = dα dt dh ds = α′

(t)

|α′

(t)_|

= 1✳

P♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦✿ ✶✳✶✳✶✵✱ β ❡stá ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✳

❊①❡♠♣❧♦ ✶✳✶✳✶✸✳ ❙❡❥❛ α(t) = (acosπt, asinπt)✳ P❛r❛♠❡tr✐③❡ α ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡

❛r❝♦✳ ❙♦❧✉çã♦✿ s= Z t t0 p

< α′

(t), α′

(t)>dt ✭✶✳✶✳✶✮

❉❡r✐✈❛♥❞♦ α(t) t❡♠♦s✿

α′

(t) = (₋πasinπt, πacosπt)✱ ❧♦❣♦

< α′

(t), α′

(t)>=π2_a2_sin2_{(πt) +π}2_a2_cos2_{(πt) =π}2_a2_✳ ❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✳✶✳✶✮✱ t❡♠♦s✿

s =Rt

t0

√

π2_a2_{dt =πa(t−t}

0)

❋❛③❡♥❞♦ t0 = 0✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ t❡♠♦s q✉❡ s=πat ❡♥tã♦ t=

s πa

❚♦♠❛♥❞♦

β(s) =α(t(s)) =acosπ s

πa

, asinπ s

πa

✱ ❧♦❣♦

β(s) = acoss

a

, asins

a

✳

❈♦♠♦ t♦❞❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ é t❛♥❣❡♥t❡ ✉♥✐tár✐❛✱ t❡♠♦s q✉❡✿

β′

(s) =

a 1 a

−sins

a, a

1 a coss a β′

(s) =₋sins

a

,coss

a

✱ ♦✉ s❡❥❛ |β′

(s)_|= 1✳

✶✳✷ ❈✉r✈❛t✉r❛ ❡ ❋ór♠✉❧❛s ❞❡ ❋r❡♥❡t✲❙❡rr❡t

◆❡st❛ s❡çã♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❝✉r✈❛s α:I _⊂R_−→R3 ♣❛r❛♠❡tr✐③❛❞❛s ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦

❞❡ ❛r❝♦✳ ▲♦❣♦✱ ♣❛r❛ ❝❛❞❛ s_∈I✱α′

(s)é ✉♠ ✈❡t♦r ✉♥✐tár✐♦✱ ♦✉ s❡❥❛✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ♥ã♦ ✈❛r✐❛

❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ✈❛r✐❛♥❞♦ ❛♣❡♥❛s ♥❛ ❞✐r❡çã♦✳ ❈♦♠♦ ❛ ❛❝❡❧❡r❛çã♦α′′

(s)✐♥❞✐❝❛ ❛ ✈❛r✐❛çã♦

❞❛ ✈❡❧♦❝✐❞❛❞❡✱ q✉❡ ♥♦ ❝❛s♦ ♥ã♦ ✈❛r✐❛✱ ❡♥tã♦ α′′

(s)♥ã♦ t❡♠ ♥❡♥❤✉♠❛ ❝♦♠♣♦♥❡♥t❡ ♣❛r❛❧❡❧❛

❛ α′

(s)✱ ♦ q✉❡ é ✈❡r❞❛❞❡ ♣♦✐s✱s❡ _|α′

(s)_|= 1✱ t❡♠♦s

< α′

, α′

>=_|α′

(t)_|= 1

❉❛í

< α′

, α′

>′

= 0

✉s❛♥❞♦ ❛ r❡❣r❛ ❞❛ ❞❡r✐✈❛çã♦ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r t❡♠♦s

2< α′

, α′′

>= 0.

■st♦ é✱ < α′

, α′′

>= 0 ♠♦str❛♥❞♦ ❝♦♠ ✐ss♦✱ q✉❡ α′ ❡

α′′ sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳

❆ss✐♠ ♣❛r❛ t♦❞♦ s _∈ I✱ t❛❧ q✉❡ α′′

(s) ₆= 0✱ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ✉♠ ✈❡t♦r ✉♥✐tár✐♦ ♥❛

❞✐r❡çã♦ ❞❡ α′′

(s)✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛ α:I _⊂R_−→R3✱ ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡

❛r❝♦ ✳ ❉❡♥♦t❛♥❞♦ ♦ ✈❡t♦r t❛♥❣❡♥t❡ ✉♥✐tár✐♦ à ❝✉r✈❛ ♥♦ ♣♦♥t♦ α(s) ♣♦r

t(s) =α′

(s)✱

❞❡✜♥✐♠♦s ❝✉r✈❛t✉r❛ ❞❡ α ♥♦ ♣♦♥t♦ α(s) ♣♦r

k(s) =_|t′

(s)_|✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱

k(s) =_|α′′

(s)_|

❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛ α :I _⊂ R _−→R3 ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐✲

♠❡♥t♦ ❞❡ ❛r❝♦ ❝♦♠ ❝✉r✈❛t✉r❛ ♣♦s✐t✐✈❛✱ ❞✐③❡♠♦s q✉❡ ♦ ✈❡t♦r❛

n(s) = α

′′

(s)

|α′′

(s)_| =

α′′

(s)

k(s)

é ❞❡♥♦♠✐♥❛❞♦ ✈❡t♦r ♥♦r♠❛❧ ❛ α ❡♠ s✳

❉❡♥♦t❛♥❞♦ ♣♦r t(s) ♦ ✈❡t♦r ✉♥✐tár✐♦ α′

(s) s❡❣✉❡ q✉❡✿

t′

(s) =k(s)n(s)

