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3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6...

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HAL Id: inria-00106823 https://hal.inria.fr/inria-00106823v2 Submitted on 27 Oct 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. 3D Sketching with profile curves Floriant Levet, Xavier Granier, Christophe Schlick To cite this version: Floriant Levet, Xavier Granier, Christophe Schlick. 3D Sketching with profile curves. International Symposium on Smart Graphic, Sep 2006, Vancouver, Canada. pp.114-125, 10.1007/11795018_11. inria-00106823v2
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Page 1: 3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6 ]. osM t h se e t se hp por a c a aer l i m e i d t o t p o e hl dy rl a o d l,es

HAL Id: inria-00106823https://hal.inria.fr/inria-00106823v2

Submitted on 27 Oct 2013

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

3D Sketching with profile curvesFloriant Levet, Xavier Granier, Christophe Schlick

To cite this version:Floriant Levet, Xavier Granier, Christophe Schlick. 3D Sketching with profile curves. InternationalSymposium on Smart Graphic, Sep 2006, Vancouver, Canada. pp.114-125, �10.1007/11795018_11�.�inria-00106823v2�

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❚❤✐s ✐s ❛ ♣r❡✲♣r✐♥t ✈❡rs✐♦♥✳ ❚❤❡ ☞♥❛❧ ✈❡rs✐♦♥ ✐s ❛✈❛✐❧❛❜❧❡ ✐♥ ✧▲❡❝t✉r❡ ◆♦t❡s

✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✧ ✲ ❱♦❧✉♠❡ ✹✵✼✸✴✷✵✵✻ ✭■❙❇◆ ✾✼✽✲✸✲✺✹✵✲✸✻✷✾✸✲✼✮✳

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✸❉ ❙❦❡t❝❤✐♥❣ ✇✐t❤ ♣r♦☞❧❡ ❝✉r✈❡s

❋�✁✂✄❛☎ ▲✆✝✆✞✱ ❳❛✝✄✆✂ ●✂❛☎✄✆✂✱ ❛☎❞ ❈✟✂✄✠✞✁✡✟✆ ☛✌✟�✄✌✍

■P❆❘✎❆ ✏✑✒❥✓✔✕ ✭■◆❘■❆ ❢✖✕✖✑✗ ✲ ✎✘❇❘■✮✙❯▼❘ ✺✽✵✵✙ ❯✚✛✜✓✑✗✛✕✢✓ ❇✒✑✣✓✘✖① ✶❀ ✤✺✶✙ ✔✒✖✑✗ ✣✓ ✥✘ ✎✛❜✢✓✑✘✕✛✒✚

✤✤✹✵✺ ❚✘✥✓✚✔✓✙ ✦✑✘✚✔✓✧★✩✪✩✫✬✯✰✳✴✷✩✰✬✻✼✾★✷✼✿❁❅★✳❂✰✷❃❄✰

❊❍❏❑❖◗❱❑❲ ■✚ ✑✓✔✓✚✕ ②✓✘✑✗✙ ✤❨ ✗❩✓✕✔❬✛✚❭ ❬✘✗ ❭✘✛✚✓✣ ✏✒✏✖✥✘✑✛✕② ✘✗ ✘✚ ✓❢✲❪✔✛✓✚✕ ✘✥✕✓✑✚✘✕✛✜✓ ✕✒ ✔✒✚✜✓✚✕✛✒✚✘✥ ✤❨ ❭✓✒♠✓✕✑✛✔ ♠✒✣✓✥✛✚❭ ❢✒✑ ✑✘✏✛✣ ✏✑✒✲✕✒✕②✏✛✚❭✙ ✘✗ ✛✕ ✘✥✥✒❫✗ ✕❬✓ ✖✗✓✑ ✕✒ ✛✚✕✖✛✕✛✜✓✥② ❭✓✚✓✑✘✕✓ ✘ ✥✘✑❭✓ ✑✘✚❭✓ ✒❢✣✛❴✓✑✓✚✕ ✗❬✘✏✓✗❵ ■✚ ✕❬✛✗ ✏✘✏✓✑✙ ❫✓ ✏✑✓✗✓✚✕ ✗✒♠✓ ✗❩✓✕✔❬✛✚❭ ✛✚✕✓✑✘✔✕✛✒✚✗❢✒✑ ✤❨ ♠✒✣✓✥✛✚❭✙ ❜✘✗✓✣ ✒✚ ✘ ✗✓✕ ✒❢ ✕❫✒ ✣✛❴✓✑✓✚✕ ❜✛✣✛♠✓✚✗✛✒✚✘✥ ✗❩✓✕✔❬✓✗✭✏✑✒❪✥✓ ✔✖✑✜✓ ✘✚✣ ✗✛✥❬✒✖✓✕✕✓ ✔✖✑✜✓✮❵ ❇② ✖✗✛✚❭ ✕❬✓✗✓ ✕❫✒ ✗❩✓✕✔❬✓✗ ✘✚✣✔✒♠❜✛✚✛✚❭ ✕❬✓♠ ❫✛✕❬ ✘ ❭✓✗✕✖✑✓ ❭✑✘♠♠✘✑✙ ✘ ✜✓✑② ✥✘✑❭✓ ✜✘✑✛✓✕② ✒❢ ✗❬✘✏✓✗✭✛✚✔✥✖✣✛✚❭ ✗❬✘✏✓✗ ❫✛✕❬ ✕✒✏✒✥✒❭✛✔✘✥ ❬✒✥✓✗✮ ✔✘✚ ❜✓ ✓✘✗✛✥② ✏✑✒✣✖✔✓✣ ❜② ✒✖✑✛✚✕✓✑✘✔✕✛✜✓ ♠✒✣✓✥✛✚❭ ✓✚✜✛✑✒✚♠✓✚✕❵q③④ ⑤⑥❖⑦❏⑧ ⑨❩✓✕✔❬✲❜✘✗✓✣ ✤❨ ▼✒✣✓✥✛✚❭

⑩ ❶❷❸❹❺❻❼❽❸❾❺❷

❿✁✠✞ ✆➀✄✠✞✄☎➁ ➁✆✁➂✆✞✂✄✌ ➂✁❞✆�✄☎➁ ✠✁➃✞➄❛✂✆✠ ✂✆➅➆✄✂✆ ❛ �✁✞ ✁➃ ✠✍✄�� ➃✂✁➂ ✞✟✆ ➆✠✆✂✱ ✞✁✂✆❛��➇ ➁✆✞ ✞✟✆ ✠✟❛✡✆ ✟✆ ➄❛☎✞✠ ✞✁ ✁➈✞❛✄☎➉ ➊✟✆ ✌✁➂✡�✆➀✄✞➇ ✁➃ ✌✁☎✝✆☎✞✄✁☎❛� ➁✆✁➂✆✞➋✂✄✌ ➂✁❞✆�✄☎➁ ✟❛✠ ✂✆✌✆☎✞�➇ �✆❞ ✞✁ ✞✟✆ ❞✆✝✆�✁✡➂✆☎✞ ✁➃ ✠✁➂✆ ❛�✞✆✂☎❛✞✄✝✆ ➂✁❞✆�✄☎➁✞✆✌✟☎✄➅➆✆✠ ✞✟❛✞ ✌❛☎ ➈✆ ✌❛✞✆➁✁✂✄➌✆❞ ❛✠ ➍➎ ➏➐➑➒➓➔→➣↔➉ ↕✄✞✟ ❛ ✠✍✆✞✌✟✄☎➁ ➂✆✞❛✡✟✁✂✱➆✠✆✂✠ ✌❛☎ ✂❛✡✄❞�➇ ➁✆☎✆✂❛✞✆ ❛ ➙➛ ✡✂✁✞✁✞➇✡✆ ✞✁ ✄��➆✠✞✂❛✞✆ ✞✟✆ ➙➛ ✁➈➜✆✌✞ ✞✟✆➇ ✟❛✝✆✄☎ ➂✄☎❞➉

➙➛ ✠✍✆✞✌✟✄☎➁ ✄✠ ❛�✠✁ ➄✆�� ✠➆✄✞✆❞ ➃✁✂ ➈✆✄☎➁ ➆✠✆❞ ✁☎ ➂✁➈✄�✆ ❞✆✝✄✌✆✠✱ ✠➆✌✟ ❛✠✠➂❛✂✞✡✟✁☎✆✠✱ ➝➛➞✠ ✁✂ ✞❛➈�✆✞ ➝❈✠➉ ➟✝✆☎ ✄➃ ✞✟✆ ✠✞✁✂❛➁✆ ❛☎❞ ✌✁➂✡➆✞❛✞✄✁☎ ✂✆✠✁➆✂✌✆✠✁➃ ✠➆✌✟ ❞✆✝✄✌✆✠ ✟❛✝✆ ➁✂✆❛✞�➇ ✄☎✌✂✆❛✠✆❞ ✄☎ ✂✆✌✆☎✞ ➇✆❛✂✠✱ ✞✟✆ �✄➂✄✞❛✞✄✁☎ ✞✁ ✁☎✆➋✟❛☎❞✆❞ ➠❛✠ ✞✟✆ ✁✞✟✆✂ ✁☎✆ ✄✠ ➆✠➆❛��➇ ✌❛✂✂➇✄☎➁ ✞✟✆ ❞✆✝✄✌✆➡ ✠✞➇�➆✠➋➈❛✠✆❞ ✄☎✞✆✂❛✌✞✄✁☎✱➂❛✍✆✠ ✄✞ ❞✄➢✌➆�✞ ✞✁ ❛❞❛✡✞ ✆➀✄✠✞✄☎➁ ➂✁❞✆�✄☎➁ ✠✁➃✞➄❛✂✆ ✞✟❛✞ ✞❛✍✆ ➃➆�� ❛❞✝❛☎✞❛➁✆✁➃ ✞✟✆ ➂✁➆✠✆➤✍✆➇➈✁❛✂❞ ✄☎✞✆✂➃❛✌✆➉

↕✄✞✟ ❛ ✠✞➇�➆✠✱ ❛ ❞✂❛➄✄☎➁ ➂✆✞❛✡✟✁✂ ❛✡✡✆❛✂✠ ➅➆✄✞✆ ☎❛✞➆✂❛� ➃✁✂ ➙➛ ➂✁❞✆�✄☎➁✞❛✠✍✠➉ ➛✄➥✆✂✆☎✞ ✠✍✆✞✌✟✄☎➁ ❛✡✡✂✁❛✌✟✆✠ ✟❛✝✆ ❛�✂✆❛❞➇ ➈✆✆☎ ✡✂✆✠✆☎✞✆❞ ❛✠ ✆➢✌✄✆☎✞ ❛☎❞✄☎✞➆✄✞✄✝✆ ✄☎✞✆✂➃❛✌✆✠ ➃✁✂ ✌✂✆❛✞✄☎➁ ❛☎❞ ✆❞✄✞✄☎➁ ➙➛ ➂✁❞✆�✠➉ ➊✟✆➇ ✟❛✝✆ ✠✟✁➄☎ ✞✟❛✞❛ ✠✟❛✡✆ ✌❛☎ ➈✆ ✂✆✌✁☎✠✞✂➆✌✞✆❞ ✆✄✞✟✆✂ ➃✂✁➂ ✠✞✂✁✍✆✠ ❛☎❞ �✄☎✆✠ ➦➧➨➩✱ ✁✂ ✌➆✂✝✆✠ ❛☎❞➁✆✠✞➆✂✆✠ ➦➫➭✱ ➧➫✱ ➫➫✱ ➯✱ ➫➲➩➉ ➳➆✄�❞✄☎➁ ➆✡✁☎ ✞✟✆ �❛✞✞✆✂ ✌➆✂✝✆✠ ❛☎❞ ➁✆✠✞➆✂✆✠ ✠➇✠✞✆➂✠✱✁➆✂ ➁✁❛� ✄✠ ✞✁ ✆➀✞✆☎❞ ✞✟✆ ❛➈✄�✄✞➇ ✁➃ ✞✟✆✠✆ ✠✍✆✞✌✟✄☎➁ ❛✡✡✂✁❛✌✟✆✠✱ ➄✟✄�✆ ✍✆✆✡✄☎➁✆➢✌✄✆☎✞ ❛☎❞ ✄☎✞➆✄✞✄✝✆ ✄☎✞✆✂❛✌✞✄✁☎➉

➊✟✆✂✆ ❛✂✆ ➃✁➆✂ ➂❛✄☎ ✌✁☎✞✂✄➈➆✞✄✁☎✠ ✄☎ ✞✟✄✠ ✡❛✡✆✂➉ ➠✄➡ ➊✟✆ ➵✂✠✞ ✁☎✆ ✄✠ ❛ ✞✆✌✟➋☎✄➅➆✆ ✞✁ ❛✌✌✁➆☎✞ ➃✁✂ ❛ ✡✂✁➵�✆ ✌➆✂✝✆ ✞✁ ✄☎➃✆✂ ❛ ➙➛ ➂✁❞✆� ➸ ☛✆✌✞✄✁☎ ➯➉ ↕✄✞✟ ✞✟✄✠❛✡✡✂✁❛✌✟✱ ✞✟✆ ➆✠✆✂ ✄✠ ☎✁✞ �✄➂✄✞✆❞ ✞✁ ➺➈�✁➈➈➇➺ ✠✟❛✡✆✠ ➈➆✞ ✌❛☎ ✆❛✠✄�➇ ✄☎✌�➆❞✆ ✠✟❛✂✡

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❡❞❣❡s✳ ▼♦r❡♦✈❡r✱ s❤❛♣❡s t❤❛t ✐♥❝❧✉❞❡ t♦♣♦❧♦❣✐❝❛❧ ❤♦❧❡s ❝❛♥ ❛❧s♦ ❜❡ ❡❛s✐❧② ❣❡♥❡r✲

❛t❡❞ ✭s❡❡ ❋✐❣✉r❡ ✶✮✳ ✭✐✐✮ ❚❤❡ s❡❝♦♥❞ ❝♦♥tr✐❜✉t✐♦♥ ✐s ❛ s❡t ♦❢ ✐♠♣r♦✈❡♠❡♥ts ♦✈❡r

t❤❡ ♦r✐❣✐♥❛❧ ❚❡❞❞② s②st❡♠ ❬✶✵❪ ④ ❙❡❝t✐♦♥ ✸✳ ✭✐✐✐✮ ❚❤❡ t❤✐r❞ ♦♥❡ ✐s ❛ ♠❡t❤♦❞ ❢♦r

❧♦❝❛❧ ❛♥❞ ❣❧♦❜❛❧ ❡❞✐t✐♦♥ ♦❢ t❤❡ s❤❛♣❡ ❜② ❛ ❧♦❝❛❧ ♦r ❣❧♦❜❛❧ ❝❤❛♥❣❡ ♦❢ t❤❡ ♣r♦☞❧❡ ④

