Robust and Efficient Delaunay triangulations of points on ...

21
HAL Id: inria-00405478 https://hal.inria.fr/inria-00405478v1 Submitted on 20 Jul 2009 (v1), last revised 17 Dec 2009 (v4) 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. Robust and Effcient Delaunay triangulations of points on or close to a sphere Manuel Caroli, Pedro Machado Manhães de Castro, Sebastien Loriot, Monique Teillaud, Camille Wormser To cite this version: Manuel Caroli, Pedro Machado Manhães de Castro, Sebastien Loriot, Monique Teillaud, Camille Wormser. Robust and Effcient Delaunay triangulations of points on or close to a sphere. [Research Report] 2009. inria-00405478v1

Transcript of Robust and Efficient Delaunay triangulations of points on ...

Page 1: Robust and Efficient Delaunay triangulations of points on ...

HAL Id: inria-00405478https://hal.inria.fr/inria-00405478v1

Submitted on 20 Jul 2009 (v1), last revised 17 Dec 2009 (v4)

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.

Robust and Efficient Delaunay triangulations of pointson or close to a sphere

Manuel Caroli, Pedro Machado Manhães de Castro, Sebastien Loriot,Monique Teillaud, Camille Wormser

To cite this version:Manuel Caroli, Pedro Machado Manhães de Castro, Sebastien Loriot, Monique Teillaud, CamilleWormser. Robust and Efficient Delaunay triangulations of points on or close to a sphere. [ResearchReport] 2009. �inria-00405478v1�

Page 2: Robust and Efficient Delaunay triangulations of points on ...

appor t

de r ech er ch e

ISS

N0

24

9-6

39

9IS

RN

INR

IA/R

R--

??

??

--F

R+

EN

G

Thème SYM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Robust and Efficient Delaunay triangulations of

points on or close to a sphere

Manuel Caroli — Pedro M. M. de Castro — Sébastien Loriot — Camille Wormser —

Monique Teillaud

N° ????

Juillet 2009

Page 3: Robust and Efficient Delaunay triangulations of points on ...
Page 4: Robust and Efficient Delaunay triangulations of points on ...

Centre de recherche INRIA Sophia Antipolis – Méditerranée2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

❘♦❜✉st ❛♥❞ ❊✣❝✐❡♥t ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥s ♦❢

♣♦✐♥ts ♦♥ ♦r ❝❧♦s❡ t♦ ❛ s♣❤❡r❡

▼❛♥✉❡❧ ❈❛r♦❧✐∗ ✱ P❡❞r♦ ▼✳ ▼✳ ❞❡ ❈❛str♦∗ ✱ ❙é❜❛st✐❡♥ ▲♦r✐♦t∗ ✱

❈❛♠✐❧❧❡ ❲♦r♠s❡r† ✱ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞∗

❚❤è♠❡ ❙❨▼ ✖ ❙②stè♠❡s s②♠❜♦❧✐q✉❡s➱q✉✐♣❡s✲Pr♦❥❡ts ●é♦♠étr✐❝❛

❘❛♣♣♦rt ❞❡ r❡❝❤❡r❝❤❡ ♥➦ ❄❄❄❄ ✖ ❏✉✐❧❧❡t ✷✵✵✾ ✖ ✶✼ ♣❛❣❡s

❆❜str❛❝t✿ ❲❡ ♣r♦♣♦s❡ t✇♦ ❛♣♣r♦❛❝❤❡s ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❉❡❧❛✉♥❛② tr✐❛♥✲❣✉❧❛t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ ❛ s♣❤❡r❡✱ ♦r ♦❢ r♦✉♥❞❡❞ ♣♦✐♥ts ❝❧♦s❡ t♦ ❛ s♣❤❡r❡✱ ❜♦t❤❜❛s❡❞ ♦♥ t❤❡ ❝❧❛ss✐❝ ✐♥❝r❡♠❡♥t❛❧ ❛❧❣♦r✐t❤♠ ✐♥✐t✐❛❧❧② ❞❡s✐❣♥❡❞ ❢♦r t❤❡ ♣❧❛♥❡✳❚❤❡ s♣❛❝❡ ♦❢ ❝✐r❝❧❡s ❣✐✈❡s t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❢♦r t❤✐s ✇♦r❦✳ ❲❡✐♠♣❧❡♠❡♥t❡❞ t❤❡ t✇♦ ❛♣♣r♦❛❝❤❡s ✐♥ ❛ ❢✉❧❧② r♦❜✉st ✇❛②✱ ❜✉✐❧❞✐♥❣ ✉♣♦♥ ❡①✲✐st✐♥❣ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠s ♣r♦✈✐❞❡❞ ❜② t❤❡ ❝❣❛❧ ❧✐❜r❛r②✳ ❚❤❡ ❡✣❝✐❡♥❝② ❛♥❞s❝❛❧❛❜✐❧✐t② ♦❢ t❤❡ ♠❡t❤♦❞ ✐s s❤♦✇♥ ❜② ❜❡♥❝❤♠❛r❦s✳