❆ ❝✉r✈❛t✉r❛ ❞❡ ✉♠❛ ❝✉r✈❛ ❡♠ ✉♠ ♣♦♥t♦ ♠❡❞❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❛ ❞✐r❡çã♦ ❞❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ ♥♦ ♣♦♥t♦❀ é ✉♠❛ ♠❡❞✐❞❛ ❞❡ q✉❛♥t♦ ❛ ❝✉r✈❛ s❡ ❝✉r✈❛✱ ✐st♦ é✱ ♦ q✉❛♥t♦ ❡❧❛ ❞✐❢❡r❡ ❞❡ ✉♠❛ r❡t❛✳ P❛r❛ ♠❡❞✐r ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♥❛ ❞✐r❡çã♦ t❛♥❣❡♥t❡ ❛tr❛✈❡s ❞♦ ✈❡t♦r t❛♥❣❡♥t❡ à ❝✉r✈❛ ✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r t❛♥❣❡♥t❡ s❡❥❛ s❡♠♣r❡ ♦ ♠❡s♠♦ ✭♦✉ ❡st❛rí❛♠♦s ♠❡❞✐♥❞♦ t❛♠❜é♠ ❛ ✈❛r✐❛çã♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r✮✱✐st♦ é✱ ❛ ❝✉r✈❛t✉r❛ ♠❡❞❡ ❛ ❛❝❡❧❡r❛çã♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ q✉❡ ♣❡r❝♦rr❡ ❛ tr❛❥❡tór✐❛ ❞❛ ❝✉r✈❛ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✉♥✐tár✐❛✳✭❋✐❣✉r❛✿ ✶✳✼✮

❋✐❣✉r❛ ✶✳✼✿ β ❝✉r✈❛✲s❡ ♠❛✐s q✉❡ α✳ ❋♦♥t❡✿ ❬✼❪

❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❙❡❥❛ α :I _−→R3 ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦

❞❡ ❛r❝♦✳ ❖ ✈❡t♦r b(s) =t(s)_×n(s) é ❝❤❛♠❛❞♦ ✈❡t♦r ❜✐♥♦r♠❛❧ ❛ α ❡♠ s✳

❙❡♥❞♦ ❛ss✐♠ t(s), n(s), b(s) é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ♣❛r❛ ♦ R3✳ ❆ ❡ss❡ r❡❢❡r❡♥❝✐❛❧

♦rt♦♥♦r♠❛❧ ❝❤❛♠❛♠♦s ❞❡ tr✐❡❞r♦ ❞❡ ❋r❡♥❡t✲❙❡rr❡t✳

❈❛❞❛ ♣❛r ❞♦ tr✐❡❞r♦ ❞❡ ❋r❡♥❡t ❞❡t❡r♠✐♥❛ ✉♠ ♣❧❛♥♦✳ ❖ ♣❧❛♥♦ ❞❡ R3 q✉❡ ❝♦♥té♠ α(s)

❡ é ♥♦r♠❛❧ ❛♦ ✈❡t♦r t(s) é ♦ ♣❧❛♥♦ ♥♦r♠❛❧ à ❝✉r✈❛ α ❡♠ s✳ ❖ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ α(s) ❡ é

♥♦r♠❛❧ ❛ b(s) é ♦ ♣❧❛♥♦ ♦s❝✉❧❛❞♦r✱ ❡ ♦ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ α(s) ❡ é ♥♦r♠❛❧ ❛n(s) é ♦ ♣❧❛♥♦

r❡t✐✜❝❛♥t❡ ❞❛ ❝✉r✈❛ α ❡♠ s✱✭❋✐❣✉r❛✿ ✶✳✽✮

❋✐❣✉r❛ ✶✳✽✿ ♣❧❛♥♦s✿♦s❝✉❧❛❞♦r✱ ♥♦r♠❛❧ ❡ r❡t✐✜❝❛♥t❡✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

❆♥❛❧✐s❛♥❞♦ ❛ ✜❣✉r❛ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡b′

(s)é ♣❛r❛❧❡❧♦ ❛n(s)✳ ❉❡ ❢❛t♦✱ s❡ ❞❡r✐✈❛♠♦s

b(s) =t(s)_×n(s) ✱ ✈❛♠♦s ❝❤❡❣❛r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦

b′

(s) = t′

(s)_×n(s) +t(s)_×n′

(s) = t(s)_×n′

(s)✱

✉♠❛ ✈❡③ q✉❡t(s)é ♣❛r❛❧❡❧♦ ❛ n(s)❙❡♥❞♦ ❛ss✐♠✱b′

(s)é ♦rt♦❣♦♥❛❧ ❛t(s)✳ ❈♦♠♦ _|b′

(s)_|= 1

✱ t❡♠♦s q✉❡ b′

(s) é ♦rt♦❣♦♥❛❧ ❛ b(s)✳ ❈♦♥❝❧✉✐♥❞♦ q✉❡ b′

(s)é ♣❛r❛r❧❡❧♦ ❛ n(s)✱ ✐st♦ é✱ b′

(s)

é ✐❣✉❛❧ ❛ n(s) ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ✉♠ ♥✉♠❡r♦ r❡❛❧ τ(s)✱ ❞❡♥♦♠✐♥❛❞♦ t♦rçã♦ ❞❛ ❝✉r✈❛ ❡♠

s✱ ❞❡✜♥✐❞♦ ♣♦r

b′

(s) =τ(s)n(s)