❙❡❝t✐♦♥ ✺✳ ✭✐✈✮ ❋✐♥❛❧❧②✱ ✇❡ ♣r❡s❡♥t s❡✈❡r❛❧ ❣❡st✉r❡✲❜❛s❡❞ ✐♥t❡r❛❝t✐♦♥s t♦ ❝r❡❛t❡ ❛

❝♦♠♣❧❡① s❤❛♣❡ ❢r♦♠ t❤❡ ♣r♦☞❧❡ ❛♥❞ s✐❧❤♦✉❡tt❡ s❦❡t❝❤❡s ④ ❙❡❝t✐♦♥ ✺✳

✷ P�✁✂✄☎✆✝ ✞☎�✟

▼❛♥② t❡❝❤♥✐q✉❡s ❛r❡ ✉s❡❞ ❜② ❛rt✐sts ✐♥ ♦r❞❡r t♦ s✉❣❣❡st t❤❡ ♦❜❥❡❝t✬s s❤❛♣❡ s✉❝❤ ❛s

❝❤❛r❛❝t❡r✐st✐❝ ❧✐♥❡s ✭♦r ❝♦♥t♦✉r ❧✐♥❡s✮ ♦r s❤❛❞❡❞ ❛r❡❛s✳ ❙✐♥❝❡ ❞r❛✇✐♥❣ ✐s ❛ ❢❛♠✐❧✐❛r

t❛s❦ ❢♦r ❛ ❧♦t ♦❢ ♣❡♦♣❧❡✱ ❭s❦❡t❝❤✐♥❣✧ ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❛s ❛ ♥❛t✉r❛❧ ❛❧t❡r♥❛t✐✈❡

❢♦r ✸❉ ♠♦❞❡❧✐♥❣ t❛s❦s✳ ❊①✐st✐♥❣ s❦❡t❝❤✐♥❣ ❛♣♣r♦❛❝❤❡s ❝❛♥ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ t❤r❡❡

❝❛t❡❣♦r✐❡s✿ t❤❡ ❧✐♥❡✲❛♥❞✲str♦❦❡ ❛♣♣r♦❛❝❤✱ t❤❡ ♣❛✐♥t✐♥❣✲♦r✲s❤❛❞✐♥❣ ❛♣♣r♦❛❝❤✱ ❛♥❞

t❤❡ ❝✉r✈❡✲❛♥❞✲❣❡st✉r❡ ❛♣♣r♦❛❝❤✳

❚❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ t❤❡ ❧✐♥❡✲❛♥❞✲str♦❦❡ ❛♣♣r♦❛❝❤ ✐s t♦ ✐♥❢❡r ❛ ✸❉ ✈♦❧✉♠❡ ❢r♦♠

❝❤❛r❛❝t❡r✐st✐❝ ❧✐♥❡s ❞r❛✇♥ ❜② ❛ ✉s❡r ❬✺✱ ✠✸❪✳ ❙✉❝❤ ❛ s♦❧✉t✐♦♥ ❝❛♥ ❡✈❡♥ ❜❡ ✐♥t❡r❛❝✲

t✐✈❡ ❬✶✺✱ ✶✽❪✳ ❲❤❡♥ ❛♠❜✐❣✉✐t✐❡s ♦❝❝✉r✱ t❤❡② ❝❛♥ ❜❡ r❡♠♦✈❡❞ ❜② ❛ ✉s❡r s❡❧❡❝t✐♦♥

♦❢ t❤❡ ❝♦rr❡❝t ♠♦❞❡❧ ✐♥t♦ ❛ ❧✐st ♦❢ ♣♦ss✐❜❧❡ r❡❝♦♥str✉❝t✐♦♥s ❬✶✹✱ ✻❪✳ ▼♦st ♦❢ t❤❡s❡

❛♣♣r♦❛❝❤❡s ❛r❡ ❧✐♠✐t❡❞ t♦ ♣♦❧②❤❡❞r❛❧ ♠♦❞❡❧s✱ ♣r❡✲❞❡☞♥❡❞ s❤❛♣❡s ♦r ♣❛r❛♠❡tr✐③❡❞

♦❜❥❡❝ts ❬✠✺❪✳ ❲✐t❤ r❡❝❡♥t ♠❡t❤♦❞s ❬✸✱ ✠✶❪✱ ♠♦r❡ ❝♦♠♣❧❡① ✸❉ ❧✐♥❡s ❝❛♥ ❜❡ s❦❡t❝❤❡❞✱

❜✉t t❤❡ ☞♥❛❧ ♠♦❞❡❧ ✐s st✐❧❧ ❧✐♠✐t❡❞ t♦ ❛ ✇✐r❡❢r❛♠❡ ♦♥❡✳ ▼♦r❡♦✈❡r✱ str♦❦❡✲❜❛s❡❞

❛♣♣r♦❛❝❤❡s r❡❞✉❝❡ t❤❡ ❧✐♠✐t❛t✐♦♥s ♦♥ ♣♦ss✐❜❧❡ ✸❉ ❧✐♥❡s ❛♥❞ ❝✉r✈❡s ❬✠✵✱ ✠❪✱ ❜✉t

t❤❡② ❛r❡ ♠♦st❧② ❧✐♠✐t❡❞ t♦ ✐❧❧✉str❛t✐♦♥ s✐♥❝❡ t❤❡② ❝❛♥♥♦t r❡❛❧❧② r❡❝♦♥str✉❝t ❛ ❢✉❧❧

✸❉ ♦❜❥❡❝t✳

❚❤❡ ♣❛✐♥t✐♥❣✲♦r✲s❤❛❞✐♥❣ ❛♣♣r♦❛❝❤ ❛❧❧♦✇s t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ ❤✐❣❤❧② ❞❡t❛✐❧❡❞

♠♦❞❡❧s✳ ❊①t❡♥❞✐♥❣ t❤❡ ✇♦r❦ ♦❢ ❲✐❧❧✐❛♠s ❬✠✹❪✱ ❖✈❡r✈❡❧❞ ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♦♠❡tr✐❝

♠♦❞❡❧✐♥❣ t❡❝❤♥✐q✉❡ ❜❛s❡❞ ♦♥ t❤❡ ♣❛✐♥t✐♥❣ ♦❢ ❛ s✉r❢❛❝❡ ❣r❛❞✐❡♥t ❬✠✠❪✳ ❙❤❛♣❡ ❡❞✐t✐♥❣

t❡❝❤♥✐q✉❡s ❜❛s❡❞ ♦♥ s❤❛❞✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❘✉s❤♠❡✐❡r ❡t ❛❧✳ ❬✶✻❪✳

▼♦r❡ r❡❝❡♥t❧②✱ ✸❉ ❤❡✐❣❤t✲☞❡❧❞ ❣❡♥❡r❛t✐♦♥ ❜② ✠❉ s❤❛❞✐♥❣ ❤❛s ❜❡❡♥ ♣r❡s❡♥t❡❞ ❜②

❑❡r❛✉tr❡t ❡t ❛❧✳ ❬✶✠❪✳

❈✉rr❡♥t❧②✱ t❤❡ ♠♦st ♣♦✇❡r❢✉❧ t❡❝❤♥✐q✉❡s ❤❛✈❡ ❜❡❡♥ ❜❛s❡❞ ♦♥ t❤❡ ❝✉r✈❡✲❛♥❞✲

❣❡st✉r❡ ❛♣♣r♦❛❝❤✳ ❚❤❡s❡ t❡❝❤♥✐q✉❡s ❛❧❧♦✇ t❤❡ ✉s❡r t♦ ❝r❡❛t❡ ❛ ❧❛r❣❡ ✈❛r✐❡t② ♦❢

❢r❡❡✲❢♦r♠ s❤❛♣❡s ❬✠✻✱ ✶✵✱ ✶✸❪ ❜② ✉s✐♥❣ ❛ ❣❡st✉r❡ ❣r❛♠♠❛r ✇❤✐❝❤ ❝♦♥✈❡rts s♦♠❡

❞r❛✇♥ ❝✉r✈❡s ✐♥t♦ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♠♦❞❡❧✐♥❣ ♦♣❡r❛t✐♦♥✿ ❡①tr✉s✐♦♥✱ ❞❡❢♦r♠❛t✐♦♥✱

❝✉tt✐♥❣✱ ❡t❝✳

❊✐t❤❡r ✈❛r✐❛t✐♦♥❛❧ s✉r❢❛❝❡s ❬✹✱ ✶✶✱ ✠✼❪✱ ✐♠♣❧✐❝✐t s✉r❢❛❝❡s ❬✶✼❪ ♦r ❝♦♥✈♦❧✉t✐♦♥

s✉r❢❛❝❡s ❬✶❪ ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❜② ❝✉r✈❡✲❛♥❞✲❣❡st✉r❡ ❛♣♣r♦❛❝❤❡s✳ ❚❤✐s ❤❛s t❤❡ ❛❞✲

✈❛♥t❛❣❡ t♦ ❣❡♥❡r❛t❡ s♠♦♦t❤ s✉r❢❛❝❡s✱ ❜✉t ❛❧s♦ ❡♠♣❤❛s✐③❡s ❛ ✧❜❧♦❜❜②✧ ❛s♣❡❝t ❢♦r

t❤❡ r❡s✉❧t✐♥❣ s❤❛♣❡s✳ ❚♦ r❡❞✉❝❡ t❤✐s ❜❧♦❜❜② ❛s♣❡❝t✱ ❚❛✐ ❡t ❛❧✳ ❬✶✾❪ ❤❛✈❡ ♣r♦♣♦s❡❞

t♦ ✉s❡ ❛ ♣r♦☞❧❡ ❝✉r✈❡✱ ❞❡☞♥❡❞ ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s✳ ✡r♦☞❧❡ ❝✉r✈❡s ❤❛✈❡ ❜❡❡♥ ✉s❡❞

♣r❡✈✐♦✉s❧② ✐♥ ❛ s❦❡t❝❤✐♥❣ ❡♥✈✐r♦♥♠❡♥t ❧✐♠✐t❡❞ t♦ ❣❡♥❡r❛❧✐③❡❞ ❝②❧✐♥❞❡rs ❬✼❪✳ ❖♥❡

♥✐❝❡ ♣r♦♣❡rt② ♦❢ ✉s✐♥❣ ✐♠♣❧✐❝✐t s✉r❢❛❝❡s ❢♦r t❤❡ ✐♥❢❡rr❡❞ ❣❡♦♠❡tr② ✐s t❤❛t t❤❡

r❡s✉❧t✐♥❣ s❤❛♣❡s ❝❛♥ ❜❡ ❡❛s✐❧② ♠❡r❣❡❞ ❜② ✉s✐♥❣ ❝❧❛ss✐❝❛❧ ❈❙● ♦♣❡r❛t♦rs✳ ❖♥ t❤❡

♦t❤❡r ❤❛♥❞✱ t❤❡ ♠❛✐♥ ❞r❛✇❜❛❝❦ ✐s t❤❛t s♦♠❡ ❡①♣❡♥s✐✈❡ t❡ss❡❧❧❛t✐♦♥ st❡♣ ❤❛s t♦

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PRIN

T✭❛✮ ✭❜✮

❋✐❣✳ ✶✳ ❆ t♦r❡ ❝r❡❛t❡❞ ✇�t❤ t✇♦ s❦❡t❝❤❡s

✭s�❧❤♦✉❡tt❡ ❝✉r✈❡ ❛♥❞ ♣r♦☞❧❡ ❝✉r✈❡✮✁

❋✐❣✳ ✷✳ ❙�❧❤♦✉❡tt❡ ❝✉r✈❡ ✭✂r❡②✮✱ r❡❝♦♥✲

str✉❝t❡❞ s❦❡❧❡t♦♥ ✭❜♦❧❞ ❜❧❛❝❦✮ ❛♥❞ ❝♦rr❡✲

s♣♦♥❞�♥✂ tr�❛♥✂✉❧❛t�♦♥ ✭❞♦t ❜❧❛❝❦✮✁

✄☎ ☎♠✆✝✞✟☎✠ ✡✞ ☛✞✌✍☎✎✡ ✡✏☎ ✑✒✎❢✓☛☎ ✔✌✡✞ ✓ ✑☎✡ ✞❢ ✡✎✔✓✌✕✝☎✑ ✡✏✓✡ ☛✓✌ ✄☎ ✑☎✌✡ ✡✞ ✡✏☎

✕✎✓✆✏✔☛✑ ✏✓✎✠✖✓✎☎✗ ✓✌✠ ☛✞♠✆✝☎① ✑✏✓✆☎✑ ☛✓✌ ✎☎q✒✔✎☎ ✓ ✝✞✡ ✞❢ ❈✘● ✞✆☎✎✓✡✔✞✌✑✙

❇✓✑☎✠ ✞✌ ✡✏☎✑☎ ✞✄✑☎✎✍✓✡✔✞✌✑✗ ✖☎ ✏✓✍☎ ✠☎✑✔✕✌☎✠ ✞✒✎ ✑✚☎✡☛✏✔✌✕ ☎✌✍✔✎✞✌♠☎✌✡

✞✌ ✡✏☎ ❢✞✝✝✞✖✔✌✕ ☛✎✔✡☎✎✔✓✙ ✛✔✎✑✡✗ ✞✒✎ ✑✟✑✡☎♠ ✖✔✝✝ ✠✔✎☎☛✡✝✟ ✔✌❢☎✎ ✓ ♠☎✑✏ ❢✎✞♠ ✓

✑☎✡ ✞❢ ☛✒✎✍☎✜✓✌✠✜✕☎✑✡✒✎☎ ☎✝☎♠☎✌✡✑✙ ■✌ ✓✠✠✔✡✔✞✌ ✡✞ ✑✆☎☎✠✗ ✒✑✔✌✕ ✓ ♠☎✑✏ ✞✢☎✎✑

✡✏☎ ✆✞✑✑✔✄✔✝✔✡✟ ✞❢ ✓✠✓✆✡✔✌✕ ✡✏☎ ✑✓♠✆✝✔✌✕ ✓☛☛✞✎✠✔✌✕ ✡✞ ✡✏☎ ✝✞☛✓✝ ☛✒✎✍✓✡✒✎☎ ✞❢ ✡✏☎

✑✒✎❢✓☛☎ ✣✔✙☎✙✗ ✔♠✆✎✞✍☎ ✡✏☎ ✆✎☎☛✔✑✔✞✌ ✖✏☎✎☎ ✌☎☎✠☎✠✤✙ ✘☎☛✞✌✠✗ ✞✒✎ ✑✟✑✡☎♠ ✖✔✝✝ ☎①✡☎✌✠