❑❡②✲✇♦r❞s✿ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr②✱ ❉❡❧❛✉♥❛② ❚r✐❛♥❣✉❧❛t✐♦♥✱ ❱♦r♦♥♦✐❉✐❛❣r❛♠✱ ❙♣❤❡r❡✱ ❙♣❛❝❡ ♦❢ ❈✐r❝❧❡s✱ ❊①❛❝t ●❡♦♠❡tr✐❝ ❈♦♠♣✉t✐♥❣✱ ❝❣❛❧

❚❤✐s ✇♦r❦ ✇❛s ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② t❤❡ ❆◆❘ ✭❆❣❡♥❝❡ ◆❛t✐♦♥❛❧❡ ❞❡ ❧❛❘❡❝❤❡r❝❤❡✮ ✉♥❞❡r t❤❡ ✏❚r✐❛♥❣❧❡s✑ ♣r♦❥❡❝t ♦❢ t❤❡ Pr♦❣r❛♠♠❡ ❜❧❛♥❝ ❆◆❘✲✵✼✲❇▲❆◆✲✵✸✶✾❤tt♣✿✴✴✇✇✇✳✐♥r✐❛✳❢r✴❣❡♦♠❡tr✐❝❛✴❝♦❧❧❛❜♦r❛t✐♦♥s✴tr✐❛♥❣❧❡s✴✳

∗ ■◆❘■❆ ❙♦♣❤✐❛ ❆♥t✐♣♦❧✐s ✕ ▼é❞✐t❡rr❛♥é❡ ❬❊♠❛✐❧✿ ④▼❛♥✉❡❧✳❈❛r♦❧✐✱ P❡❞r♦✳▼❛❝❤❛❞♦✱❙❡❜❛st✐❡♥✳▲♦r✐♦t✱ ▼♦♥✐q✉❡✳❚❡✐❧❧❛✉❞⑥❅s♦♣❤✐❛✳✐♥r✐❛✳❢r❪

† ❊❚❍ ❩ür✐❝❤✱ ❙✇✐t③❡r❧❛♥❞ ❬❊♠❛✐❧✿ ❈❛♠✐❧❧❡✳❲♦r♠s❡r❅✐♥❢✳❡t❤③✳❝❤❪

Page 5: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ❞❡ ❉❡❧❛✉♥❛② r♦❜✉st❡ ❡t ❡✣❝❛❝❡✱ ♣♦✉r