❝♦♠♦ n =b_×t✱ t❡♠♦s q✉❡✿

n′

(s) = b′

×t+b_×t′

= τ n_×t+b_×kn

= ₋τ b₋kt

❆ss✐♠ ♣♦❞❡♠♦s ❝❤❡❣❛r ❛ ❝♦♥❝❧✉sã♦ q✉❡ ♦ ❚r✐❡❞r♦ ❞❡ ❋r❡♥❡t✲❙❡rr❡t ❡ ✉♠❛ ❝✉r✈❛ ♣❛r❛✲ ♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡ ❝♦♠ ❝✉r✈❛t✉r❛ ♣♦s✐t✐✈❛ sã♦ ♦s ✈❡t♦r❡s t(s)✱ n(s) ❡

b(s) q✉❡ s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿





t′

(s)

n′

(s)

b′

(s)



= 



0 k(s) 0

−k(s) 0 ₋τ(s) 0 τ(s) 0



 



t(s)

n(s)

b(s)





t′

(s) = 0t(s) +k(s)n(s) + 0b(s)

n′

(s) = ₋k(s)t(s) + 0n(s)₋τ(s)b(s)

b′

(s) = 0t(s) +τ(s)n(s) + 0b(s)

❋✐❣✉r❛ ✶✳✾✿ ❚r✐❡❞r♦ ❞❡ ❋r❡♥❡t✲❙❡rr❡t✳ ❋♦♥t❡✿ ●♦♦❣❧❡ ✐♠❛❣❡♥s

❊①❡♠♣❧♦ ✶✳✷✳✹✳ ❖❜t❡r ♦ tr✐❡❞r♦ ❞❡ ❋r❡♥❡t✲❙❡rr❡t✱ ❛ ❝✉r✈❛t✉r❛ ❡ ❛ t♦rçã♦ ❞❛ ❤é❧✐❝❡ ♣❛r❛✲ ♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦

α(s) =

acos√ s

a2_+b2, asin

s

√

a2_+b2,

bs

√

a2_+b2

✱ s_∈R,

♦♥❞❡ ✱ a >0 é ✉♠❛ ❝♦♥st❛♥t❡✳

α′

(s) = √ 1

a2_+b2

−asin√ s

a2_+b2, acos

s

√

a2_+b2, b

α′′

(s) = −a

a2_+b2

cos_√ s

a2_+b2,sin

s

√

a2_+b2,0

k(s) =_|α′′

(s)_|= a

a2_+b2.

P♦rt❛♥t♦✱

n(s) = α

′′

(s)

k(s) =

−cos_√ s

a2_+b2,−sin

s

√

a2_+b2,0

,

b(s) = t(s)_×n(s) = √ 1

a2_+b2

bsin√ s

a2_+b2,−bcos

s

√

a2_+b2, a

,

b′

(s) = _√ b

a2_+b2

cos_√ s

a2_+b2,sin

s

√

a2_+b2,0

,

τ(s) =< b′

(s), n(s)>=_−√ b

a2_+b2

❈❛♣ít✉❧♦ ✷

❙✉♣❡r❢í❝✐❡s r❡❣✉❧❛r❡s

❙❡❣✉✐r❡♠♦s ♦ ♠❡s♠♦ ♣❡♥s❛♠❡♥t♦ ❛❞♦t❛❞♦✱ q✉❛♥❞♦ ❢❛❧❛♠♦s ❞❡ ❝✉r✈❛✱ ♣❛r❛ ❢❛❧❛r ❞❡ s✉♣❡r❢í❝✐❡s✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❡①♣❧♦r❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠❡tr✐❝❛s ❧♦❝❛✐s ❞❡ s✉♣❡r✲ ❢í❝✐❡s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ R3✳ ❈♦♥❝❡✐t♦s ❝♦♠♦ ♣❛r❛♠❡tr✐③❛çã♦✱ ♣❧❛♥♦ t❛♥❣❡♥t❡✱ ❢♦r♠❛s

❢✉♥❞❛♠❡♥t❛✐s ❡ ✐s♦♠❡tr✐❛s s❡rã♦ tr❛t❛❞♦s✱ r❡str✐♥❣✐♥❞♦ ❛♦ ❡st✉❞♦ ❞❡ s✉♣❡r❢í❝✐❡s q✉❡ ❡♠ ❝❛❞❛ ♣♦♥t♦ ❛❞♠✐t❡♠ ✉♠ ♣❧❛♥♦ t❛♥❣❡♥t❡✱ ♣❛r❛ ✐ss♦ ❛s r❡❢❡rê♥❝✐❛s ❬✶❪✱ ❬✸❪✱ ❬✺❪✱ ❬✽❪✱ ❬✶✵❪ ❡ ❬✶✷❪ ❞❛rá ❛♦ ❧❡✐t♦r ✉♠ ♠❛♦✐r ❛♣r♦❢✉♥❞❛♠❡♥t♦✳ ❆ss✉♠✐r❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s x, y, z ❡♠ R3 ❡ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❛♣❧✐❝❛çã♦

X(u, v) = (x(u, v), y(u, v), z(u, v)),

❞❡ ❞✉❛s ✈❛r✐á✈❡✐su, v q✉❡ ✈❛r✐❛♠ ❡♠ ✉♠ ❛❜❡rt♦U _⊂R2✱ ♦♥❞❡ ❝❛❞❛ ♣❛r(u, v)_∈U, X(u, v)

❞❡t❡r♠✐♥❛ ✉♠ ♣♦♥t♦ ❞❡ R3✳

✷✳✶ P❛r❛♠❡tr✐③❛çã♦✱ ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ❡ ✈❡t♦r t❛♥✲

❣❡♥t❡

❉❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛s ❝✉r✈❛s✱ é ♣♦ssí✈❡❧ ❛ ♣❛rt✐r ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦❜t❡r♠♦s ✉♠❛ ❝✉r✈❛ ♥ã♦ r❡t❛✱ ✐st♦ é✱ ❞❡❢r♦♠❛♥❞♦✲❛✳ ◆♦ ❝❛s♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞♦ R3✱

♣♦❞❡♠♦s ✐♠❛❣✐♥❛r ❛❧❣✉♠❛s ❞❡❧❛s ❝♦♠♦ ❞❡❢♦r♠❛çõ❡s ❞♦ ♣❧❛♥♦✱ ♣♦r ❡①❡♠♣❧♦ ✉♠ ❝✐❧✐♥❞r♦✱ ❛ ❢❛✐①❛ ❞❡ ▼♦❜✐✉s✱ ❡♥tr❡ ♦✉tr♦s✱ ♥♦s ❞❛rã♦ ❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞♦ q✉❡ ❝❤❛♠❛r❡♠♦s s✉♣❡r❢í❝✐❡s✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ S _⊂R3 é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✱s❡ ♣❛r❛ ❝❛❞❛ p _∈S✱

❡①✐st❡ ✉♠ ❛❜❡rt♦ V _⊂ R3 ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ X : U _⊂ R2 _→ V _∩S ❞❡✜♥✐❞❛ ♥✉♠ ❛❜❡rt♦ U

❞❡ R2✱ t❛❧ q✉❡✿

✐✮Xé ❞✐❢❡r❡♥❝✐á✈❡❧✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ❡s❝r❡✈❡r♠♦sX(u, v) = (x(u, v), y(u, v), z(u, v))✱ (u, v)_∈U ❛s ❢✉♥çõ❡s ❞❡x(u, v)✱ y(u, v)❡z(u, v)t❡♠ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❝♦♥tí♥✉❛s ❞❡ t♦❞❛s

❛s ♦r❞❡♥s❀

✐✐✮X é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ ❈♦♠♦X é ❝♦♥t✐♥✉❛ ✭♣♦r ✐✮✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ t❡♠ ✐♥✈❡rs❛

X−₁

:R3 _−→R2 q✉❡ t❛♠❜é♠ é ❝♦♥tí♥✉❛❀

✐✐✐✮✭❈♦♥❞✐çã♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡✮ P❛r❛ t♦❞♦ q _∈ U✱ ❛ ❞✐❢❡r❡♥❝✐❛❧ dXq : R2 _−→ R3 é

✐♥❥❡t✐✈❛✳

❈♦♥s✐❞❡r❛♥❞♦✲s❡ ❛ ❛♣❧✐❝❛çã♦X(u, v)❞❡✜♥✐❞❛ ❛❝✐♠❛✱ às ✈❛r✐á✈❡✐su, v sã♦ ♦s ♣❛râ♠❡tr♦s

❞❛ s✉♣❡r❢í❝✐❡✳ ❖ tr❛ç♦ ❞❛ s✉♣❡r❢í❝✐❡ é ♦ s✉❜❝♦♥❥✉♥t♦ S ❞❡ R3 ♦❜t✐❞♦ ♣❡❧❛ ✐♠❛❣❡♠ ❞❛

❛♣❧✐❝❛çã♦ X✳ ❆ ❡ss❛ ❛♣❧✐❛❝❛çã♦ X✱ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ♣❛r❛♠❡tr✐③❛çã♦ ❧♦❝❛❧ ♦✉ s✐st❡♠❛

❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ p ✭❋✐❣✉r❛✿ ✷✳✶✮✳ ❆ ✈✐③✐♥❤❛♥ç❛ V _∩S é ❝❤❛♠❛❞❛ ❞❡ ✈✐③✐♥❤❛♥ç❛

❝♦♦r❞❡♥❛❞❛ ❞❡ ❙ ❡♠ p✳

❋✐❣✉r❛ ✷✳✶✿ ❙✉♣❡r❢í❝✐❡✳❋♦♥t❡✿ ❬✺❪

❖❜sr✈❛çõ❡s✿

✶✳ ❆ ❝♦♥❞✐çã♦ ✐✮ ♥♦s ♣❡r♠✐t❡ ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❧❛♥♦ t❛♥❣❡♥t❡

✷✳ ❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♥✈❡rs❛ ♥❛ ❝♦♥❞✐çã♦ ✐✐✮ s❡r✈❡ ♣❛r❛ ♣r♦✈❛r q✉❡ ❝❡rt♦s ❝♦♥❝❡✐t♦s ❞❡♣❡♥❞❡♠ ❛♣❡♥❛s ❞♦ ♣♦♥t♦ P _∈S✱ ✐st♦ é✳ ✐♥❞❡♣❡♥❞❡♠ ❞❛ ♣❛r❛♠❡tr✐③❛çã♦ X :U _⊂

R2 _→V _∩S ❝♦♠ P _∈V _∩S✳

✸✳ ❆ ❝♦♥❞✐çã♦ ✐✐✐✮ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡S✳

❊♠ ✐✐✐✮✱ t❡♠♦s q✉❡ ∂X

∂u ❡

∂X

∂v sã♦ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡X❝❛❧❝✉❧❛❞❛s

♥♦ ♣♦♥t♦ q = (u0, v0) ✱ q✉❡ ❝♦rr❡s♣♦♠❞❡♠ às ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ JX(u, v)✱ ♦✉ s❡❥❛✱ ❛ ♠❛tr✐③ ❞❡ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ♥❛s ❜❛s❡s ❝❛♥ô♥✐❝❛s ❞❡ R2 ❡ R3✳

JX(u, v) =



    

∂x ∂u

∂x ∂v ∂y ∂u

∂y ∂v ∂z ∂u

∂z ∂v



    