✡✏☎ ✕✎✓♠♠✓✎ ✠☎✥✌☎✠ ✔✌ ❚☎✠✠✟ ❬✦✵❪ ✄✟ ☛✎☎✓✡✔✌✕ ✓ ✆✎✞✥✝☎ ☛✒✎✍☎ ❢✞✎ ✡✏☎ ✕☎✞♠☎✡✎✔☛

♠✞✠☎✝ ✡✏✓✡ ☛✓✌ ✄☎ ☎✠✔✡☎✠ ☎✔✡✏☎✎ ✕✝✞✄✓✝✝✟ ✞✎ ✝✞☛✓✝✝✟✙

✸ ✧★✩★✪✫✬ ✫✯✯✪✰✫✴✹

✺✑ ✑✓✔✠ ✓✄✞✍☎✗ ✞✒✎ ✓✆✆✎✞✓☛✏ ✔✑ ✄✓✑✔☛✓✝✝✟ ✓✌ ☎①✡☎✌✑✔✞✌ ✞❢ ✡✏☎ ❚☎✠✠✟ ✑✟✑✡☎♠✗ ✓✌✠

✡✏✒✑ ✒✑☎✑ ✓ ✑✔♠✔✝✓✎ ❢✞✒✎✜✑✡☎✆ ✆✎✞☛☎✑✑ ✓✑ ✡✏☎ ✞✌☎ ✆✎✞✆✞✑☎✠ ✄✟ ■✕✓✎✓✑✏✔ ☎✡ ✓✝ ❬✦✵❪✿

✦✙ ✺ ✑✔✝✏✞✒☎✡✡☎ ☛✒✎✍☎ ✔✑ ✑✚☎✡☛✏☎✠ ✄✟ ✡✏☎ ✒✑☎✎ ✓✌✠ ✡✏☎✌ ✑✓♠✆✝☎✠✙

✻✙ ✺ ☛✞✌✑✡✎✓✔✌☎✠ ❉☎✝✓✒✌✓✟ ✡✎✔✓✌✕✒✝✓✡✔✞✌ ✣❈❉❚✤ ✔✑ ☛✞♠✆✒✡☎✠ ❢✎✞♠ ✡✏☎ ✑✔✝✏✞✒✜

☎✡✡☎ ✑✓♠✆✝☎✑✗ ✓✌✠ ✒✑☎✠ ✡✞ ☎①✡✎✓☛✡ ✓ ✑✚☎✝☎✡✞✌ ✣✑☎☎ ✛✔✕✒✎☎ ✻✤✙

✼✙ ✛✞✎ ☎✓☛✏ ✑✚☎✝☎✡✞✌ ✆✞✔✌✡✗ ✓✌ ☎✝☎✍✓✡✔✞✌ ✠✔✑✡✓✌☛☎ ✔✑ ☛✞♠✆✒✡☎✠✗ ✓✑ ✡✏☎ ✓✍☎✎✓✕☎

✠✔✑✡✓✌☛☎ ✡✞ ☛✞✌✌☎☛✡☎✠ ✑✔✝✏✞✒☎✡✡☎ ✑✓♠✆✝☎✑✙

✽✙ ❊✓☛✏ ✔✌✡☎✎✌✓✝ ☎✠✕☎ ✞❢ ✡✏☎ ✡✎✔✓✌✕✒✝✓✡✔✞✌ ✔✑ ✑✓♠✆✝☎✠✗ ✓✌✠ ✡✏☎ ☛✞✎✎☎✑✆✞✌✠✔✌✕

✆✞✔✌✡✑ ✓✎☎ ✡✎✓✌✑✝✓✡☎✠ ✓☛☛✞✎✠✔✌✕ ✡✞ ✡✏☎ ☎✝☎✍✓✡✔✞✌ ✠✔✑✡✓✌☛☎✗ ✡✞ ☛✎☎✓✡☎ ✡✏☎ ✥✌✓✝

♠☎✑✏ ✍☎✎✡✔☛☎✑✙

❈✞♠✆✓✎☎✠ ✡✞ ✡✏☎ ✞✎✔✕✔✌✓✝ ❚☎✠✠✟ ✔♠✆✝☎♠☎✌✡✓✡✔✞✌✗ ✞✒✎ ✑✟✑✡☎♠ ✆✎✞✆✞✑☎✑ ✡✖✞ ✔♠✜

✆✎✞✍☎♠☎✌✡✑ ✞❢ ✡✏✔✑ ❢✞✒✎✜✑✡☎✆ ✆✎✞☛☎✑✑✙ ✛✔✎✑✡✗ ✡✏☎ ☎✝☎✍✓✡✔✞✌ ✑✡☎✆ ✔✑ ✑✝✔✕✏✡✝✟ ♠✞✠✔✥☎✠✗

✔✌ ✞✎✠☎✎ ✡✞ ✓☛☛✞✒✌✡ ❢✞✎ ✌✞✌✜☛✞✌✍☎① ✆✎✞✥✝☎ ☛✒✎✍☎✑✙ ✘☎☛✞✌✠✗ ✖☎ ✆✎✞✆✞✑☎ ✓ ✄☎✡✡☎✎

☎✑✡✔♠✓✡✔✞✌ ✞❢ ✡✏☎ ✌✞✎♠✓✝ ✍☎☛✡✞✎ ✓✡ ✡✏☎ ✎☎✑✒✝✡✔✌✕ ✍☎✎✡✔☛☎✑✗ ✔✌ ✞✎✠☎✎ ✡✞ ✞✄✡✓✔✌ ✓

✑♠✞✞✡✏☎✎ ✓✆✆☎✓✎✓✌☛☎ ✞❢ ✡✏☎ ✥✌✓✝ ✼❉ ✞✄❥☎☛✡✙

✾❀❁ ❂❃❄❅❍❏❑▲▼ ▲◆ ❑▼❏❄❖▼❍❃ P▲❑▼❏◗

❲✔✡✏ ❚☎✠✠✟✗ ♠☎✑✏ ✍☎✎✡✔☛☎✑ ✓✎☎ ☛✎☎✓✡☎✠ ✄✟ ✑✓♠✆✝✔✌✕ ✡✏☎ ✔✌✡☎✎✌✓✝ ☎✠✕☎✑ ✞❢ ✡✏☎

✡✎✔✓✌✕✒✝✓✡✔✞✌ ✞✌ ✡✏☎ ✑✔✝✏✞✒☎✡✡☎ ✆✝✓✌☎ ✣❘❯❳❱

❘❯❨ ✤ ✓✌✠ ☎✝☎✍✓✡✔✌✕ ✡✏☎♠ ✓☛☛✞✎✠✔✌✕ ✡✞

Page 6: 3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6 ]. osM t h se e t se hp por a c a aer l i m e i d t o t p o e hl dy rl a o d l,es

PRE-

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T❱

▲✈

▲③♠�①

✁✂✄▲✈

✆✝✞

✝✟

P♣

✡❛☛

✡❛☛✝✞

✌s

✏r♦✑❧❡ ✏r♦❥❡❝t❡❞ ❙✒✓✔❧✐♥❣ ✏r♦✑❧❡ ✏r♦❥❡❝t❡❞ ❙✒✓✔❧✐♥❣

✭✕✮ ❈✖✗✘✙✚ ✛✜✖✢✣✙ ✤✉✜✘✙ ✭❚✙✥✥②✮ ✭❜✮ ◆✖✗✲✤✖✗✘✙✚ ✛✜✖✢✣✙ ✤✉✜✘✙

❋✦✧✳ ✸✳ ❉✐★❡r❡♥❝❡ ✩❡t✇❡❡♥ ✒ ❝♦♥✪❡✫ ✒♥❞ ✒ ♥♦✬❝♦♥✪❡✫ ✔r♦✑❧❡ ❝✯r✪❡✰

✱ ✴✵✶✴✷✹✱✶ ✺✶✻✼✹✽ ✴✷✶✾✽ ✿❀✽✽ ❁✵❂✷✶✽ ❃✿✱❄❄❅ ❯❆❢✻✶❇✷❆✱❇✽✹❊● ❇❤✵❀ ✴✱❆ ✻❆✹❊ ❍✻✶❦ ❢✻✶

✴✻❆✾✽■ ✺✶✻✼✹✽ ✴✷✶✾✽❀ ✿✵❅✽❅● ✺✶✻✼✹✽ ✴✷✶✾✽❀ ❇❤✱❇ ✴✱❆ ❏✽ ✽■✺✶✽❀❀✽❑ ✱❀ ✱ ❤✽✵❂❤❇▼✼✽✹❑❄❅

❲❤✽❆ ✷❀✵❆❂ ❆✻❆▼✴✻❆✾✽■ ✺✶✻✼✹✽ ✴✷✶✾✽❀● ❍✽ ✺✶✻✺✻❀✽ ❇✻ ✴✻❖✺✷❇✽ ✱ ❀✽❇ ✻❢ ❀✱❖✺✹✽

✺✻✵❆❇❀ ✻❢ ❇❤✽ ✺✶✻✼✹✽ ✴✷✶✾✽ ◗❘ ❬ ✿◗❘❭ ❪ ◗❘❫ ❄● ✱❆❑ ✷❀✽ ❇❤✽❀✽ ❀✱❖✺✹✽❀ ✱✹✻❆❂ ✽✱✴❤

✵❆❇✽✶❆✱✹ ✽❑❂✽ ❇✻ ✻❏❇✱✵❆ ❇❤✽ ✺✻❀✵❇✵✻❆ ✻❢ ❇❤✽ ✴✻✶✶✽❀✺✻❆❑✵❆❂ ❖✽❀❤ ✾✽✶❇✵✴✽❀ ◗ ❬

✿◗❴❪◗❵❪ ◗❫❄❅ q❤✽ ❆✽❍ ✽✹✽✾✱❇✵✻❆ ✻❢ ✱ ✼❆✱✹ ❃④ ✾✽✶❇✽■ ✵❀ ❂✵✾✽❆ ❏❊⑤

◗❫ ❬ ✿◗⑥⑦⑧

⑨⑩❴❫ ❄◗

❘❫ ❪

❍❤✽✶✽ ⑧⑨⑩❴❫ ✵❀ ❇❤✽ ❖✱■✵❖✷❖ ❤✽✵❂❤❇ ✻❢ ❇❤✽ ✺✶✻✼✹✽ ✴✷✶✾✽ ✿❀✽✽ ❁✵❂✷✶✽ ❃✿❏❄❄❅ q❤✵❀

✱✺✺✶✻✱✴❤ ❂✷✱✶✱❆❇✽✽❀ ❇❤✱❇ ❇❤✽ ✾✱✹✷✽ ✻❢ ❇❤✽ ❤✵❂❤✽❀❇ ✺✻❀❀✵❏✹✽ ✽✹✽✾✱❇✵✻❆ ✵❀ ❀❇✵✹✹ ✽❶✷✱✹

❇✻ ❇❤✽ ✻✶✵❂✵❆✱✹ ✽✹✽✾✱❇✵✻❆ ◗⑥ ✻❢ ❇❤✽ ✴✻✶✶✽❀✺✻❆❑✵❆❂ ❀❦✽✹✽❇✻❆ ✺✻✵❆❇❅ ❷✻❇✽ ❇❤✱❇● ❑✽▼

✺✽❆❑✵❆❂ ✻❆ ❇❤✽ ✺✶✻✼✹✽ ✴✷✶✾✽● ❇❤✽ ✽✹✽✾✱❇✵✻❆ ❑✵❀❇✱❆✴✽ ❖✱❊ ❏✽ ❆✷✹✹ ❢✻✶ ❀✻❖✽ ❀❦✽✹✽❇✻❆

✺✻✵❆❇❀● ❍❤✵✴❤ ❖✽✱❆ ❇❤✱❇ ❇✻✺✻✹✻❂✵✴✱✹ ❤✻✹✽❀ ❖✱❊ ❏✽ ✴✶✽✱❇✽❑ ✱✹✻❆❂ ❇❤✽ ❀❦✽✹✽❇✻❆● ✻❆

❇❤✽ ✶✽❀✷✹❇✵❆❂ ❃④ ❖✻❑✽✹❅

❲✽ ✱✹❀✻ ❆✽✽❑ ❇✻ ✽❆❀✷✶✽ ❇❤✱❇ ❇❤✽ ✼✶❀❇ ❀✱❖✺✹✽ ❢✻✶ ❇❤✽ ✺✶✻✼✹✽ ✴✷✶✾✽ ❑✻✽❀ ✴✻✶✶✽▼

❀✺✻❆❑ ❇✻ ✱ ❀✵✹❤✻✷✽❇❇✽ ✺✻✵❆❇❅ q❤✵❀ ✴✱❆ ❏✽ ✻❏❇✱✵❆✽❑ ❏❊ ✴✻❖✺✷❇✵❆❂ ❇❤✽ ❇❍✻ ✻❇❤✽✶

✴✻✻✶❑✵❆✱❇✽❀ ◗❴ ✱❆❑ ◗❵ ✱❀⑤

◗❴ ❬ ◗⑥❴ ❸ ❹❴ ❺ ✿◗

❘❭ ⑦⑧❭❄ ✱❆❑ ◗❵ ❬ ◗

⑥❵ ❸ ❹❵ ❺ ✿◗

❘❭ ⑦⑧❭❄ ❻

❷✻❇✽ ❇❤✱❇ ❍✵❇❤ ✴✻❆✾✽■ ✺✶✻✼✹✽ ✴✷✶✾✽❀ ✱❀ ✵❆ q✽❑❑❊● ❍✽ ✱✹❍✱❊❀ ❤✱✾✽ ❇❤✽ ❢✻✹✹✻❍✵❆❂

✴✻❆❑✵❇✵✻❆⑤ ⑧❭ ❬ ⑧⑨⑩❴❭ ✿❀✽✽ ❁✵❂✷✶✽ ❃✿✱❄❄❅

❼❽❾ ❿➀➁➂➃➄➅➆➇ ➈➉➊ ➋➁➁➊➋➂➋➆➌➊ ➃➍ ➈➉➊ ➀➃➎➊➏

➐❆✽ ✹✵❖✵❇✱❇✵✻❆ ✻❢ ❇❤✽ ✻✶✵❂✵❆✱✹ q✽❑❑❊ ✱✺✺✶✻✱✴❤ ✵❀ ❇❤✽ ✹✱✴❦ ✻❢ ❀❖✻✻❇❤❆✽❀❀ ✻❢ ❇❤✽

✶✽❀✷✹❇✵❆❂ ❀❤✱✺✽❅ q✻ ❂✽❇ ✱ ✺✹✽✱❀✱❆❇ ✱✺✺✽✱✶✱❆✴✽● ✱ ❀❖✻✻❇❤✵❆❂ ❀❇✽✺ ✵❀ ❂✽❆✽✶✱✹✹❊

✶✽❶✷✵✶✽❑ ➑➒➓● ❍❤✵✴❤ ❀✹✻❍❀ ❑✻❍❆ ❇❤✽ ❀❦✽❇✴❤✵❆❂ ✺✶✻✴✽❀❀❅ ➔❆ ✻✷✶ ✱✺✺✶✻✱✴❤● ❍✽ ✺✶✻✺✻❀✽