❞❡s ♣♦✐♥ts s✉r ❧❛ s♣❤èr❡ ♦✉ ♣r♦❝❤❡ ❞✬❡❧❧❡

❘és✉♠é ✿ ◆♦✉s ♣r♦♣♦s♦♥s ❞❡✉① ❢❛ç♦♥s ❞❡ ❝❛❧❝✉❧❡r ❧❛ tr✐❛♥❣✉❧❛t✐♦♥ ❞❡❉❡❧❛✉♥❛② ❞✬✉♥ ❡♥s❡♠❜❧❡ ❞❡ ♣♦✐♥ts q✉✐ ❛♣♣❛rt✐❡♥♥❡♥t s♦✐t à ❧❛ s♣❤èr❡✱ s♦✐tà s♦♥ ✈♦✐s✐♥❛❣❡✳ ❈❡s ❞❡✉① ♠ét❤♦❞❡s r❡♣♦s❡♥t s✉r ❧✬❛❧❣♦r✐t❤♠❡ ✐♥❝ré♠❡♥t❛❧❝❧❛ss✐q✉❡✱ t❡❧ q✉✬✐❧ ❛ été ❝réé à ❧✬♦r✐❣✐♥❡ ♣♦✉r ❝❛❧❝✉❧❡r ❧❡s tr✐❛♥❣✉❧❛t✐♦♥s ❞❡❉❡❧❛✉♥❛② ♣❧❛♥❛✐r❡s✳ ▲❡ ❝❛❞r❡ ♠❛t❤é♠❛t✐q✉❡ ❝❧❛ss✐q✉❡ ❥✉st✐✜❛♥t ❝❡tt❡ ❛♣✲♣r♦❝❤❡ ❡st r❛♣♣❡❧é✱ à ❧✬❛✐❞❡ ❞❡ ❧✬❡s♣❛❝❡ ❞❡s ❝❡r❝❧❡s✳ ❈❡s ❞❡✉① ❛♣♣r♦❝❤❡s ♦♥tété ✐♠♣❧❛♥té❡s ❞❡ ❢❛ç♦♥ r♦❜✉st❡ ❡♥ s✬❛♣♣✉②❛♥t s✉r ❧❡s ❛❧❣♦r✐t❤♠❡s ❣é♥ér✐q✉❡s❢♦✉r♥✐s ♣❛r ❧❛ ❜✐❜❧✐♦t❤èq✉❡ ❈●❆▲✳ ❉❡s t❡sts ❝♦♠♣❛r❛t✐❢s ♠♦♥tr❡♥t ❧✬❡✣❝❛❝✐té❞❡ ♥♦s ✐♠♣❧❛♥t❛t✐♦♥s s✉r ❞❡s ❥❡✉① ❞❡ ❞♦♥♥é❡s ❞❡ t❛✐❧❧❡ ✈❛r✐é❡✳

▼♦ts✲❝❧és ✿ ●é♦♠étr✐❡ ❛❧❣♦r✐t❤♠✐q✉❡✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ❞❡ ❉❡❧❛✉♥❛②✱ ❉✐❛✲❣r❛♠♠❡ ❞❡ ❱♦r♦♥♦ï✱ ❙♣❤èr❡✱ ❊s♣❛❝❡ ❞❡s ❝❡r❝❧❡s✱ ❈❛❧❝✉❧ ❣é♦♠étr✐q✉❡ ❡①❛❝t❡✱❝❣❛❧

Page 6: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✸

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ❝❣❛❧ ♣r♦❥❡❝t ❬❝❣❛❪ ♣r♦✈✐❞❡s ✉s❡rs ✇✐t❤ ❛ ♣✉❜❧✐❝ ❞✐s❝✉ss✐♦♥ ♠❛✐❧✐♥❣ ❧✐st✱✇❤❡r❡ t❤❡② ❛r❡ ✐♥✈✐t❡❞ t♦ ♣♦st q✉❡st✐♦♥s ❛♥❞ ❡①♣r❡ss t❤❡✐r ♥❡❡❞s✳ ❚❤❡r❡ ❛r❡r❡❝✉rr✐♥❣ r❡q✉❡sts ❢♦r ❛ ♣❛❝❦❛❣❡ ❝♦♠♣✉t✐♥❣ t❤❡ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ♦r✐ts ❞✉❛❧✱ t❤❡ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠✱ ♦❢ ♣♦✐♥ts ♦♥ ❛ s♣❤❡r❡✳ ❚❤✐s ✐s ✉s❡❢✉❧ ❢♦r ✈❛r✐✲♦✉s ❛♣♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥s s✉❝❤ ❛s ❣❡♦❧♦❣②✱ ❣❡♦❣r❛♣❤✐❝ ✐♥❢♦r♠❛t✐♦♥ s②st❡♠s ♦rstr✉❝t✉r❛❧ ♠♦❧❡❝✉❧❛r ❜✐♦❧♦❣②✱ t♦ ♥❛♠❡ ❛ ❢❡✇✳ ❆♥ ❡❛s② ❛♥❞ st❛♥❞❛r❞ ❛♥s✇❡r❝♦♥s✐sts ✐♥ ❝♦♠♣✉t✐♥❣ t❤❡ ✸❉ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡ ♣♦✐♥ts✱ ✇❤✐❝❤ ✐s ♥♦t♦r✐♦✉s❧②❡q✉✐✈❛❧❡♥t ❬❇r♦✽✵✱ ❙✉❣✵✷❪✳ ❚❤❡ ❝♦♥✈❡① ❤✉❧❧ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ♣♦♣✉❧❛r str✉❝✲t✉r❡s ✐♥ ❝♦♠♣✉t❛t✐♦♥❛❧ ❣❡♦♠❡tr② ❬❞❇✈❑❖❙✵✵✱ ❇❨✾✽❪❀ ♦♣t✐♠❛❧ ❛❧❣♦r✐t❤♠s❛♥❞ ❡✣❝✐❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❬❤✉❧✱ q❤✉❪✳