❂

∂X ∂u

∂X ∂v

P♦r ❝♦♠♦❞✐❞❛❞❡ ✈❛♠♦s ❞❡♥♦t❛r ∂X

∂u ♣♦r Xu ❡

∂X

∂v ♣♦r Xv✳

❙❡♥❞♦ ❛ss✐♠ ❛ ❝♦♥❞✐çã♦iii) ✭❞❡✜♥✐çã♦✿ ✷✳✶✳✶✮ s❡ ❡q✉✐✈❛❧❡ às s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

✶✳ ❖s ✈❡t♦r❡s ❞❛ ❝♦❧✉♥❛ ✱Xu❡Xv✱ ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s❀ ✷✳ Xu×Xv 6= 0❀

✸✳ ❆ ♠❛tr✐③ JX(u, v) t❡♠ ♣♦st♦ ✷✱ ♦✉ s❡❥❛ ✉♠ ❞♦s ♠❡♥♦r❡s ❞❡t❡r♠✐♥❛♥t❡s ❏❛❝♦❜✐❛♥♦s

❞❡ ♦r❞❡♠ ✷

∂(x, y)

∂(u, v) ❂

   ∂x ∂u ∂x ∂v ∂y ∂u ∂y ∂v   ✱

∂(y, z)

∂(u, v) ❂

   ∂y ∂u ∂y ∂v ∂z ∂u ∂z ∂v   ✱

∂(x, z)

∂(u, v) ❂

   ∂x ∂u ∂x ∂v ∂z ∂u ∂z ∂v   

❞❡✈❡ s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ❡♠ q= (u0, v0)✳

❊♥tã♦ ❛ ❝♦♥❞✐çã♦iii)❡①❝❧✉✐ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡①✐st✐r ✧❜✐❝♦s✧❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r

❡ ❛ss✐♠ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣❧❛♥♦ t❛♥❣❡t❡ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ S✳✭❋✐❣✉r❛✿ ✷✳✷✮

❋✐❣✉r❛ ✷✳✷✿ ❖ ❝♦♥❡ ❙ ♥ã♦ ♣♦ss✉✐ ♣❛r❛♠❡tr✐③❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ✈ért✐❝❡ ♣✳ ❋♦♥t❡✿ ❬✺❪

❊①❡♠♣❧♦ ✷✳✶✳✷✳ ❆ ❡s❢❡r❛ ✉♥✐tár✐❛S2 _{= (x, y, z)∈R}3 _:x2_+y2_+z2 _{= 1 é ✉♠❛ s✉♣❡r❢í❝✐❡} r❡❣✉❧❛r✳

▼♦str❛r❡♠♦s q✉❡ X1 : U ⊂ R2 −→ R3✱ ❞❛❞❛ ♣♦r X1(u, v) = (u, v, p

1₋(u2_+v2₎₎✱ ❝♦♠ U = (u, v)_∈R2 :u2_+v2 _{<1 é ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ S}2_{✳ P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡} ❛ ✐♠❛❣❡♠ ❞❛ ♣❛r❛♠❡tr✐③❛çã♦ X1 é ❛ ♣❛rt❡ ❛❜❡rt❛ ❞❡ S2 ❛❝✐♠❛ ❞♦ ♣❧❛♥♦ ①②✳ ❆s ❢✉♥çõ❡s ❝♦♠♣♦♥❡♥t❡s ❞❡ X1 sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ✱ ❥á q✉❡ t❡♠♦s q✉❡ u2+v2 <1✱ ❧♦❣♦ ❛ ❝♦♥❞✐çã♦ ✭✐✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r é ✈❡r✐✜❝❛❞❛✳

❙❡❥❛ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r(x, y, z)_∈X1(U)⊂S2✱ ❝♦♠X1(U)⊂S2 = (x, y, z)⊂S2 :x2+y2 <1

❡ z > 0 ❡ ✜③❡r♠♦s X−₁

1 (x, y, z) 7−→ (x, y) t❡♠♦s u ❡ v ❜❡♠ ❞❡✜♥✐❞♦s ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛

♣♦r u =x ❡ v =y ✱ ❧♦❣♦ X1 é ❜✐❥❡t✐✈❛ ✳ ❊ X

−1

1 é ❛ ♣r♦❥❡çã♦ ❞❡ X1(U)⊆S❡♠ U ✱ q✉❡ é ❝♦♥tí♥✉❛ ❡ ✈❡r✐✜❝❛ ❛ ❝♦♥❞✐çã♦ ii) ❞❛ ❞❡✜♥✐çã♦ ❞❡ s✉♣❡r✜❝✐❡ r❡❣✉❧❛r✳

P❛r❛ ✈❡r✐✜❝❛r ❛ ❝♦♥❞✐çã♦✐✐✐✮ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ ❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛

JX1(u, v) =

    1 0 0 1 −u p

1₋(u2_+v2

−v

p

1₋(u2_+v2 

  ✱

t❡♠ ♣♦st♦ ✷✳

P❛r❛ ❝♦❜r✐r t♦❞❛ ❡s❢❡r❛ ❞❡✈❡♠♦s ✉t✐❧✐③❛r ❛s ♣♦ssí✈❡✐s ♣❛r❛♠❡tr✐③❛çõ❡s s✐♠✐❧❛r❡s ❛X1 ❡ t♦❞❛s ❡ss❛s ❞❡✈❡♠ ✈❡r✐✜❝❛r ❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱♣♦r ❡①❡♠♣❧♦X2(u, v) = (u, v,

p

1₋(u2_+v2₎₎ ✈❛r✐❛♥❞♦ ❛s ✐♠❛❣❡♥s✳✭❋✐❣✉r❛✿ ✷✳✸✮

❋✐❣✉r❛ ✷✳✸✿ P❛r❛♠❡tr✐③❛çõ❡s ❧♦❝❛✐s ❞❛ ❊s❢❡r❛✳ ❋♦♥t❡✿ ❬✼❪

❆ss✐♠ ✈❡r✐✜❝❛♠♦s q✉❡ ❛ ❡s❢❡r❛ é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✸✳ ❙❡ f(u, v) é ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ♦♥❞❡ (u, v) _∈U ✱ ❛❜❡rt♦