❇✻ ❑✵✶✽✴❇✹❊ ✽❆❤✱❆✴✽ ❇❤✽ ✱✺✺✽✱✶✱❆✴✽ ❑✷✶✵❆❂ ❇❤✽ ✴✶✽✱❇✵✻❆ ✻❢ ❇❤✽ ✾✽✶❇✵✴✽❀● ❏❊ ✷❀✵❆❂

✱❆ ✱❑▼❤✻✴❦ ✺✶✻✴✽❀❀ ❇✻ ✽❀❇✵❖✱❇✽ ❆✻✶❖✱✹ ✾✽✴❇✻✶❀❅ q❤✵❀ ✺✶✻✴✽❀❀ ✵❀ ❏✱❀✽❑ ✻❆ ❇❤✽

❢✻✹✹✻❍✵❆❂ ✻❏❀✽✶✾✱❇✵✻❆❀⑤ ✿✵❄ ✱ ✾✵❀✷✱✹ ✴✻❆❇✵❆✷✵❇❊ ✵❀ ✶✽❶✷✵✶✽❑ ✻❆ ❇❤✽ ❀✵✹❤✻✷✽❇❇✽ ✺✻✵❆❇❀

Page 7: 3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6 ]. osM t h se e t se hp por a c a aer l i m e i d t o t p o e hl dy rl a o d l,es

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T◆♦

◆s

◆s

◆♦

✭❛✮ ❖♥ �✐❧❤✁✉❡tt❡ ♣❧❛♥❡

❩✄ ❚

✭❜✮ ❖♥ ♣r✁☞❧❡ ❝✉r✈❡

❋☎❣✳ ✹✳ ●❡✁♠❡tr✐❝ ❝✁♥☞✆✉r❛t✐✁♥ ❢✁r ♥✁r♠❛❧ ✆❡♥❡r❛t✐✁♥

✝✞✟✠✟✱ ✡☛✠ ✌✍✎❥✠✏✡✞✎✑ ✎✒ ✡☛✠ ✑✎✍✓✔✕ ✎✑ ✡☛✠ ✖✞✕☛✎✗✠✡✡✠ ✌✕✔✑✠ ☛✔✖ ✡✎ ✘✠ ✏✎✕✕✞✑✠✔✍

✇✞✡☛ ✡☛✠ ✑✎✍✓✔✕ ✙✠✏✡✎✍ ✡✎ ✡☛✠ ✖✞✕☛✎✗✠✡✡✠✚❀ ✝✞✞✚ ✔✡ ✑✎✑✲✠①✡✍✠✓✔✕ ✖❦✠✕✠✡✎✑ ✌✎✞✑✡✖✱

✡☛✠ ✌✍✎❥✠✏✡✞✎✑ ✎✒ ✡☛✠ ✑✎✍✓✔✕ ✎✑ ✡☛✠ ✖✞✕☛✎✗✠✡✡✠ ✌✕✔✑✠ ☛✔✖ ✡✎ ✘✠ ✏✎✕✕✞✑✠✔✍ ✇✞✡☛ ✡☛✠

✑✎✍✓✔✕ ✙✠✏✡✎✍ ✡✎ ✡☛✠ ✖❦✠✕✠✡✎✑✟ ✝✞✞✞✚ ✡☛✠ ✑✎✍✓✔✕ ✡✎ ✡☛✠ ✖✗✍✒✔✏✠ ☛✔✖ ✡✎ ✘✠ ✎✍✡☛✎✛✎✑✔✕

✡✎ ✡☛✠ ✌✍✎✜✕✠ ✏✗✍✙✠✟

✢✗✍ ✑✎✍✓✔✕ ✙✠✏✡✎✍ ✏✎✓✌✗✡✔✡✞✎✑ ✞✖ ✡✎✡✔✕✕② ✕✎✏✔✕✱ ✇☛✞✏☛ ✓✠✔✑✖ ✡☛✔✡ ✞✡ ✏✔✑ ✘✠

❞✎✑✠ ✖✞✓✗✕✡✔✑✠✎✗✖✕② ✡✎ ✡☛✠ ✙✠✍✡✠① ✛✠✑✠✍✔✡✞✎✑✟ ■✑ ✎✍❞✠✍ ✡✎ ✛✗✔✍✔✑✡✠✠ ✡☛✔✡ ✡☛✠ ✡✇✎

✜✍✖✡ ✏✎✑❞✞✡✞✎✑✖ ✔✍✠ ✒✗✕✜✕✕✠❞✱ ✔ ✕✞✑✠✔✍ ✞✑✡✠✍✌✎✕✔✡✞✎✑ ✞✖ ❞✎✑✠ ✘✠✡✇✠✠✑ ✡☛✠ ✑✎✍✓✔✕

✡✎ ✡☛✠ ✖✞✕☛✎✗✠✡✡✠✣✦✤✥ ✔✑❞ ✡☛✠ ✑✎✍✓✔✕ ✡✎ ✡☛✠ ✖❦✠✕✠✡✎✑

✣✦✤✧✱ ✎✑ ✡☛✠ ✖✞✕☛✎✗✠✡✡✠ ✌✕✔✑✠

✔✑❞ ✔✕✎✑✛ ✠✔✏☛ ✞✑✡✠✍✑✔✕ ✠❞✛✠ ✝✖✠✠ ★✞✛✗✍✠ ✩ ✒✎✍ ✡☛✠ ✑✎✡✔✡✞✎✑✖✚✿

✤✪ ❂ ✝✶ ✣ ✫✚✤✧✪ ✰ ✫✤✥

✪ ✔✑❞ ✤✬ ❂ ✝✶ ✣ ✫✚✤✧✬ ✰ ✫✤✥

✬ ✯

✴✎✡✠ ✡☛✔✡✱ ✒✎✍ ✡☛✠ ✠①✡✍✠✓✔✕ ✌✎✞✑✡✖ ✎✒ ✡☛✠ ✖❦✠✕✠✡✎✑✱ ✇✠ ✖✠✡ ✡☛✠ ✑✎✍✓✔✕ ✔✖ ✡☛✠

❞✞✍✠✏✡✞✎✑ ✎✒ ✡☛✠ ✞✑✡✠✍✑✔✕ ✠❞✛✠✿✣✦✤✧ ❂

✣✦✵ ✟ ❆✖ ✡☛✠ ✑✎✍✓✔✕ ☛✔✖ ✡✎ ✘✠ ✎✍✡☛✎✛✎✑✔✕ ✡✎

✡☛✠ ✌✍✎✜✕✠✱ ✇✠ ☛✔✙✠ ✡☛✠ ✏✎✑❞✞✡✞✎✑✣✦✤ ✷

✣✦✸ ❂ ✺✱ ✇☛✠✍✠

✣✦✸ ✞✖ ✡☛✠ ✡✔✑✛✠✑✡ ✡✎ ✡☛✠

✌✍✎✜✕✠ ✝✖✠✠ ★✞✛✗✍✠ ✩✚✟ ✻☛✞✖ ✕✠✔❞✖ ✡✎

✤③ ❂ ✝✣✸✪✼✸③✚ ✝✤✪✵✪ ✰ ✤✬✵✬✚✯

✻☛✠ ✕✔✖✡ ✖✡✠✌ ✞✖ ✖✞✓✌✕② ✔ ✗✑✞✡✲✕✠✑✛✡☛ ✑✎✍✓✔✕✞✽✔✡✞✎✑ ✎✒ ✡☛✠ ✍✠✖✗✕✡✞✑✛ ✙✠✏✡✎✍ ✝✤✪✾✤✬✾ ✤③✚✟

❁ P❃❄❅❇❈ ❉❈❊❈❃❍❏❑❄❊

▲▼◗ ❘❙❯❲❳❨❲❨❳ ❬ ❭❪❫❴❵❙ q④❪⑤❙

❆✖ ✖✔✞❞ ✔✘✎✙✠✱ ✞✑ ✎✍❞✠✍ ✡✎ ✏✍✠✔✡✠ ✔ ✛✠✎✓✠✡✍✞✏ ✓✎❞✠✕ ✇✞✡☛ ✎✗✍ ✖❦✠✡✏☛✞✑✛ ✠✑✙✞✍✎✑✲

✓✠✑✡✱ ✡☛✠ ✗✖✠✍ ☛✔✖ ✡✎ ❞✠✖✞✛✑ ✔ ✖✠✏✎✑❞ ✖❦✠✡✏☛✱ ✡☛✠ ✌✍✎✜✕✠ ✏✗✍✙✠✱ ✞✑ ✔❞❞✞✡✞✎✑ ✡✎

✡☛✠ ✗✖✗✔✕ ✖✞✕☛✎✗✠✡✡✠ ✏✗✍✙✠✟ ❆✖ ✓✔✑② ✖☛✔✌✠✖ ✞✑✏✕✗❞✠ ✖✎✓✠ ✖②✓✓✠✡✍✞✠✖✱ ✡☛✠ ✗✖✠✍

✓✔② ✇✞✖☛ ✡✎ ❞✍✔✇ ✎✑✕② ✔ ✌✔✍✡ ✎✒ ✞✡✟ ✻☛✠ ✖②✖✡✠✓ ✌✠✍✓✞✡✖ ✡✎ ❞✍✔✇ ✠✞✡☛✠✍ ✔ ⑥✗✔✍✡✠✍

✎✒ ✡☛✠ ✏✗✍✙✠✱ ✏✎✓✌✕✠✡✠❞ ✇✞✡☛ ✔ ❞✎✗✘✕✠ ✖②✓✓✠✡✍② ✝✖✠✠ ★✞✛✗✍✠ ⑦✝✔✚✚ ✎✍ ✔ ☛✔✕✒ ✎✒

✡☛✠ ✏✗✍✙✠✱ ✏✎✓✌✕✠✡✠❞ ✇✞✡☛ ✔ ✖✞✑✛✕✠ ✖②✓✓✠✡✍② ✝✖✠✠ ★✞✛✗✍✠ ⑦✝✏✚✚✟

❆✖ ✏✔✑ ✘✠ ✖✠✠✑ ✞✑ ★✞✛✗✍✠ ⑦✝✔✚ ✔✑❞ ★✞✛✗✍✠ ⑦✝✏✚✱ ✡☛✠ ✏✗✍✙✠✖ ✖❦✠✡✏☛✠❞ ✘② ✗✖✠✍✖

✓✔② ✘✠ ✗✑✏✕✎✖✠❞✟ ■✑ ✎✍❞✠✍ ✡✎ ✛✠✑✠✍✔✡✠❞ ✖✎✕✞❞ ✝✞✟✠✟✱ ✏✕✎✖✠❞✚ ✎✘❥✠✏✡✖✱ ✡☛✠ ✌✍✎✜✕✠

Page 8: 3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6 ]. osM t h se e t se hp por a c a aer l i m e i d t o t p o e hl dy rl a o d l,es

PRE-

PRIN

T✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮

❋✐❣✳ ✺✳ ✭❛✮✲✭❝✮ s❤♦✇ t✇♦ ♣r♦☞❧❡ ❝✉r✈❡s s❦❡t❝❤❡❞ ❜② ✉s❡rs t❤❛t ❤❛✈❡ ❜❡❡♥ ❝♦♠♣❧❡t❡❞ ❜②

s②♠♠❡tr②� ✭❜✮✲✭❞✮ s❤♦✇ t❤❡ ❝♦rr❡s♣♦♥❞✁♥✂ ❛✉t♦♠❛t✁❝❛❧❧② ❝❧♦s❡❞ ❛♥❞ s♠♦♦t❤❡❞ ❝✉r✈❡s�

✭❛✮ ■♥♣✉t s❦❡t❝❤ ♦❢ t❤❡

✉s❡r�

✭❜✮ ❙♠♦♦t❤❡❞ ♣r♦☞❧❡

❝✉r✈❡�

✭❝✮ ❘❡s✉❧t✁♥✂ ♠♦❞❡❧�

❋✐❣✳ ✻✳ ✄r♦♠ t❤❡ ✁♥♣✉t ♣r♦☞❧❡ ❝✉r✈❡ t♦ t❤❡ r❡s✉❧t✁♥✂ ✸❉ ♠♦❞❡❧�

☎✆✝✞✟ ✠✡ ☛✆✌✍✎☛✌✠☎☛✏✏✑ ☎✏✍✡✟✒ ☛✓✌✟✝ ✡✔✟✌☎✕✠✖✗✱ ✟✠✌✕✟✝ ✘✑ ✙✝✍❥✟☎✌✠✖✗ ✠✌✡ ✡✌☛✝✌✠✖✗

☛✖✒ ✟✖✒✠✖✗ ✙✍✠✖✌✡ ✍✖ ✌✕✟ ☛①✟✡ ✚✡✟✟ ✛✠✗✆✝✟ ✜✚✘✢✢✱ ✍✝ ✘✑ ✏✠✖✔✠✖✗ ✠✌✡ ✣✝✡✌ ☛✖✒ ✏☛✡✌

✙✍✠✖✌✡ ✚✡✟✟ ✛✠✗✆✝✟ ✜✚✒✢✢✱ ☛☎☎✍✝✒✠✖✗ ✌✍ ✡✍✎✟ ✒✠✡✌☛✖☎✟ ☎✝✠✌✟✝✠✍✖✤

✹✥✷ ✦✧★★✩✪✫✬✯ ✩✪✰ ✴✵★✶✼✰ ✽✾✵✿✰

❆✡ ☛ ✕☛✖✒❀✒✝☛❁✖ ✡✔✟✌☎✕ ☎☛✖ ✘✟ ✞✟✝✑ ✖✍✠✡✑✱ ✠✌ ✠✡ ✆✡✆☛✏✏✑ ✎✍✝✟ ✙✏✟☛✡☛✖✌ ✌✍ ✡✎✍✍✌✕

✍✆✌ ✕✠✗✕ ✓✝✟q✆✟✖☎✑ ✓✟☛✌✆✝✟✡ ✘✟✓✍✝✟ ✠✖✓✟✝✝✠✖✗ ✌✕✟ ❂❃ ✡✕☛✙✟✤ ❖✖✟ ✟☛✡✑ ✡✍✏✆✌✠✍✖ ✠✡ ✌✍