❆♥♦t❤❡r ❢r✉✐t❢✉❧ ✇❛② t♦ ❝♦♠♣✉t❡ ❉❡❧❛✉♥❛② ♦♥ ❛ s♣❤❡r❡ ❝♦♥s✐sts ♦❢ r❡✇♦r❦✲✐♥❣ ❦♥♦✇♥ ❛❧❣♦r✐t❤♠s ❞❡s✐❣♥❡❞ ❢♦r ❝♦♠♣✉t✐♥❣ tr✐❛♥❣✉❧❛t✐♦♥s ✐♥ R

2✳ ❘❡♥❦❛❛❞❛♣ts t❤❡ ❞✐st❛♥❝❡ ✐♥ t❤❡ ♣❧❛♥❡ t♦ ❛ ❣❡♦❞❡s✐❝ ❞✐st❛♥❝❡ ♦♥ ❛ s♣❤❡r❡ ❛♥❞ tr✐❛♥✲❣✉❧❛t❡s ♣♦✐♥ts ♦♥ ❛ s♣❤❡r❡ ❬❘❡♥✾✼❪ t❤r♦✉❣❤ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ✢✐♣♣✐♥❣ ❛❧❣♦r✐t❤♠❢♦r ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥s ✐♥ R

2 ❬▲❛✇✼✼❪✳ ❆s ❛ ❜②✲♣r♦❞✉❝t ♦❢ t❤❡✐r ❛❧❣♦✲r✐t❤♠ ❢♦r ❛rr❛♥❣❡♠❡♥ts ♦❢ ❝✐r❝✉❧❛r ❛r❝s✱ ❋♦❣❡❧ ❡t ❛❧✳ ❝❛♥ ❝♦♠♣✉t❡ ❱♦r♦♥♦✐❞✐❛❣r❛♠s ✐♥ t❤❡ r❡str✐❝t❡❞ ❝❛s❡ ♦❢ ♣♦✐♥ts ❧②✐♥❣ ❡①❛❝t❧② ♦♥ t❤❡ s♣❤❡r❡ ❛♥❞❤❛✈✐♥❣ r❛t✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡s ❬❋❙❍✵✽✱ ❋❙✱ ❇❋❍+✵✾❜✱ ❇❋❍+✵✾❛❪✳ ❯s✐♥❣ t✇♦✐♥✈❡rs✐♦♥s ❛❧❧♦✇s ◆❛ ❡t ❛❧✳ t♦ r❡❞✉❝❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠♦❢ s✐t❡s ♦♥ ❛ s♣❤❡r❡ t♦ ❝♦♠♣✉t✐♥❣ t✇♦ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠s ✐♥ R

2 ❬◆▲❈✵✷❪✱ ❜✉t♥♦ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐s ❛✈❛✐❧❛❜❧❡✳ ◆♦t❡ t❤❛t t❤✐s ♠❡t❤♦❞ ❛ss✉♠❡s t❤❛t ❞❛t❛♣♦✐♥ts ❛r❡ ❧②✐♥❣ ❡①❛❝t❧② ♦♥ ❛ s♣❤❡r❡✳