❞❡ R2✱ ❡♥tã♦ ♦ ❣rá✜❝♦ ❞❛ ❛♣❧✐❝❛çã♦ X(u, v) = (u, v, f(u, v)) é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✳

❉❡♠♦♥str❛çã♦✳ ❆ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ X ❞❡❝♦rr❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s

❞❡ X sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❆ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ ❞❡X é ✐❣✉❛❧ ❛

J =





1 0 0 1

fu fv



✳

❊①❡♠♣❧♦ ✷✳✶✳✹✳ ❙❡❥❛♠ P0 ∈R3 ❡ a, b∈R3 ✈❡t♦r❡s ▲■✳ ❊♥tã♦ ♦ ♣❧❛♥♦

π=_{p0+λa+µb|λ, µ∈R}

q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ p0 ❡ é ♣❛r❛❧❡❧♦ ❛♦s ✈❡t♦r❡s a ❡ b✱ é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✳ ❉❡ ❢❛t♦✱ s❡❥❛ N =a_×b ♦ ✈❡t♦r ♥♦r♠❛❧ ❛♦ ♣❧❛♥♦ π✳ ❊♥tã♦

π =_{p_∈R3_|< p₋p0, N >= 0}

❙❡♥❞♦ N = (A, B, C)✱ t❡♠♦s q✉❡

π =_{(x, y, z)_∈R3_|Ax+By+Cz =D_}✱♦♥❞❡ D=< p0, N >

❈♦♠♦ N ₆= (0,0,0)✱ t❡♠♦s q✉❡ A₆= 0 ♦✉ B ₆= 0 ♦✉ C ₆= 0✳

❙❡ C ₆= 0✱ ♣♦r ❡①❡♠♣❧♦✱ π é ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ f :R2 _−→R ❞❛❞❛ ♣♦r✿

Z =f(x, y) = D−Ax−By

C ✳

❙❡♥❞♦ ❛ss✐♠✱ π é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✳✭❋✐❣✉r❛✿ ✷✳✹✮

❋✐❣✉r❛ ✷✳✹✿ ❖ ♣❧❛♥♦ π ✈✐st♦ ❝♦♠♦ ❣r❛✜❝♦ ❞❡f(x, y)✳ ❋♦♥t❡✿ ❬✺❪

❊①❡♠♣❧♦ ✷✳✶✳✺✳ ❖ P❛r❛❜♦❧ó✐❞❡ ❡❧í♣t✐❝♦

P =

(x, y, z)_∈R3_|z = x

2

a2 +

y2

b2

✱

♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s ✱ é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✳✭❋✐❣✉r❛✿ ✷✳✺✮

❉❡ ❢❛t♦✱ P =Graf(f)✱ ♦♥❞❡f :R2 _−→R é ❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❛❞❛ ♣♦r

f(x, y) = x

2

a2 +

y2

b2

❋✐❣✉r❛ ✷✳✺✿ P❛r❛❜♦❧ó✐❞❡ ❊❧í♣t✐❝♦ P ❣rá✜❝♦ ❞❡f(x, y)✳ ❋♦♥t❡✿ ❬✺❪

❊①❡♠♣❧♦ ✷✳✶✳✻✳ ❖ P❛r❛❜♦❧ó✐❞❡ ❍✐♣❡r❜ó❧✐❝♦

H =

(x, y, z)_∈R3_|z = y

2

b2 −

x2

a2

✱

♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s ✱t❛♠❜é♠ é ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r✳✭❋✐❣✉r❛✿ ✷✳✻✮

❈♦♠ ❡❢❡✐t♦✱ H é ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ f :R2 _−→R ❞❛❞❛ ♣♦r

f(x, y) = y

2

b2 −

x2

a2

❋✐❣✉r❛ ✷✳✻✿ P❛r❛❜♦❧ó✐❞❡ ❍✐♣❡r❜ó❧✐❝♦ P ❣rá✜❝♦ ❞❡ f(x, y)✳ ❋♦♥t❡✿ ❬✺❪

✷✳✶✳✶ ❈✉r✈❛s ❈♦♦r❞❡♥❛❞❛s

❆s ❝✉r✈❛s ♥❛ s✉♣❡r❢í❝✐❡ q✉❡ s❡ ♦❜t❡♠ ❢❛③❡♥❞♦ ✉♠ ❞♦s ♣❛râ♠❡tr♦su♦✉v ❝♦♥st❛♥t❡✱ sã♦

❝❤❛♠❛❞❛s ❞❡ ❈✉r✈❛s ❈♦♦r❞❡♥❛❞❛s ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ X(u, v)✳

❙❡ X : U _⊂ R2 _−→ R3 _{é ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❧♦❝❛❧ ❞❛ s✉♣❡r❢í❝✐❡✱ ❡♥tã♦ ✜①❛♥❞♦ ✉♠} ♣♦♥t♦ (u0, v0)∈U✱ ❛s ❝✉r✈❛s

u_−→X(u, v0)

v _−→X(u0, v)

sã♦ ❛s ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ❞❡X ❡♠ (u0, v0)✳ ❖s ✈❡t♦r❡sXu(u0, v0) ❡Xv(u0, v0) sã♦ ♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s às ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ✭❋✐❣✳✷✳✼✮✳