✆✡✟ ✡✍✎✟ ☛✙✙✝✍①✠✎☛✌✠✍✖ ✡✙✏✠✖✟ ✚✠✖ ✍✆✝ ✠✎✙✏✟✎✟✖✌☛✌✠✍✖✱ ❁✟ ✆✡✟ ✌✕✟ ◆❯❄❇❅❈❈

✏✠✘✝☛✝✑❊✢ ✘✆✌ ☛✖✑ ✏✍❁❀✙☛✡✡ ✣✏✌✟✝✠✖✗ ✌✟☎✕✖✠q✆✟ ✎☛✑ ✘✟ ✟✎✙✏✍✑✟✒✤

✛✠✗✆✝✟ ● ✡✕✍❁✡ ✌✕✟ ✝✟✏✟✞☛✖☎✟ ✍✓ ✍✆✝ ✠✒✟☛✤ ❚✕✟ ✙✝✍✣✏✟ ☎✆✝✞✟ ✡✔✟✌☎✕✟✒ ✘✑ ✌✕✟

✆✡✟✝ ✠✡ ✗✠✞✟✖ ✠✖ ✛✠✗✆✝✟ ●✚☛✢✤ ❍✌ ✠✡ ✞✟✝✑ ✖✍✠✡✑ ❁✠✌✕ ☛ ✏✍✌ ✍✓ ✡✎☛✏✏ ✒✠✡☎✍✖✌✠✖✆✠✌✠✟✡✤ ❚✕✟

✝✟☎✍✖✡✌✝✆☎✌✟✒ ☎✆✝✞✟ ✠✡ ✡✕✍❁✖ ✠✖ ✛✠✗✆✝✟ ●✚✘✢✤ ❲✟ ☎☛✖ ✡✟✟ ✌✕☛✌ ✌✕✟ ✒✠✡☎✍✖✌✠✖✆✠✌✠✟✡

✕☛✞✟ ✒✠✡☛✙✙✟☛✝✟✒ ☛✖✒ ✌✕☛✌ ✌✕✟ ☎✆✝✞✟ ✠✡ ✖✍❁ ✞✟✝✑ ✡✎✍✍✌✕✤ ❚✕✟ ✝✟✡✆✏✌✠✖✗ ❂❃ ✎✍✒✟✏

✍✖ ✛✠✗✆✝✟ ●✚☎✢ ✒✍✟✡ ✖✍✌ ✙✝✟✡✟✖✌ ✌✕✟ ✝✍✆✗✕❀☛✖✒❀✒✠✝✌✑ ☛✙✙✟☛✝☛✖☎✟ ✍✓✌✟✖ ☛✡✡✍☎✠☛✌✟✒

❁✠✌✕ ✡✔✟✌☎✕❀✘☛✡✟✒ ✌✍✍✏✡✤

▼✍✝✟✍✞✟✝✱ ✡✠✖☎✟ ❁✟ ☛✝✟ ✆✡✠✖✗ ✙☛✝☛✎✟✌✝✠☎ ☎✆✝✞✟✡✱ ❁✟ ☎☛✖ ✟☛✡✠✏✑ ☎✍✖✌✝✍✏ ✌✕✟

✒✟✖✡✠✌✑ ✍✓ ✌✕✟✠✝ ✡☛✎✙✏✠✖✗✱ ☛☎☎✍✝✒✠✖✗ ✌✍ ✌✕✟ ✏✍☎☛✏ ☎✆✝✞☛✌✆✝✟ ✚✝✟✓✟✝ ✌✍ ❅✟☎✌✠✍✖ ❂ ✓✍✝

✌✕✟ ✌✝☛✖✡✓✍✝✎☛✌✠✍✖ ✓✝✍✎ ✌✕✟ ✡☛✎✙✏✠✖✗ ✌✍ ✌✕✟ ✎✟✡✕ ✞✟✝✌✠☎✟✡✢✤

❏ ❤tt♣❑▲▲❧✁❜♥✉r❜s�s♦✉r❝❡❢♦r✂❡�♥❡t

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T✭❛✮ ❯s❡r ❤❛s ❞r❛✇♥ ❛ s❦❡t❝❤ ♦❢ s❡❧❡❝t✐♦♥✳ ✭❜✮ ❙❡❧❡❝t❡❞ ♣♦✐♥ts ❛r❡ ✐♥ ❜❧❛❝❦✳

❋�❣✁ ✼✁ ▲♦❝❛❧ s❡❧❡❝t✐♦♥ ♦❢ ❛ ♠♦❞❡❧✳

✺ ■✂✄☎✆✝✞✄✟✠✂

❚✡☛ ☞☛✌✍✎✏ ✑✒ ✓✡☛ ✔✕✑✖✗☛ ✘✉✕✈☛ ✙☞☞✌ ✙ ✏☛✚ ✌✓☛✔ ✍✏ ✓✡☛ ✍✏✓☛✕✙✘✓✍✑✏ ✗✑✑✔ ✛☛✓✚☛☛✏

✉✌☛✕✌ ✙✏☞ ✓✡☛ ✌✜☛✓✘✡✍✏✎ ✙✔✔✗✍✘✙✓✍✑✏✢ ◆✑✚ ✓✡☛② ✡✙✈☛ ✓✑ ☞✕✙✚ ✓✚✑ ✌✜☛✓✘✡☛✌ ✍✏ ✑✕☞☛✕

✓✑ ✘✕☛✙✓☛ ✙ ✣✑☞☛✗ ✘✑✏✓✕✙✕② ✓✑ ✣✑✌✓ ✑✒ ✓✡☛ ✌✜☛✓✘✡✍✏✎ ✙✔✔✗✍✘✙✓✍✑✏✌ ✓✡✙✓ ✕☛✘✑✏✌✓✕✉✘✓

✙ ✣✑☞☛✗ ✑✏✗② ✚✍✓✡ ✓✡☛ ✌✍✗✡✑✉☛✓✓☛ ✌✜☛✓✘✡✢ ✤✑ ✉✌☛✕✌ ✡✙✈☛ ✑✏☛ ✣✑✕☛ ✎☛✌✓✉✕☛ ✓✑ ☞✑ ✍✏

✑✕☞☛✕ ✓✑ ✎☛✏☛✕✙✓☛ ✙ ✣✑☞☛✗✢ ❇✉✓ ✓✡✍✌ ✎☛✌✓✉✕☛ ✙✗✗✑✚✌ ✓✡☛ ✘✕☛✙✓✍✑✏ ✑✒ ✣✑✕☛ ☞☛✓✙✍✗✌ ✑✏

✓✡☛ ✌✉✕✒✙✘☛✱ ✙✏☞ ✍✏ ✑✏✗② ✓✚✑ ✎☛✌✓✉✕☛✌✱ ✍✓ ✍✌ ✏✑✚ ✔✑✌✌✍✛✗☛ ✓✑ ✘✕☛✙✓☛ ✣✑✕☛ ✘✑✣✔✗☛①

✌✡✙✔☛✌✢

✤✍✏✘☛ ✉✌☛✕✌ ✡✙✈☛ ☞☛✌✍✎✏☛☞ ✓✡☛ ✌✍✗✡✑✉☛✓✓☛ ✙✏☞ ✔✕✑✖✗☛ ✘✉✕✈☛✌ ✙✏☞ ✓✡☛ ✣✑☞☛✗

✌✡✙✔☛ ✚✙✌ ✍✏✒☛✕✕☛☞✱ ✉✌☛✕✌ ✘✙✏ ✎✗✑✛✙✗✗② ☛☞✍✓ ✓✡✍✌ ✌✡✙✔☛ ✛② ✣✑☞✍✒②✍✏✎ ✓✡☛ ✔✕✑✖✗☛

✘✉✕✈☛✢ ❲✡☛✏ ✙ ✉✌☛✕ ✌✜☛✓✘✡☛✌ ✙ ✏☛✚ ✔✕✑✖✗☛ ✘✉✕✈☛✱ ✓✡☛ ✣✑☞☛✗ ✌✡✙✔☛ ✍✌ ☞☛✒✑✕✣☛☞

✑✏✲✓✡☛✲✥② ✍✏ ✑✕☞☛✕ ✓✑ ✣✙✓✘✡ ✓✡✍✌ ✏☛✚ ✔✕✑✖✗☛ ✦✌☛☛ ✧✍✎✉✕☛ ✽★✢ ✧✑✕ ✙ ✣✑✕☛ ✍✏✓✉✍✓✍✈☛

✍✏✓☛✕✙✘✓✍✑✏✱ ✙ ✌✡✙✔☛ ✘✙✏ ✛☛ ☞✍✕☛✘✓✗② ✍✏✒☛✕☛☞ ✒✕✑✣ ✓✡☛ ✌✍✗✡✑✉☛✓✓☛ ✛② ✉✌✍✏✎ ✙ ☞☛✒✙✉✗✓

✔✕✑✖✗☛✱ ✙✏☞ ✓✡☛✏✱ ✓✡✍✌ ✔✕✑✖✗☛ ✘✙✏ ✛☛ ✎✗✑✛✙✗✗② ☛☞✍✓☛☞✢

▼✑✕☛✑✈☛✕✱ ✚☛ ✡✙✈☛ ☞☛✈☛✗✑✔☛☞ ✙ ✗✑✘✙✗ ☛☞✍✓✍✑✏✢ ❚✡✍✌ ✍✏✘✕☛✙✌☛✌ ✓✡☛ ✕✙✏✎☛ ✑✒

✔✑✌✌✍✛✗☛ ✌✡✙✔☛✌ ✛② ☛✏✙✛✗✍✏✎ ✓✡☛ ✘✑✣✛✍✏✙✓✍✑✏ ✑✒ ☞✍✩☛✕☛✏✓ ✔✕✑✖✗☛ ✘✉✕✈☛✌ ✙✗✑✏✎ ✓✡☛

✕☛✌✉✗✓✍✏✎ ✌✉✕✒✙✘☛✢ ✪✏ ✑✕☞☛✕ ✓✑ ✙✔✔✗② ✙ ✘✡✙✏✎☛ ✑✏✗② ✗✑✘✙✗✗②✱ ✉✌☛✕✌ ✡✙✈☛ ✓✑ ✖✕✌✓

✌☛✗☛✘✓ ✙ ✕☛✎✍✑✏ ✑✒ ✓✡☛ ✌✉✕✒✙✘☛✢ ❚✡☛ ✌②✌✓☛✣ ☞☛✓☛✕✣✍✏☛✌ ✚✡✍✘✡ ✍✏✓☛✕✏✙✗ ☛☞✎☛✌ ✡✙✈☛

✛☛☛✏ ✌☛✗☛✘✓☛☞ ✦✌☛☛ ✧✍✎✉✕☛ ✫✦✛★★ ✒✕✑✣ ✙ ✉✌☛✕✲☞✕✙✚✏ ✌✜☛✓✘✡ ✑✏ ✓✡☛ ✌✘✕☛☛✏ ✦✌☛☛

✧✍✎✉✕☛ ✫✦✙★★✢ ❚✡☛✏✱ ✉✌☛✕✌ ✌✜☛✓✘✡ ✙ ✏☛✚ ✔✕✑✖✗☛ ✘✉✕✈☛ ✙✏☞ ✓✡☛ ✸❉ ✔✑✌✍✓✍✑✏✌ ✑✒ ✓✡☛

✔✑✍✏✓✌ ✑✏ ✓✡☛ ✌☛✗☛✘✓☛☞ ✍✏✓☛✕✏✙✗ ☛☞✎☛✌ ✦✍✢☛✢✱ ✙ ✔✙✕✓ ✑✒ ✓✡☛ ✑✛❥☛✘✓ ✌✡✙✔☛★ ✙✕☛ ✘✡✙✏✎☛☞

✍✏ ✑✕☞☛✕ ✓✑ ✣✙✓✘✡ ✓✡☛ ✏☛✚ ✔✕✑✖✗☛✢

✻ ❘☎✬✯✰✄✬ ✝✂✴ ✴✟✬✞✯✬✬✟✠✂

❚✡☛ ✔✕✑✓✑✓②✔☛ ✌②✌✓☛✣ ✚✙✌ ✚✕✍✓✓☛✏ ✍✏ ✓✡☛ ❈✵✵ ✗✙✏✎✉✙✎☛ ✉✌✍✏✎ ✓✡☛ ❖✔☛✏●✶

✗✍✛✕✙✕② ✓✑ ✕☛✏☞☛✕ ✣✑☞☛✗✌ ✙✏☞ ✓✡☛ ◗✓ ✗✍✛✕✙✕② ✓✑ ☞☛✌✍✎✏ ✓✡☛ ✍✏✓☛✕✒✙✘☛✢ ❆✗✗ ☛①✙✣✔✗☛

✣✑☞☛✗✌ ✌✡✑✚✏ ✍✏ ✓✡✍✌ ✔✙✔☛✕ ✚☛✕☛ ✛✉✍✗✓ ✚✍✓✡ ✓✡✍✌ ✔✕✑✓✑✓②✔☛ ✌②✌✓☛✣ ✕✉✏✏✍✏✎ ✑✏ ✙

✌✓✙✏☞✙✕☞ ☞☛✌✜✓✑✔ P❈✢

❖✉✕ ✒✕✙✣☛✚✑✕✜ ✙✗✗✑✚✌ ✉✌☛✕✌ ✓✑ ✘✕☛✙✓☛ ✙ ✚✍☞☛ ✕✙✏✎☛ ✑✒ ☞✍✩☛✕☛✏✓ ✣✑☞☛✗✌ ✛② ✑✏✗②

☞✕✙✚✍✏✎ ✓✚✑ ✌✜☛✓✘✡☛✌✢ ❇② ✉✌✍✏✎ ✙ ☞☛✒✙✉✗✓ ✘✍✕✘✉✗✙✕ ✔✕✑✖✗☛ ✘✉✕✈☛✱ ✙✏② ✷❚☛☞☞②✲✗✍✜☛✷

✌✡✙✔☛ ✘✙✏ ✛☛ ✑✛✓✙✍✏☛☞ ✦✌☛☛ ✧✍✎✉✕☛ ✽✦✙★ ✘✑✗✉✣✏★✢ ❇② ✑✩☛✕✍✏✎ ✓✡☛ ✔✑✌✌✍✛✍✗✍✓② ✓✑

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T

✭❛✮ ✭❜✮ ✭❝✮

❋✐❣✳ ✽✳ ●❧♦❜❛❧ ❡❞�t�♦♥ ♦❢ t✇♦ ♠♦❞❡❧s✁ ✭❛✮ s❤♦✇s t❤❡ ❝r❡❛t�♦♥ ♦❢ t✇♦ ♠♦❞❡❧s ✭♦♥❡ ✇�t❤

❛ ❝�r❝✉❧❛r s�❧❤♦✉❡tt❡ ❝✉r✈❡ ❛♥❞ t❤❡ ♦t❤❡r ✇�t❤ ❛ ♠♦r❡ ❝♦♠♣❧❡① ♦♥❡✮ ✇�t❤ t❤❡ ❞❡❢❛✉❧t

❝�r❝✉❧❛r ♣r♦☞❧❡ ❝✉r✈❡✁ ✭❜✮✲✭❝✮ s❤♦✇ t❤❡ ❡☛❡❝ts ♦❢ t❤❡ ✂❧♦❜❛❧ ❡❞�t�♦♥ ♦❢ t❤❡ ♣r♦☞❧❡ ❝✉r✈❡✁