❆s ✇❡ ❛r❡ ♠♦t✐✈❛t❡❞ ❜② ❛♣♣❧✐❝❛t✐♦♥s✱ ✇❡ t❛❦❡ ♣r❛❝t✐❝❛❧ ✐ss✉❡s ✐♥t♦ ❛❝❝♦✉♥t❝❛r❡❢✉❧❧②✳ ❲❤✐❧❡ ❞❛t❛ ♣♦✐♥ts ❧②✐♥❣ ❡①❛❝t❧② ♦♥ t❤❡ s♣❤❡r❡ ❝❛♥ ❜❡ ♣r♦✈✐❞❡❞❡✐t❤❡r ❜② ✉s✐♥❣ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s r❡♣r❡s❡♥t❡❞ ❜② ❛ ♥✉♠❜❡r t②♣❡ ❝❛♣❛❜❧❡♦❢ ❤❛♥❞❧✐♥❣ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡rs ❡①❛❝t❧②✱ ♦r ❜② ✉s✐♥❣ s♣❤❡r✐❝❛❧ ❝♦♦r❞✐♥❛t❡s✱ ✐♥♣r❛❝t✐❝❡ ❞❛t❛✲s❡ts ✐♥ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ✇✐t❤ ❞♦✉❜❧❡ ♣r❡❝✐s✐♦♥ ❛r❡ ♠♦st❝♦♠♠♦♥✳ ■♥ t❤✐s s❡tt✐♥❣✱ t❤❡ ❞❛t❛ ❝♦♥s✐sts ♦❢ r♦✉♥❞❡❞ ♣♦✐♥ts t❤❛t ❞♦ ♥♦t❡①❛❝t❧② ❧✐❡ ♦♥ t❤❡ s♣❤❡r❡✱ ❜✉t ❝❧♦s❡ t♦ ✐t✳

■♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ ♣r♦♣♦s❡ t✇♦ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s t♦ ❤❛♥❞❧❡ s✉❝❤ r♦✉♥❞❡❞❞❛t❛✳ ❇♦t❤ ❛♣♣r♦❛❝❤❡s ❛❞❛♣t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ✐♥❝r❡♠❡♥t❛❧ ❛❧❣♦r✐t❤♠ ❬❇♦✇✽✶❪t♦ t❤❡ ❝❛s❡ ♦❢ ♣♦✐♥ts ♦♥✱ ♦r ❝❧♦s❡ t♦ t❤❡ s♣❤❡r❡✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t✐❝❡t❤❛t✱ ❡✈❡♥ t❤♦✉❣❤ ❞❛t❛ ♣♦✐♥ts ❛r❡ r♦✉♥❞❡❞✱ ✇❡ ❢♦❧❧♦✇ t❤❡ ❡①❛❝t ❣❡♦♠❡tr✐❝❝♦♠♣✉t❛t✐♦♥ ♣❛r❛❞✐❣♠ ♣✐♦♥❡❡r❡❞ ❜② ❈✳ ❑✳ ❨❛♣ ❬❨❉✾✺❪✳ ■♥❞❡❡❞✱ ✐t ✐s ♥♦✇ ✇❡❧❧✉♥❞❡rst♦♦❞ t❤❛t s✐♠♣❧② r❡❧②✐♥❣ ♦♥ ✢♦❛t✐♥❣ ♣♦✐♥t ❛r✐t❤♠❡t✐❝ ❢♦r ❛❧❣♦r✐t❤♠s♦❢ t❤✐s t②♣❡ ✐s ❜♦✉♥❞ t♦ ❢❛✐❧ ✭s❡❡ ❬❑▼P+✵✽❪ ❢♦r ✐♥st❛♥❝❡✮✳

• ❚❤❡ ✜rst ❛♣♣r♦❛❝❤ ✭❙❡❝t✐♦♥ ✹✳✶✮ ❝♦♥s✐❞❡rs ❛s ✐♥♣✉t t❤❡ ♣r♦❥❡❝t✐♦♥s ♦❢t❤❡ r♦✉♥❞❡❞ ❞❛t❛ ♣♦✐♥ts ♦♥t♦ t❤❡ s♣❤❡r❡✳ ❚❤❡✐r ❝♦♦r❞✐♥❛t❡s ❛r❡ ❛❧❣❡❜r❛✐❝♥✉♠❜❡rs ♦❢ ❞❡❣r❡❡ t✇♦✳ ❚❤❡ ❛♣♣r♦❛❝❤ ❝♦♠♣✉t❡s t❤❡ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥♦❢ t❤❡s❡ ♣♦✐♥ts ❡①❛❝t❧② ❧②✐♥❣ ♦♥ t❤❡ s♣❤❡r❡✳