❋✐❣✉r❛ ✷✳✼✿ ❱❡t♦r❡s t❛♥❣❡♥t❡s ❛s ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s✳ ❋♦♥t❡✿ ❬✺❪

❊①❡♠♣❧♦ ✷✳✶✳✼✳ ◆❛ ❡s❢❡r❛ S2 _{❛s ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ❝♦♠ ✉♠❛ ❝❡rt❛ ♣❛r❛♠❡tr✐③❛çã♦}

X(u, v) sã♦ ❞❛❞❛s ♣❡❧♦s ♣❛r❛❧❡❧♦s ❡ ♣❡❧♦s ♠❡r✐❞✐❛♥♦s ❞❛ ❡s❢❡r❛✳✭❋✐❣✉r❛✿ ✷✳✽✮

❋✐❣✉r❛ ✷✳✽✿ ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ♥❛ ❡s❢❡r❛✳ ❋♦♥t❡✿ ❬❄❪

❊①❡♠♣❧♦ ✷✳✶✳✽✳ ◆♦ ❝❛s♦ ❞❡ ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❛s ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s s❡❣✉✐♥❞♦ ✉♠❛ ❝❡rt❛ ♣❛r❛♠❡tr✐③❛çã♦✱ sã♦ s❡❣♠❡♥t♦s ❞❡ r❡t❛s ♣❛r❛❧❡❧❛s ❛♦ ❡✐①♦ ❞♦s z ❡ ❝ír❝✉❧♦s ♣❛r❛❧❡❧♦s

❛♦ ♣❧❛♥♦ xy✳✭❋✐❣✉r❛✿ ✷✳✾✮

❋✐❣✉r❛ ✷✳✾✿ ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ♥♦ ❝✐❧✐♥❞r♦✳ ❋♦♥t❡✿ ❬❄❪

❊①❡♠♣❧♦ ✷✳✶✳✾✳ ◆♦ ❝❛s♦ ❞❛ s✉♣❡r❢í❝✐❡ s❡r ♦ ❣r❛✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f✱ ❛s ❝✉r✈❛s ❝♦♦r✲

❞❡♥❛❞❛s sã♦ ♦❜t✐❞❛s ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞♦s ♣❧❛♥♦s ♣❛r❛❧❡❧♦s ❛♦s ♣❧❛♥♦s ❝♦♦r❞❡♥❛❞♦s yz ❡ xz

❝♦♠ G(f)❀ ♦❜s❡r✈❡ q✉❡ ❡ss❛s ❝✉r✈❛s ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦rt♦❣♦♥❛✐s❀✭❋✐❣✉r❛✿ ✷✳✶✶✮

❋✐❣✉r❛ ✷✳✶✵✿ s✉♣❡r❢í❝✐❡ s❡♥❞♦ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ❋♦♥t❡✿ ❬❄❪

✷✳✶✳✷ P❧❛♥♦ ❚❛♥❣❡♥t❡

❈♦♠♦ ❥á ❤❛✈✐❛ ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ❛ ❝♦♥❞✐çã♦iii)❞❛ ❞❡✜♥✐çã♦ ❞❡ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r

♥♦s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❛S❡♠p✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦p_∈S✱ ♦ ❝♦♥❥✉♥t♦

❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s às ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ❞❡ S q✉❡ ♣❛ss❛♠ ♣♦r p✱ ❝♦♥st✐t✉❡♠ ✉♠

♣❧❛♥♦✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r TpS✳

❉❡✜♥✐çã♦ ✷✳✶✳✶✵✳ ❙❡❥❛♠ S _⊂R3 ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r ❡ p_∈S✳ ❉✐③❡♠♦s q✉❡ ✉♠ ✈❡t♦r v ❞❡ R3 é ✉♠ ✈❡t♦r t❛♥❣❡♥t❡ ❛ S ❡♠ p s❡ v = α′

(0)✱ ♦♥❞❡ α : (₋ε, ε) _−→ S é ✉♠❛

❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ S ❡♠ 0 ❡ α(0) =p✳

❊♥t❡♥❞❡♠♦s ♣❡❧❛ ❞❡✜♥✐çã♦ q✉❡ ✉♠ ✈❡t♦r t❛♥❣❡♥t❡ ❛S ❡♠ ✉♠ ♣♦♥t♦p_∈S ❝♦♠♦ s❡♥❞♦

♦ ✈❡t♦r α′

(0) ❞❡ ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛α : (₋ε, ε)_−→S ✱ ❝♦♠ α(0) =p✳

❙❡❥❛♠✿

✶✳ X :U _⊆R2 _−→S✱ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ S❀

✷✳ β : (₋ε, ε)_−→U✱ ❝✉r✈❛ ❡♠ ❯❀

✸✳ α: (₋ε, ε)_−→S✱ ❝✉r✈❛ ❡♠ S ♣❛ss❛♥❞♦ ♣♦r p✱ ♦♥❞❡α =X_◦β✳

❙❡X é ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ S ❡ β é ✉♠❛ ❝✉r✈❛ ❞❡U t❛❧ q✉❡α=X_◦β✱ ❡♥tã♦ ❡ss❡

✈❡t♦r α′

(0) é ❞❛❞♦ ♣♦r✿

α(0) =X_◦β(0)

α′

(0) =dXα(0)α(0)α′(0) =

d

dt(X◦β(0))✳

P♦rt❛♥t♦✱ ❛ ❞❡r✐✈❛❞❛dXq♠❛♣❡✐❛ ♦s ✈❡t♦r❡s ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❝✉r✈❛s ♣❛ss❛♥❞♦ ♣♦rβ(0) =

q ❡♠ ✈❡t♦r❡s ✈❡❧♦❝✐❞❛❞❡ ❞❛s s✉❛s r❡s♣❡❝t✐✈❛s ✐♠❛❣❡♥s ❡♠p=X(q)✳✭❋✐❣✉r❛✿ ✷✳✶✶✮

❋✐❣✉r❛ ✷✳✶✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧α=X_◦β✳ ❋♦♥t❡✿ ❬✺❪