✄☎✆✝✞✟ ✠☎✟ ✡✌✍✎✏✟ ✄✑✌✒✟✱ ✓✟ ✞✌✟✆✠✏② ✔✝✄✌✟✆✕✟ ✠☎✟ ✒✆✌✔✟✠② ✍✖ ✕☎✆✡✟✕ ✠☎✆✠ ✄✆✝ ✗✟

✞✟✝✟✌✆✠✟✘ ✙✕✟✟✱ ✖✍✌ ✔✝✕✠✆✝✄✟✱ ✠☎✟ ✚✍✓✟✌✡✍✠ ✍✝ ✛✔✞✑✌✟ ✾✜✢

✛✔✞✑✌✟ ✣ ✔✏✏✑✕✠✌✆✠✟✕ ✠☎✟ ✡✌✔✝✄✔✡✏✟ ✍✖ ✞✏✍✗✆✏ ✟✘✔✠✔✍✝✱ ✆✕ ✘✔✕✄✑✕✕✟✘ ✔✝ ❙✟✄✠✔✍✝ ✺✢

✛✔✌✕✠ ✓✟ ✕✠✆✌✠ ✖✌✍✤ ✆ ❭✗✏✍✗✗②✧ ✍✗❥✟✄✠ ✗② ✘✌✆✓✔✝✞ ✆ ✕✔✏☎✍✑✟✠✠✟ ✄✑✌✒✟ ✄✍✤✗✔✝✟✘

✓✔✠☎ ✆ ✄✔✌✄✑✏✆✌ ✡✌✍✎✏✟ ✄✑✌✒✟ ✙✛✔✞✑✌✟ ✣✙✆✜✥✑✡✜✢ ❚☎✟ ✡✌✍✎✏✟ ✄✑✌✒✟ ✄✆✝ ✠☎✟✝ ✗✟

✔✝✠✟✌✆✄✠✔✒✟✏② ✤✍✘✔✎✟✘ ✙✛✔✞✑✌✟ ✣✙✗✜✥✑✡ ✍✌ ✣✙✄✜✥✑✡✜ ✑✝✠✔✏ ✠☎✟ ✑✕✟✌ ✔✕ ✕✆✠✔✕✎✟✘ ✓✔✠☎

✠☎✟ ✌✟✕✑✏✠✔✝✞ ✕☎✆✡✟✢

❚☎✟ ✤✍✘✔✎✄✆✠✔✍✝ ✍✖ ✠☎✟ ✡✌✍✎✏✟ ✄✑✌✒✟ ✤✆② ✆✏✕✍ ✗✟ ✆✡✡✏✔✟✘ ✍✝✏② ✍✝ ✆ ✡✆✌✠

✍✖ ✠☎✟ ✕☎✆✡✟✱ ✆✕ ✡✌✟✕✟✝✠✟✘ ✔✝ ✛✔✞✑✌✟ ✶✹✢ ❚☎✟ ✔✝✔✠✔✆✏ ✤✍✘✟✏ ✓✆✕ ✆ ✕✡☎✟✌✟✢ ❆✖✠✟✌

✕✟✏✟✄✠✔✝✞ ✆ ✡✆✌✠ ✍✖ ✠☎✟ ✤✍✘✟✏ ✙✔✝ ✍✑✌ ✄✆✕✟✱ ✠☎✟ ✌✔✞☎✠ ☎✆✏✖ ✍✖ ✠☎✟ ✕✡☎✟✌✟✜✱ ✆ ✝✟✓

✡✌✍✎✏✟ ✄✑✌✒✟ ✔✕ ✘✟✕✔✞✝✟✘ ✙✕✟✟ ✛✔✞✑✌✟ ✶✹✙✆✜✜ ✆✝✘ ✆✡✡✏✔✟✘ ✍✝ ✠☎✟ ✕✟✏✟✄✠✟✘ ✡✆✌✠✢ ❚☎✟

✌✟✕✑✏✠✔✝✞ ✕☎✆✡✟ ✔✕ ✡✌✟✕✟✝✠✟✘ ✔✝ ✛✔✞✑✌✟ ✶✹✙✗✜ ✆✝✘ ✛✔✞✑✌✟ ✶✹✙✄✜✢

▼✍✌✟ ✄✍✤✡✏✟✦ ✕☎✆✡✟✕ ✄✆✝ ✗✟ ✍✗✠✆✔✝✟✘ ✗② ✑✕✔✝✞ ✤✍✌✟ ✄✍✤✡✏✟✦ ✡✌✍✎✏✟ ✄✑✌✒✟✕

✆✕ ✔✝ ✛✔✞✑✌✟ ✶✸✢ ❚☎✟ ☎✆✤✤✟✌ ✍✖ ✛✔✞✑✌✟ ✶✸✙✆✜ ✓✍✑✏✘ ✌✟q✑✔✌✟ ✆ ✏✍✠ ✍✖ ✕✑✌✖✆✄✟

Page 11: 3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6 ]. osM t h se e t se hp por a c a aer l i m e i d t o t p o e hl dy rl a o d l,es

PRE-

PRIN

T✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮

❋✐❣✳ ✾✳ ❈r❡❛t�♦♥ ♦❢ ❛ ✌♦✇❡r♣♦t✁ ❚❤�s ♠♦❞❡❧ ✇❛s ❞❡s�✂♥❡❞ ✇�t❤ t❤❡ ♣r♦☞❧❡ ❝✉r✈❡ s❤♦✇♥

�♥ ✭❛✮ ❝♦♠❜�♥❡❞ ✇�t❤ ❛ ❝�r❝✉❧❛r s�❧❤♦✉❡tt❡ ❝✉r✈❡✁ ✭❜✮ ❛♥❞ ✭❝✮ s❤♦✇ t❤❡ ♣♦t ❢r♦♠ ❞�☛❡r❡♥t

♣♦�♥ts ♦❢ ✈�❡✇ ✇❤�❧❡ ✭❞✮ ♣r❡s❡♥ts ❛ ❝✉t✲✈�❡✇✁ ◆♦t❡ t❤❛t t❤❡ ♣♦t �s ❛❝t✉❛❧❧② ❛ ✂❡♥✉s ✶

s✉r❢❛❝❡✁

✭❛✮ ✭❜✮

❋✐❣✳ ✄✵✳ ❆ ♠♦❞❡❧ ✇�t❤ ❛ ♠♦r❡ ❝♦♠♣❧❡①

s�❧❤♦✉❡tt❡ ❝✉r✈❡✁

✭❛✮ ✭❜✮

❋✐❣✳ ✄✄✳ ❈r❡❛t�♦♥ ♦❢ ❛ ❞♦♦r ❤❛♥❞❧❡ ✇�t❤ ♦♥❧②

t✇♦ ❝✉r✈❡s✁

☎✆✝✞✟✠✡✍✞✆✎✏ ✞✎ ✏✍✡✍✑✒✆✓✒✍✔✑✒✡✕✍ ✏❦✑✍✠✔✞✎✖ ✏✗✏✍✑☎✏✘ ❇✙✍✱ ✚✞✍✔ ✆✙✕ ✏✗✏✍✑☎✱ ✚✑

✠✕✑✡✍✑✝ ✞✍ ✚✞✍✔ ✆✎✛✗ ✟✜✑ ✏❦✑✍✠✔✑✏✘ ❲✑ ✟✕✏✍ ✏❦✑✍✠✔✑✝ ✍✔✑ ✔✡☎☎✑✕ ✏✞✛✔✆✙✑✍✍✑ ✠✙✕✜✑✘

❲✑ ✍✔✑✎ ✝✞✝ ✡ ✟✕✏✍ ✛✆✠✡✛ ✏✑✛✑✠✍✞✆✎ ✆✓ ✞✍✏ ✔✑✡✝ ✡✎✝ ✏❦✑✍✠✔✑✝ ✡ ✏q✙✡✕✑ ✢✕✆✟✛✑ ✠✙✕✜✑✘

✣✞✎✡✛✛✗ ✚✑ ✏✑✛✑✠✍✑✝ ✍✔✑ ✢✆✞✎✍✏ ✆✓ ✞✍✏ ✢✆☎☎✑✛ ✡✎✝ ✏❦✑✍✠✔✑✝ ✡ ✢✕✆✟✛✑ ✠✙✕✜✑ ✚✞✍✔ ✡

✠✕✑✡✏✑ ✤q✙✞✍✑ ✏✞☎✞✛✡✕ ✍✆ ✍✔✑ ✢✕✆✟✛✑ ✠✙✕✜✑ ✆✓ ✣✞✖✙✕✑ ✽✒✙✢✕✞✖✔✍✥✘

▲✦✧✦★✩★✦✪✫✬

❲✑ ✏✍✞✛✛ ✔✡✜✑ ✏✆☎✑ ✢✕✆✯✛✑☎✏ ✞✎ ✍✔✑ ✏❦✑✍✠✔✑✝ ✏✔✡✢✑✱ ☎✆✏✍✛✗ ✝✙✑ ✍✆ ✍✔✑ ✏❦✑✛✑✍✆✎

✡✎✝ ✞✎✍✑✕✎✡✛ ✢✆✞✎✍✏ ✠✕✑✡✍✞✆✎✏ ✤✍✔✑✏✑ ✢✆✞✎✍✏ ✓✆✛✛✆✚ ✍✔✑ ✞✎✍✑✕✎✡✛ ✑✝✖✑✏✥✘ ✰✏ ✠✡✎ ✯✑

✏✑✑✎ ✞✎ ✣✞✖✙✕✑ ✴✺✱ ✑✜✑✎ ✚✞✍✔ ✏✞☎✢✛✑ ✏✞✛✔✆✙✑✍✍✑ ✠✙✕✜✑✏✱ ✍✔✑ ✏❦✑✛✑✍✆✎✏ ✠✕✑✡✍✑✝ ✔✡✜✑

✡ ✛✆✍ ✆✓ ✏☎✡✛✛ ✝✞✏✠✆✎✍✞✎✙✞✍✞✑✏✱ ✛✑✡✝✞✎✖ ✍✆ ✡✎✎✆✗✞✎✖ ✔✑✞✖✔✍ ✝✞✷✑✕✑✎✠✑✏ ✓✆✕ ✍✔✑✞✕

✢✆✞✎✍✏ ✤✡✏ ✠✡✎ ✯✑ ✏✑✑✎ ✞✎ ✣✞✖✙✕✑ ✴✸ ✡✎✝ ✣✞✖✙✕✑ ✽✒✯✆✍✍✆☎✥✘

❙✞✎✠✑ ✍✔✑ ✠✕✑✡✍✞✆✎ ✆✓ ✍✔✑ ☎✑✏✔ ✜✑✕✍✞✠✑✏ ✓✆✛✛✆✚✏ ✍✔✑ ✞✎✍✑✕✎✡✛ ✑✝✖✑✏✱ ✍✔✑ ✏✡☎✢✛✞✎✖

✆✓ ✍✔✑ ☎✆✝✑✛ ✡✎✝ ✍✔✑ ✕✑✏✙✛✍✞✎✖ ✍✕✞✡✎✖✙✛✡✍✞✆✎ ✠✡✎ ✯✑ ✙✎✝✑✕✒✏✡☎✢✛✑✝ ✞✎ ✠✑✕✍✡✞✎

✕✑✖✞✆✎✏ ✆✓ ✍✔✑ ☎✆✝✑✛✘ ❇✑✠✡✙✏✑ ☎✆✕✑ ✍✔✡✎ ✆✎✑ ✞✎✍✑✕✎✡✛ ✑✝✖✑ ✠✡✎ ✑✎✝ ✆✎ ✆✎✑ ✆✙✍✛✞✎✑

✢✆✞✎✍✱ ✍✔✑ ✕✑✏✙✛✍✞✎✖ ✍✕✞✡✎✖✙✛✡✍✞✆✎ ✞✏ ✓✡✕ ✓✕✆☎ ✯✑✞✎✖ ✑q✙✞✛✡✍✑✕✡✛ ✤✚✞✍✔ ✛✆✍✏ ✆✓ ✍✔✞✎

✍✕✞✡✎✖✛✑✏✥ ✡✎✝ ✍✔✙✏ ✠✆✎✏✞✏✍✑✎✍ ✑✜✡✛✙✡✍✞✆✎ ✆✓ ✎✆✕☎✡✛ ✜✑✠✍✆✕✏ ☎✡✗ ✯✑ ✝✞✹✠✙✛✍✘

❯✏✞✎✖ ✍✔✑ ✡✢✢✕✆✡✠✔ ✝✑✏✠✕✞✯✑✝ ✞✎ ❬✽❪ ☎✡✗ ✯✑ ✡ ✏✆✛✙✍✞✆✎ ✍✆ ✖✑✎✑✕✡✍✑ ✯✑✍✍✑✕

✏❦✑✛✑✍✆✎✏ ✚✞✍✔✆✙✍ ✍✔✑ ✎✑✑✝ ✍✆ ✙✏✑ ✡ ✻❉✼✘ ■✍ ✚✆✙✛✝ ✕✑q✙✞✕✑ ✝✑✜✑✛✆✢✞✎✖ ✡ ✎✑✚

✡✛✖✆✕✞✍✔☎ ✍✆ ✠✆☎✢✙✍✑ ✞✎✍✑✕✎✡✛ ✢✆✞✎✍✏✘

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PRIN

T✭❛✮ ✭❜✮

❋✐❣✳ ✶✷✳ ❆ ✈❛s❡ ❝r❡❛t❡❞ ✇�t❤ t✇♦ s❦❡t❝❤❡s✁

✭❛✮ ✭❜✮

❋✐❣✳ ✶✸✳ ▼♦r❡ ❝♦♠♣❧❡① ♠♦❞❡❧s ❝r❡❛t❡❞

✇�t❤ ❧♦❝❛❧ ✭❛✮ ❛♥❞ ✂❧♦❜❛❧ ✭❜✮ ❡❞�t�♦♥✁ ❚❤❡

r❡s✉❧t�♥✂ s❤❛♣❡ �♥ ✭❛✮ �s ❛ ❤❛♠♠❡r ❛♥❞ �♥

✭❜✮ �s ❛ ❧❡tt❡r ● �♥ ✄❉✁

✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮

❋✐❣✳ ✶✹✳ ■❧❧✉str❛t�♦♥ ♦❢ t❤❡ ❧♦❝❛❧ ❡❞�t�♦♥✁ ❚❤❡ ♣r♦☞❧❡ ❝✉r✈❡ ❛♣♣❧�❡❞ ♦♥ t❤❡ s❡❧❡❝t❡❞ ♣♦�♥t