• ❚❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ✭❙❡❝t✐♦♥ ✹✳✷✮ ❝♦♥s✐❞❡rs ❝✐r❝❧❡s ♦♥ t❤❡ s♣❤❡r❡ ❛s✐♥♣✉t✳ ❚❤❡ r❛❞✐✉s ♦❢ ❛ ❝✐r❝❧❡ ✭✇❤✐❝❤ ❝❛♥ ❛❧t❡r♥❛t✐✈❡❧② ❜❡ s❡❡♥ ❛s ❛ ✇❡✐❣❤t❡❞

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 7: Robust and Efficient Delaunay triangulations of points on ...

✹ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

♣♦✐♥t✮ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥t t♦ t❤❡ s♣❤❡r❡✳ ❚❤❡❛♣♣r♦❛❝❤ ❝♦♠♣✉t❡s t❤❡ ✇❡✐❣❤t❡❞ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ t❤❡s❡ ❝✐r❝❧❡s ♦♥t❤❡ s♣❤❡r❡✱ ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✱ ✇❤✐❝❤ ✐s t❤❡ ❞✉❛❧ ♦❢ t❤❡▲❛❣✉❡rr❡ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ♦♥ t❤❡ s♣❤❡r❡ ❬❙✉❣✵✷❪ ❛♥❞ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡r♦✉♥❞❡❞ ❞❛t❛ ♣♦✐♥ts✳

❚❤❡s❡ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ r♦✉♥❞❡❞ ❞❛t❛ ❛r❡ s✉♣♣♦rt❡❞ ❜② t❤❡ s♣❛❝❡ ♦❢❝✐r❝❧❡s ❬❇❡r✽✼✱ ❉▼❚✾✷❪ ✭❙❡❝t✐♦♥ ✸✮✳

❚❤❡ s❛♠❡ ❛❧❣♦r✐t❤♠ ❜❡✐♥❣ ✉s❡❞ ❢♦r ❜♦t❤ ❛♣♣r♦❛❝❤❡s✱ ♦♥❧② t❤❡ ✐♥t❡r♣r❡✲t❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛ ❛♥❞ t❤❡ ✇❛② ✐♥ ✇❤✐❝❤ ❜❛s✐❝ ❝♦♠♣✉t❛t✐♦♥s ♦♥ t❤❡♠ ❛r❡♣❡r❢♦r♠❡❞ ♠♦❞✐❢② t❤❡ ❜❡❤❛✈✐♦r ❛♥❞ ♣♦ss✐❜❧② t❤❡ r❡s✉❧t ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✿❙♦♠❡ ❝✐r❝❧❡s ♠❛② ❜❡ ❤✐❞❞❡♥ ✐♥ ❛ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥✳ ❲❡ s❤♦✇ ✐♥ ❙❡❝✲t✐♦♥ ✹✳✷ t❤❛t t❤✐s ❝❛♥ ❤❛♣♣❡♥ ♦♥❧② ✐❢ t❤❡ r♦✉♥❞❡❞ ♣♦✐♥ts ❛r❡ ❡①tr❡♠❡❧② ❝❧♦s❡t♦ ♦♥❡ ❛♥♦t❤❡r✳ ❚❤✉s✱ ✉♥❞❡r s♦♠❡ s❛♠♣❧✐♥❣ ❝♦♥❞✐t✐♦♥s✱ t❤❡ t✇♦ ❛♣♣r♦❛❝❤❡s❝♦♠♣✉t❡ t❤❡ s❛♠❡ tr✐❛♥❣✉❧❛t✐♦♥ ✭❡①❝❡♣t ♠❛②❜❡ ✐♥ ❞❡❣❡♥❡r❛t❡ ❝❛s❡s ❢♦r ✇❤✐❝❤t❤❡ ❝♦♥s✐❞❡r❡❞ tr✐❛♥❣✉❧❛t✐♦♥s ❛r❡ ❦♥♦✇♥ ♥♦t t♦ ❜❡ ✉♥✐q✉❡❧② ❞❡✜♥❡❞✮✱ ❛♥❞ ✐♥❣❡♥❡r❛❧✱ ❛❧❧ ♣♦✐♥ts ❛♣♣❡❛r ✐♥ t❤❡ r❡s✉❧t ♦❢ ❜♦t❤ ❛♣♣r♦❛❝❤❡s✳