❖s ✈❡t♦r❡s Xu(u0, v0) ❡ Xv(u0, v0) sã♦ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ X ❡♠ (u0, v0)✱ ❥á q✉❡ sã♦ t❛♥❣❡♥t❡s às ❝✉r✈❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ X✳

❉❡✜♥✐çã♦ ✷✳✶✳✶✶✳ ❖ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❛ ✉♠❛ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r S ❡♠ p é ♦ ❝♦♥❥✉♥t♦ ❞❡

✈❡t♦r❡s t❛♥❣❡♥t❡ ❞❛s ❝✉r✈❛s ❡♠ S ♣❛ss❛♥❞♦ ♣♦r p✱ ❝♦♠ p_∈S ✱ ❞❡♥♦t❛❞♦ ♣♦r TpS✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✶✷✳ ❙❡❥❛X :U _−→R3 ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡S✱❝♦♥t❡♥❞♦

✉♠ ♣♦♥t♦p✱ ❡ s❡❥❛(u, v)❝♦♦r❞❡♥❛❞❛s ❞❡U✳ ❖ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛S ❡♠pé ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

❞❡ R3 ❣❡r❛❞♦s ♣❡❧♦s ✈❡t♦r❡s Xu ❡ Xv t❛❧ q✉❡ X(u0, v0) = p✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ α ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡♠ S ❝♦♠

α(t) =X(u(t), v(t))✳

❉❡r✐✈❛♥❞♦ α✱ ♣❡❧❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛✱ t❡♠♦s✿ α′

(t) =Xuu

′

(t) +Xvv

′

(t)

❆ss✐♠✱ α′

(t)é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ✈❡t♦r❡s Xu ❡ Xv✳

P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛❧q✉❡r ✈❡t♦r ❞♦R3 ❣❡r❛❞♦ ♣♦rXu ❡Xv é ❞❛ ❢♦r♠❛aXu+bXv ♣❛r❛ ❛❧❣✉♥s ❡s❝❛❧❛r❡s a ❡ b✳ ❊s♣❡❝✐✜❝❛♥❞♦ ✉♠❛ ❝✉r✈❛ α ♣♦r✿

α(t) = X(u0+at, v0+bt)

t❡♠♦s q✉❡ α é ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡♠ p_∈S✱ s❡♥❞♦ ❛ss✐♠✱

α′

(t) = aXu+bXv

■ss♦ ♠♦str❛ q✉❡ ❝❛❞❛ ✈❡t♦r ♥♦ ❡s♣❛ç♦ ❞❡ S ❣❡r❛❞♦ ♣♦r Xu ❡ Xv é ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❡♠ p ❞❡ ✉♠❛ ❝✉r✈❛ ❞❡ S✳

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ s✉♣❡r❢í❝✐❡ r❡❣✉❧❛r ✱Xu ❡ Xv sã♦ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ P♦rt❛♥t♦✱ s❡❣✉❡✲s❡ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r q✉❡ TpS é ✉♠ ♣❧❛♥♦ ❞❡ R3✱ ❣❡r❛❞♦ ♣♦r Xu ❡

Xv✳✭❋✐❣✉r❛✿ ✷✳✶✷✮

❋✐❣✉r❛ ✷✳✶✷✿ ❖ ♣❧❛♥♦ t❛♥❣❡♥t❡ TpS✳❋♦♥t❡✿ ❬✶✵❪

❉❡✜♥✐çã♦ ✷✳✶✳✶✸✳ ❉❛❞❛ ✉♠❛ s✉♣❡r❢í❝✐❡ X(u, v) ❡♠ ✉♠ ♣♦♥t♦ p_∈U _⊂R3✱ ❞✐③❡♠♦s q✉❡

✉♠ ✈❡t♦r ❞❡ R3 é ♥♦r♠❛❧ ❛ X(u, v) ❡♠ p s❡ ❡❧❡ é ♦rt♦❣♦♥❛❧ ❛♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❡♠ p✱ ♦✉

s❡❥❛✱ s❡ é ♦rt♦❣♦♥❛❧ ❛ t♦❞♦s ♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ X(u, v) ♥♦ ♣♦♥t♦ p✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦

✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ N ♥♦ ♣♦♥t♦ p ♣♦r✿

N = Xu×Xv

|Xu×Xv |

✷✳✷ Pr✐♠❡✐r❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧

▼❡❞✐r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡✱ é ✉♠❛ ❝✉r✐♦s✐❞❛❞❡ ❣❡♦♠étr✐❝❛ q✉❡ q✉❛❧q✉❡r ❤❛❜✐t❛♥t❡ ❞❡ss❛ s✉♣❡r❢í❝✐❡ ❞❡s❡❥❛r✐❛ s❛❜❡r✳ ❊✈✐❞❡♥t❡♠❡♥t❡ ❡ss❛ ❞✐stâ♥❝✐❛ ♥ã♦ s❡rá ❛ ♠❡s♠❛ ❞❛ ♠❡❞✐❞❛ ♣♦r ✉♠ ❤❛❜✐t❛♥t❡ ❞♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ♣♦✐s ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❞á ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✱ ❣❡r❛❧♠❡♥t❡ ♥ã♦ ❡stá ❝♦♥t✐❞♦ ♥❛ s✉♣❡r❢í❝✐❡✳✭❋✐❣✉r❛✿ ✷✳✶✸✮