�s ✂�✈❡♥ �♥ ✭❛✮✁ ✭❜✲❝✮ s❤♦✇s t✇♦ ✈�❡✇s ♦❢ t❤❡ r❡s✉❧t�♥✂ ♠♦❞❡❧✁ ■♥ ✭❞✮✱ ♠❡s❤ ✈❡rt�❝❡s ❛r❡

s❤♦✇♥✁

✼ ❈☎✆✝✞✟✠✡☎✆ ☛✆✌ ✍✟✎✟✏✑ ✒☎✏✓

✔✕ ✖✗✘✙ ✚✛✚✜✢✣ ✤✜ ✗✛✥✜ ✚✢✜✙✜✕✖✜✦ ✕✜✤ ✙✧✜✖★✗✘✕✩ ✘✕✖✜✢✛★✖✘✪✕✙ ✫✪✢ ✖✗✜ ★✢✜✛✖✘✪✕ ✪✫

★✪✬✚✯✜✰ ✴✵ ✙✗✛✚✜✙ ✙✖✛✢✖✘✕✩ ✫✢✪✬ ✛ ✢✜✦✺★✜✦ ✕✺✬✻✜✢ ✪✫ ✙✧✜✖★✗✜✦ ★✺✢✥✜✙✽ ✾ ✩✜✕✜✢✛✯

✕✪✕✿★✪✕✥✜✰ ✚✢✪❀✯✜ ★✺✢✥✜ ★✛✕ ✻✜ ★✪✬✻✘✕✜✦ ✤✘✖✗ ✛✕ ✛✢✻✘✖✢✛✢② ✙✘✯✗✪✺✜✖✖✜ ★✺✢✥✜ ✖✪

✩✜✕✜✢✛✖✜ ✛ ✯✛✢✩✜ ✥✛✢✘✜✖② ✪✫ ✢✜✙✺✯✖✘✕✩ ✴✵ ✙✗✛✚✜✙✽

❲✜ ✗✛✥✜ ✛✯✙✪ ✚✢✜✙✜✕✖✜✦ ✙✪✬✜ ✘✦✜✛✙ ✖✪ ✘✬✚✢✪✥✜ ✖✗✜ ✬✜✙✗ ✩✜✕✜✢✛✖✘✪✕ ✫✢✪✬ ✖✗✜

✙✘✯✗✪✺✜✖✖✜ ✛✕✦ ✚✢✪❀✯✜ ★✺✢✥✜✙✣ ✻② ✯✪★✛✯✯② ✜✙✖✘✬✛✖✘✕✩ ✖✗✜ ✕✪✢✬✛✯ ✥✜★✖✪✢ ✫✪✢ ✜✛★✗

✬✜✙✗ ✥✜✢✖✜✰✽ ❁✘✕✛✯✯②✣ ✛✫✖✜✢ ✖✗✜ ✘✕✘✖✘✛✯ ✬✜✙✗ ✩✜✕✜✢✛✖✘✪✕✣ ✙✪✬✜ ✘✕✖✜✢✛★✖✘✥✜ ✜✦✘✖✘✕✩✣

✜✘✖✗✜✢ ✯✪★✛✯ ✪✢ ✩✯✪✻✛✯✣ ✬✛② ✻✜ ✺✙✜✦ ✖✪ ❀✕✜✯② ✛✦❥✺✙✖ ✖✗✜ ❀✕✛✯ ✙✗✛✚✜ ✪✫ ✖✗✜ ✪✻❥✜★✖✽

❙✘✕★✜ ✪✺✢ ✛✚✚✢✪✛★✗ ✘✙ ✻✛✙✜✦ ✪✕ ✖✗✜ ✘✕✘✖✘✛✯ ❂✜✦✦② ✙②✙✖✜✬✣ ✛✯✯ ✖✗✜ ✪✖✗✜✢ ✩✜✙✖✺✢✜

✛✕✦ ★✺✢✥✜ ✘✕✖✜✢✛★✖✘✪✕✙ ✤✪✺✯✦ ✻✜ ✙✖✘✯✯ ✺✙✛✻✯✜✽ ❲✜ ✗✛✥✜✕❃✖ ★✺✢✢✜✕✖✯② ✘✬✚✯✜✬✜✕✖✜✦

✛✯✯ ✪✫ ✖✗✜✬ ✘✕ ✪✺✢ ✚✢✪✖✪✖②✚✜ ✙✪✫✖✤✛✢✜✽ ❁✺✢✖✗✜✢✬✪✢✜✣ ✛ ✯✪★✛✯ ✘✕✖✜✢✚✪✯✛✖✘✪✕ ✪✫ ✚✢✪❀✯✜

❋✐❣✳ ✶❄✳ ❅❉❚ ♦❢ ❞�❇❡r❡♥t s�❧❤♦✉❡tt❡ ❝✉r✈❡s✁ ❊❡ ❝❛♥ s❡❡ t❤❡ s❦❡❧❡t♦♥s ❞�s❝♦♥t�♥✉�t�❡s✁

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PRIN

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❝✉r✈❡s ❜❡t✇❡❡♥ s❡❧❡❝t❡❞ ③♦♥❡s ✇♦✉❧❞ ♣r♦✈✐❞❡ ❛ s♠♦♦t❤❡r tr❛♥s✐t✐♦♥✳ ❚❤✐s ❝❛♥ ❜❡

✐♥t❡❣r❛t❡❞ ❜② ✉s✐♥❣ ❝✉r✈❡✲♠♦r♣❤✐♥❣✳

❚❤❡ ♠❛✐♥ r❡♠❛✐♥✐♥❣ ♣r♦❜❧❡♠✱ ❛s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❙❡❝t✐♦♥ ✻✱ ✐s t❤❡ ❣❡♥❡r❛t✐♦♥

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

❧❡❛❞ t♦ ✈✐s✉❛❧ ❛rt✐❢❛❝ts✱ ❡✈❡♥ ✇✐t❤ ♦✉r ✐♠♣r♦✈❡❞ ♥♦r♠❛❧ ❝♦♠♣✉t❛t✐♦♥✳ ❲❡ ❜❡❧✐❡✈❡

t❤❛t ❛ ♠♦r❡ r♦❜✉st ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ s❦❡❧❡t♦♥ ❬✽❪✱ ✇✐t❤ ❛♥ ❛❞❛♣t✐✈❡ s❛♠♣❧✐♥❣ ♦❢

t❤❡ s✐❧❤♦✉❡tt❡ ♠❛② ♣r♦✈✐❞❡ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠✳

❘�✁�✂�✄☎�✆

✶✝ ❆✝ ❆✞✟①✟✠ ▲✝ ❇✡☛☞✌✟✠ ▼✝P✝ ❈✡✍✎✠ ✡✍✏ ❱✝ ●✡✎✞✏☛✡☞✝ ✑✌✡✒✟ ✓✔✏✟✞✞✎✍✕ ✖✗ ✘✙✟☞✚✌✎✍✕

✛✘✎✍✕ ✚✔✍✜✔✞✛☞✎✔✍ ✘✛☛✢✡✚✟✘✝ ■✍ ✣✤✥✦✧✥ ★✩✤✪✫✦✥✬ ✭✮✫✯✩✰ ✣✤✪✴✩✬✵✠ ❖✚☞✔✖✟☛ ✷✸✸✺✝✷✝ ❉✝ ❇✔✛☛✕✛✎✕✍✔✍✠ ▼✝✹P✝ ❈✡✍✎✠ ✡✍✏ ●✝ ❉☛✟☞☞✡✙✎✘✝ ❉☛✡✼✎✍✕ ✢✔☛ ■✞✞✛✘☞☛✡☞✎✔✍ ✡✍✏ ❆✍✍✔✹

☞✡☞✎✔✍ ✎✍ ✾❉✝ ✿✯❀✪❁✰✴✩ ★✩✤✪✫✦✥✬ ❋✯✩❁❀ ✭✣✩✯✥❂ ❊❁✩✯❃✩✤✪✫✦✥✬ ❄❅❅❍✵✠ ✷✸❏✾❑◆✶✶◗④✶✷✷✠

✑✟✒☞✟✓✖✟☛ ✷✸✸✶✝✾✝ ❯✔✍✡☞✌✡✍ ▼✝ ❈✔✌✟✍✠ ▲✟✟ ▼✡☛✙✔✘✎✡✍✠ ❳✔✖✟☛☞ ❈✝ ❩✟✞✟❨✍✎✙✠ ❯✔✌✍ ❭✝ ❫✛✕✌✟✘✠ ✡✍✏

❳✔✍✟✍ ❇✡☛❨✟✞✝ ❆✍ ✎✍☞✟☛✢✡✚✟ ✢✔☛ ✘✙✟☞✚✌✎✍✕ ✾✏ ✚✛☛✜✟✘✝ ■✍ ✮❴❵❥ q⑤⑤⑥ ✣✩✯✥✴✴⑦✦⑧❃✬ ✯⑨

✰✫✴ ❍⑤⑤⑤ ✬⑩❀✪✯✬✦❁❀ ✯⑧ ❴⑧✰✴✩✤✥✰✦❶✴ ❵❥ ❃✩✤✪✫✦✥✬✠ ✒✡✕✟✘ ✶❷④✷✶✠ ❸✟✼ ❹✔☛✙✠ ❸❹✠ ❺✑❆✠

✶❻❻❻✝ ❆❈▼ P☛✟✘✘✝◗✝ ❆✝ ❈✛✍✔✠ ❈✝ ❼✘✒✟☛✡✍❽✚✡✠ P✝ ❳✔✓✡ ❈✡✜✡✞✚✡✍☞✎✠ ✡✍✏ ❳✝ ❭✡☛✎✡✘✝ ✾❉ ❭☛✟✟✹✢✔☛✓ ▼✔✏✟✞✎✍✕

✼✎☞✌ ❱✡☛✎✡☞✎✔✍✡✞ ✑✛☛✢✡✚✟✘✝ ■✍ ❾✮✿★ ✭❿✯❁✩⑧✤➀ ✣✤✪✴✩✬✵ ❄❅❅➁✠ ✒✡✕✟✘ ❻④✶➂✠ ❯✡✍✛✡☛✗

✷✸✸✺✝✺✝ ▲✝ ❼✕✕✞✎✠ ❈✝ ✗✝ ❫✘✛✠ ❇✝ ❉✝ ❇☛➃✛✏✟☛✞✎✍✠ ✡✍✏ ●✝ ❼✞✖✟☛✝ ■✍✢✟☛☛✎✍✕ ✾❉ ✓✔✏✟✞✘ ✢☛✔✓

✢☛✟✟✌✡✍✏ ✘✙✟☞✚✌✟✘ ✡✍✏ ✚✔✍✘☞☛✡✎✍☞✘✝ ✿✯❀✪❁✰✴✩➄➅✦⑦✴⑦ ❥✴✬✦❃⑧ ✭❿✿➅❥✵✠ ✷❻❏✷❑◆✶✸✶④

✶✶✷✠ ❭✟✖☛✛✡☛✗ ✶❻❻❷✝➂✝ ▼✝ ❯✝ ❭✔✍✘✟✚✡✠ ❆✝ ❭✟☛☛✟✎☛✡✠ ✡✍✏ ❯✝ ❆✝ ❯✔☛✕✟✝ ➆✔✼✡☛✏✘ ✾❉ ▼✔✏✟✞✎✍✕ ✛✘✎✍✕ ✑✙✟☞✚✌✟✘

✡✍✏ ❳✟☞☛✎✟✜✡✞✝ ■✍ ❊❁✩✯❃✩✤✪✫✦✥✬ ❾✯✩➇✬✫✯✪ ✯⑧ ✮➇✴✰✥✫➄➈✤✬✴⑦ ❴⑧✰✴✩⑨✤✥✴✬ ✤⑧⑦ ➉✯⑦✴➀✦⑧❃✠

❆✛✕✛✘☞ ✷✸✸◗✝❷✝ ❈✎✍✏✗ ●☛✎✓✓ ✡✍✏ ❯✔✌✍ ❫✛✕✌✟✘✝ ■✓✒✞✎✚✎☞ ✕✟✍✟☛✡✞✎❨✟✏ ✚✗✞✎✍✏✟☛✘ ✛✘✎✍✕ ✒☛✔➊✞✟ ✚✛☛✜✟✘✝

■✍ ❴❀✪➀✦✥✦✰ ✮❁✩⑨✤✥✴✬✠ ✒✡✕✟✘ ✾✾④◗✶✠ ❯✛✍✟ ✶❻❻➋✝ ❈☛✟✡☞✎✍✕ ✘✼✟✟✒ ✘✛☛✢✡✚✟✘ ✛✘✎✍✕ ✘✙✟☞✚✌✹

✎✍✕✝➋✝ ▲✝ ❫✟✝ ➅ ✿✯❀✪✤✩✦✬✯⑧ ✯⑨ ❥✴⑨✯✩❀✤➌➀✴ ✿✯⑧✰✯❁✩ ➉✴✰✫✯⑦✬ ✤⑧⑦ ➉✯⑦✴➀ ➈✤✬✴⑦ ➅✪✪✩✯✤✥✫

➍✬✦⑧❃ ✮➇✴➀✴✰✯⑧ ⑨✯✩ ✮✫✤✪✴ ➎✴✥✯❶✴✩⑩ ❋✩✯❀ ❴❀✤❃✴✬✝ P✌❉ ☞✌✟✘✎✘✠ ❺✍✎✜✟☛✘✎☞✗ ✔✢ ❈✎✍✚✎✍✹

✍✡☞✎✠ ✷✸✸✾✝❻✝ ➆✝ ■✕✡☛✡✘✌✎ ✡✍✏ ❯✝ ❭✝ ❫✛✕✌✟✘✝ ✑✓✔✔☞✌ ✓✟✘✌✟✘ ✢✔☛ ✘✙✟☞✚✌✹✖✡✘✟✏ ✢☛✟✟✢✔☛✓ ✓✔✏✟✞✎✍✕✝

■✍ ✮❴❵❥ q❅❵⑥ ✣✩✯✥✴✴⑦✦⑧❃✬ ✯⑨ ✰✫✴ ❄❅❅❵ ✬⑩❀✪✯✬✦❁❀ ✯⑧ ❴⑧✰✴✩✤✥✰✦❶✴ ❵❥ ❃✩✤✪✫✦✥✬✠ ✒✡✕✟✘

✶✾❻④✶◗✷✠ ❸✟✼ ❹✔☛✙✠ ❸❹✠ ❺✑❆✠ ✷✸✸✾✝ ❆❈▼ P☛✟✘✘✝✶✸✝ ➆✝ ■✕✡☛✡✘✌✎✠ ✑✝ ▼✡☞✘✛✔✙✡✠ ✡✍✏ ❫✝ ➆✡✍✡✙✡✝ ➆✟✏✏✗◆ ✡ ✘✙✟☞✚✌✎✍✕ ✎✍☞✟☛✢✡✚✟ ✢✔☛ ✾✏