■♥ ❙❡❝t✐♦♥ ✺✱ ✇❡ s❤♦✇ t❤❛t t❤❡ ❣❡♥❡r✐❝✐t② ♦❢ ❝❣❛❧ ❬❋❚✵✻❪ ❛❧❧♦✇s t♦ r❡✉s❡❛ ❧❛r❣❡ ♣❛rt ♦❢ t❤❡ ❡①✐st✐♥❣ ♣❛❝❦❛❣❡ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s ✐♥ R

2 t♦ ✐♠♣❧❡♠❡♥t♦✉r t✇♦ ❛♣♣r♦❛❝❤❡s✳ ❲❡ ♣r❡s❡♥t ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ♦♥ ✈❡r② ❧❛r❣❡ ❞❛t❛✲s❡ts✱s❤♦✇✐♥❣ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ♦✉r ❛♣♣r♦❛❝❤✳ ❲❡ ❝♦♠♣❛r❡ ♦✉r ❝♦❞❡ t♦ s♦❢t✇❛r❡❞❡s✐❣♥❡❞ ❢♦r ❝♦♠♣✉t✐♥❣ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥s ♦♥ t❤❡ s♣❤❡r❡✱ ❛♥❞ t♦ ❝♦♥✈❡①❤✉❧❧ s♦❢t✇❛r❡ ❬❍❙✵✾✱ P❚✵✾❜✱ ❤✉❧✱ s✉❣✱ q❤✉✱ ❘❡♥✾✼✱ ❋❙❪✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡✱r♦❜✉st♥❡ss✱ ❛♥❞ s❝❛❧❛❜✐❧✐t② ♦❢ ♦✉r ❛♣♣r♦❛❝❤❡s s❤♦✇ t❤❡✐r ❛❞❞❡❞ ✈❛❧✉❡✳

✷ ❉❡✜♥✐t✐♦♥s ❛♥❞ ◆♦t❛t✐♦♥

▲❡t ✉s ✜rst r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ✐♥ R2✱ ❛❧s♦

❦♥♦✇♥ ❛s ✇❡✐❣❤t❡❞ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥✳ ❆ ❝✐r❝❧❡ c ✇✐t❤ ❝❡♥t❡r p ∈ R2

❛♥❞ sq✉❛r❡❞ r❛❞✐✉s r2 ✐s ❝♦♥s✐❞❡r❡❞ ❡q✉✐✈❛❧❡♥t❧② ❛s ❛ ✇❡✐❣❤t❡❞ ♣♦✐♥t ❛♥❞ ✐s❞❡♥♦t❡❞ ❜② c = (p, r2)✳ ❚❤❡ ♣♦✇❡r ♣r♦❞✉❝t ♦❢ c = (p, r2) ❛♥❞ c′ = (p′, r′2) ✐s❞❡✜♥❡❞ ❛s ♣♦✇(c, c′) = ‖pp′‖2 − r2 − r′2✱ ✇❤❡r❡ ‖pp′‖ ❞❡♥♦t❡s t❤❡ ❊✉❝❧✐❞❡❛♥❞✐st❛♥❝❡ ❜❡t✇❡❡♥ p ❛♥❞ p′✳ ❈✐r❝❧❡s c ❛♥❞ c′ ❛r❡ ♦rt❤♦❣♦♥❛❧ ✐✛ ♣♦✇(c, c′) = 0✳■❢ ♣♦✇(c, c′) > 0 ✭✐✳❡✳ t❤❡ ❞✐s❦s ❞❡✜♥❡❞ ❜② c ❛♥❞ c′ ❞♦ ♥♦t ✐♥t❡rs❡❝t✱ ♦r t❤❡❝✐r❝❧❡s ✐♥t❡rs❡❝t ✇✐t❤ ❛♥ ❛♥❣❧❡ str✐❝t❧② s♠❛❧❧❡r t❤❛♥ π