✢☛✟✟✢✔☛✓ ✏✟✘✎✕✍✝ ■✍ ✮❴★★➎➅✣➏ q⑤⑤⑥ ✣✩✯✥✴✴⑦✦⑧❃✬ ✯⑨ ✰✫✴ ❄➐✰✫ ✤⑧⑧❁✤➀ ✥✯⑧⑨✴✩✴⑧✥✴

✯⑧ ✿✯❀✪❁✰✴✩ ❃✩✤✪✫✦✥✬ ✤⑧⑦ ✦⑧✰✴✩✤✥✰✦❶✴ ✰✴✥✫⑧✦➑❁✴✬✠ ✒✡✕✟✘ ◗✸❻④◗✶➂✠ ❸✟✼ ❹✔☛✙✠ ❸❹✠

❺✑❆✠ ✶❻❻❻✝ ❆❈▼ P☛✟✘✘➒❆✏✏✎✘✔✍✹➓✟✘✞✟✗ P✛✖✞✎✘✌✎✍✕ ❈✔✝✶✶✝ ❖✝ ➔✡☛✒✟✍✙✔✠ ❯✝ ❫✛✕✌✟✘✠ ✡✍✏ ❳✝ ❳✡✘✙✡☛✝ ❭☛✟✟✹✢✔☛✓ ✘✙✟☞✚✌✎✍✕ ✼✎☞✌ ✜✡☛✎✡☞✎✔✍✡✞

✎✓✒✞✎✚✎☞ ✘✛☛✢✡✚✟✘✝ ■✍ ❊➍➎→★➎➅✣➏❴✿✮ q❅❄✠ ✷✸✸✷✝✶✷✝ ❇✟☛☞☛✡✍✏ ➔✟☛✡✛☞☛✟☞✠ ➣✡✜✎✟☛ ●☛✡✍✎✟☛✠ ✡✍✏ ❆✚✌✎✞✞✟ ❇☛✡↔✛✟✞✡✎☛✟✝ ■✍☞✛✎☞✎✜✟ ✘✌✡✒✟ ✓✔✏✹

✟✞✎✍✕ ✖✗ ✘✌✡✏✎✍✕ ✏✟✘✎✕✍✝ ■✍ ❴⑧✰✴✩⑧✤✰✦✯⑧✤➀ ✮⑩❀✪✯✬✦❁❀ ✯⑧ ✮❀✤✩✰ ★✩✤✪✫✦✥✬✠ ✜✔✞✛✓✟

✾➂✾➋ ✔✢ ↕✴✥✰❁✩✴ ➙✯✰✴✬ ✦⑧ ✿✯❀✪❁✰✴✩ ✮✥✦✴⑧✥✴✠ ✒✡✕✟✘ ✶➂✾④✶❷◗✝ ✑✒☛✎✍✕✟☛✹❱✟☛✞✡✕ ●✓✖❫✠

✡✛✕ ✷✸✸✺✝✶✾✝ ✑✝ ❖✌✼✡✏✡✠ ❭✝ ❸✎✟✞✘✟✍✠ ➔✝ ❸✡✙✡❨✡✼✡✠ ✡✍✏ ➆✝ ■✕✡☛✡✘✌✎✝ ❆ ✑✙✟☞✚✌✎✍✕ ■✍☞✟☛✢✡✚✟ ✢✔☛

▼✔✏✟✞✎✍✕ ☞✌✟ ■✍☞✟☛✍✡✞ ✑☞☛✛✚☞✛☛✟✘ ✔✢ ✾❉ ✑✌✡✒✟✘✝ ■✍ ✣✩✯✥❂ ✮❀✤✩✰ ★✩✤✪✫✦✥✬✠ ✒✡✕✟✘

◗❻④✺❷✝ ✑✒☛✎✍✕✟☛✹❱✟☛✞✡✕✠ ❯✛✞✗ ✷✸✸✾✝

Page 14: 3D Sketching with profile curves - Inria - Accueil · p os l se i b ocer os cisur nt t n 1 4 [, 6 ]. osM t h se e t se hp por a c a aer l i m e i d t o t p o e hl dy rl a o d l,es

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✶✹✳ ❏✳ P✳ P❡r❡✐r❛✱ ❱✳ ❆✳ ❇r❛♥❝♦✱ ❏✳ ❆✳ ❏♦r❣❡✱ ◆✳ ❋✳ ❙✐❧✈❛✱ ❚✳ ❉✳ ❈❛r❞♦s♦✱ ❛♥❞ ❋✳ ◆✉♥❡s

❋❡rr❡✐r❛✳ ❈❛s❝❛❞✐♥❣ ❘❡❝♦❣♥✐③❡rs ❢♦r ❆♠❜✐❣✉♦✉s ❈❛❧❧✐❣r❛♣❤✐❝ ■♥t❡r❛❝t✐♦♥✳ ■♥ ❊�✁✂✲

✄✁☎✆✝✞✟✠ ❲✂✁❦✠✝✂✆ ✂✡ ☛❦☞✌✟✝✲✍☎✠☞✎ ✏✡✌☞✁✑☎✟☞✠ ☎✡✎ ▼✂✎☞✒✞✡✄✱ ❆✉❣✉st ✷✵✵✹✳

✶✺✳ ❉✳ P✉❣❤✳ ❉❡s✐❣♥✐♥❣ s♦❧✐❞ ♦❜❥❡❝ts ✉s✐♥❣ ✐♥t❡r❛❝t✐✈❡ s✓❡t❝❤ ✐♥t❡r♣r❡t❛t✐♦♥✳ ■♥ ✔✁✂✟✕

☛✏✸✖ ✬✾✗✱ ♣❛❣❡s ✶✶✼④✶✷✻✳ ❆❈✘ Pr❡ss✱ ✶✙✙✷✳

✶✻✳ ❍✳ ❘✉s❤♠❡✐r✱ ❏✳ ●♦♠❡s✱ ▲✳ ❇❛❧♠❡❧❧✐✱ ❋✳ ❇❡r♥❛r❞✐✱ ❛♥❞ ●✳ ❚❛✉❜✐♥✳ ■♠❛❣❡✚❇❛s❡❞

❖❜❥❡❝t ✛❞✐t✐♥❣✳ ■♥ ✔✁✂✟✕ ✸✖✏▼ ✬✜✸✱ ♣❛❣❡s ✷✵④✷✽✱ ❖❝t♦❜❡r ✷✵✵✢✳

✶✼✳ ❘✳ ❙❝❤♠✐❞t✱ ❇✳ ✣②✈✐❧❧✱ ✘✳❈✳ ❙♦✉s❛✱ ❛♥❞ ❏✳❆✳ ❏♦r❣❡✳ ❙❤❛♣❡s❤♦♣✿ ❙✓❡t❝❤✚❜❛s❡❞

s♦❧✐❞ ♠♦❞❡❧✐♥❣ ✇✐t❤ ❜❧♦❜tr❡❡s✳ ■♥ ✗✡✎ ❊�✁✂✄✁☎✆✝✞✟✠ ❲✂✁❦✠✝✂✆ ✂✡ ☛❦☞✌✟✝✲✍☎✠☞✎

✏✡✌☞✁✑☎✟☞✠ ☎✡✎ ▼✂✎☞✒✞✡✄✱ ♣❛❣❡s ✺✢④✻✷✱ ✷✵✵✺✳

✶✽✳ ❆✳ ❙❤❡s❤ ❛♥❞ ❇✳ ❈❤❡♥✳ ❙✘❆❘❚P❆P✛❘④❆♥ ■♥t❡r❛❝t✐✈❡ ❛♥❞ ❯s❡r✚❢r✐❡♥❞❧② ❙✓❡t❝❤✐♥❣

❙②st❡♠✳ ✤✂✥✆�✌☞✁ ✦✁☎✆✝✞✟✠ ✧✂✁�✥ ✭✔✁✂✟✕ ❊�✁✂✄✁☎✆✝✞✟✠ ✗✜✜★✮✱ ✷✹✩✢✪✱ ❙❡♣t❡♠❜❡r

✷✵✵✹✳

✶✙✳ ❈✳✚▲✳ ❚❛✐✱ ❍✳ ❩❤❛♥❣✱ ❛♥❞ ❈✳✚❑✳ ❋♦♥❣✳ Pr♦t♦t②♣❡ ♠♦❞❡❧✐♥❣ ❢r♦♠ s✓❡t❝❤❡❞ s✐❧❤♦✉❡tt❡s

❜❛s❡❞ ♦♥ ❝♦♥✈♦❧✉t✐♦♥ s✉r❢❛❝❡s✳ ✤✂✥✆�✌☞✁ ✦✁☎✆✝✞✟✠ ✧✂✁�✥ ✭❊�✁✂✄✁☎✆✝✞✟✠ ✬✜★✮✱ ✷✵✵✹✳

✷✵✳ ❖✳ ❚♦❧❜❛✱ ❏✳ ❉♦rs❡②✱ ❛♥❞ ▲✳ ✘❝✘✐❧❧❛♥✳ ❙✓❡t❝❤✐♥❣ ✇✐t❤ ♣r♦❥❡❝t✐✈❡ ✷❉ str♦✓❡s✳ ■♥

✔✁✂✟✕ ✫✏☛✯ ✬✾✾✱ ♣❛❣❡s ✶✹✙④✶✺✼✳ ❆❈✘ Pr❡ss✱ ✶✙✙✙✳

✷✶✳ ❙✳ ❚s❛♥❣✱ ❘✳ ❇❛❧❛✓r✐s❤♥❛♥✱ ❑✳ ❙✐♥❣❤✱ ❛♥❞ ❆✳ ❘❛♥❥❛♥✳ ❆ s✉❣❣❡st✐✈❡ ✐♥t❡r❢❛❝❡ ❢♦r

✐♠❛❣❡ ❣✉✐❞❡❞ ✢❞ s✓❡t❝❤✐♥❣✳ ■♥ ✤✰✏ ✬✜★✴ ✔✁✂✟☞☞✎✞✡✄✠ ✂✑ ✌✝☞ ☛✏✦✤✰✏ ✟✂✡✑☞✁☞✡✟☞ ✂✡

✰�✥☎✡ ✑☎✟✌✂✁✠ ✞✡ ✟✂✥✆�✌✞✡✄ ✠❀✠✌☞✥✠✱ ♣❛❣❡s ✺✙✶④✺✙✽✱ ◆❡✇ ❨♦r✓✱ ◆❨✱ ❯❙❆✱ ✷✵✵✹✳

❆❈✘ Pr❡ss✳

✷✷✳ ❈✳ ✣✳ ❆✳ ✘✳ ✈❛♥ ❖✈❡r✈❡❧❞✳ P❛✐♥t✐♥❣ ❣r❛❞✐❡♥ts✿ ❢r❡❡✚❢♦r♠ s✉r❢❛❝❡ ❞❡s✐❣♥ ✉s✐♥❣ s❤❛❞✚

✐♥❣ ♣❛tt❡r♥s✳ ■♥ ✔✁✂✟✕ ✦✁☎✆✝✞✟✠ ✞✡✌☞✁✑☎✟☞ ✬✾❁✱ ♣❛❣❡s ✶✺✶④✶✺✽✳ ❈❛♥❛❞✐❛♥ ■♥❢♦r♠❛t✐♦♥

Pr♦❝❡ss✐♥❣ ❙♦❝✐❡t②✱ ✶✙✙✻✳

✷✢✳ P✳ ❆✳ ❈✳ ❱❛r❧❡②✱ ❨✳ ❚❛✓❛❤❛s❤✐✱ ❏✳✘✐t❛♥✐✱ ❛♥❞ ❍✳❙✉③✉✓✐✳ ❆ ❚✇♦✚❙t❛❣❡ ❆♣♣r♦❛❝❤

❢♦r ■♥t❡r♣r❡t✐♥❣ ▲✐♥❡ ❉r❛✇✐♥❣s ♦❢ ❈✉r✈❡❞ ❖❜❥❡❝ts✳ ■♥ ❊�✁✂✄✁☎✆✝✞✟✠ ❲✂✁❦✠✝✂✆ ✂✡

☛❦☞✌✟✝✲✍☎✠☞✎ ✏✡✌☞✁✑☎✟☞✠ ☎✡✎ ▼✂✎☞✒✞✡✄✱ ❆✉❣✉st ✷✵✵✹✳

✷✹✳ ▲✳ ✣✐❧❧✐❛♠s✳ ✢❉ ♣❛✐♥t✳ ■♥ ✔✁✂✟✕ ☛✏✸✖ ✬✾✜✱ ♣❛❣❡s ✷✷✺④✷✢✢✳ ❆❈✘ Pr❡ss✱ ✶✙✙✵✳

✷✺✳ ❈❤❡♥ ❨❛♥❣✱ ❉❛♥❛ ❙❤❛r♦♥✱ ❛♥❞ ✘✐❝❤✐❡❧ ✈❛♥ ❞❡ P❛♥♥❡✳ ❙✓❡t❝❤✚❜❛s❡❞ ♠♦❞❡❧✐♥❣ ♦❢

♣❛r❛♠❡t❡r✐③❡❞ ♦❜❥❡❝ts✳ ■♥ ❊�✁✂✄✁☎✆✝✞✟✠ ❲✂✁❦✠✝✂✆ ✂✡ ☛❦☞✌✟✝✲✍☎✠☞✎ ✏✡✌☞✁✑☎✟☞✠ ☎✡✎

▼✂✎☞✒✞✡✄✱ ♣❛❣❡s ✻✢④✼✷✱ ✷✵✵✺✳

✷✻✳ ❘✳ ❈✳ ❩❡❧❡③♥✐✓✱ ❑✳ P✳ ❍❡r♥❞♦♥✱ ❛♥❞ ❏✳ ❋✳ ❍✉❣❤❡s✳ ❙❑✛❚❈❍✿ ❛♥ ✐♥t❡r❢❛❝❡ ❢♦r

s✓❡t❝❤✐♥❣ ✢❉ s❝❡♥❡s✳ ■♥ ✔✁✂✟✕ ☛✏✦✦❂❃✔✰ ✬✾❁✱ ♣❛❣❡s ✶✻✢④✶✼✵✳ ❆❈✘ Pr❡ss✱ ❏✉❧②

✶✙✙✻✳

✷✼✳ ❘✳ ❩❡♥✓❛ ❛♥❞ P✳ ❙❧❛✈✐✓✳ ◆❡✇ ❞✐♠❡♥s✐♦♥ ❢♦r s✓❡t❝❤❡s✳ ■♥ ☛✤✤✦ ✬✜✸✴ ✔✁✂✟☞☞✎✞✡✄✠ ✂✑

✌✝☞ ❄✾✌✝ ✠✆✁✞✡✄ ✟✂✡✑☞✁☞✡✟☞ ✂✡ ✤✂✥✆�✌☞✁ ✄✁☎✆✝✞✟✠✱ ♣❛❣❡s ✶✺✼④✶✻✢✱ ◆❡✇ ❨♦r✓✱ ◆❨✱

❯❙❆✱ ✷✵✵✢✳ ❆❈✘ Pr❡ss✳


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