2 ✮✱ ✇❡ s❛② t❤❛t c ❛♥❞c′ ❛r❡ s✉❜♦rt❤♦❣♦♥❛❧✳ ■❢ ♣♦✇(c, c′) < 0✱ t❤❡♥ ✇❡ s❛② t❤❛t c ❛♥❞ c′ ❛r❡ s✉♣❡r✲♦rt❤♦❣♦♥❛❧ ✭s❡❡ ❋✐❣✉r❡ ✶✮✳ ❚❤r❡❡ ❝✐r❝❧❡s ✇❤♦s❡ ❝❡♥t❡rs ❛r❡ ♥♦t ❝♦❧❧✐♥❡❛r ❤❛✈❡❛ ✉♥✐q✉❡ ❝♦♠♠♦♥ ♦rt❤♦❣♦♥❛❧ ❝✐r❝❧❡✳

▲❡t S ❜❡ ❛ s❡t ♦❢ ❝✐r❝❧❡s✳ ●✐✈❡♥ t❤r❡❡ ❝✐r❝❧❡s ♦❢ S✱ ci = (pi, r2i ), i ∈

I ⊂ N ❛♥❞ |I| = 3✱ ✇❤♦s❡ ❝❡♥t❡rs ❛r❡ ♥♦t ❝♦❧❧✐♥❡❛r✱ ❧❡t TI ❜❡ t❤❡ tr✐✲❛♥❣❧❡ ✇❤♦s❡ ✈❡rt✐❝❡s ❛r❡ t❤❡ t❤r❡❡ ❝❡♥t❡rs pi, i ∈ I✳ ❲❡ ❞❡✜♥❡ t❤❡ ♦r✲

t❤♦❣♦♥❛❧ ❝✐r❝❧❡ ♦❢ TI ❛s t❤❡ ❝✐r❝❧❡ t❤❛t ✐s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ t❤r❡❡ ❝✐r❝❧❡sci, i ∈ I✳ TI ✐s s❛✐❞ t♦ ❜❡ r❡❣✉❧❛r ✐❢ ❢♦r ❛♥② ❝✐r❝❧❡ c ∈ S✱ t❤❡ ♦rt❤♦❣✲♦♥❛❧ ❝✐r❝❧❡ ♦❢ TI ❛♥❞ c ❛r❡ ♥♦t s✉♣❡r♦rt❤♦❣♦♥❛❧✳ ❆ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥

■◆❘■❆

Page 8: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✺

π/2 < π/2 > π/2

orthogonal: pow(so, s1) = 0 suborthogonal: pow(s0, s1) > 0 superorthogonal: pow(s0, s1) < 0

❋✐❣✉r❡ ✶✿ ❖rt❤♦❣♦♥❛❧✱ s✉❜♦rt❤♦❣♦♥❛❧ ❛♥❞ s✉♣❡r♦rt❤♦❣♦♥❛❧ ❝✐r❝❧❡s ✐♥ R2✳

RT (S) ✐s ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡ ❝❡♥t❡rs ♦❢ t❤❡ ❝✐r❝❧❡s ♦❢ S✐♥t♦ r❡❣✉❧❛r tr✐❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡s❡ ❝❡♥t❡rs✳ ❙❡❡ ❋✐❣✉r❡ ✷ ❢♦r ❛♥ ❡①❛♠♣❧❡✳

❋✐❣✉r❡ ✷✿ ❘❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥♦❢ ❛ s❡t ♦❢ ❝✐r❝❧❡s ✐♥ t❤❡ ♣❧❛♥❡✭t❤❡✐r ♣♦✇❡r ❞✐❛❣r❛♠ ✐s s❤♦✇♥❞❛s❤❡❞✮

❚❤❡ ❞✉❛❧ ♦❢ t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ✐s❦♥♦✇♥ ❛s t❤❡ ♣♦✇❡r ❞✐❛❣r❛♠✱ ✇❡✐❣❤t❡❞

❱♦r♦♥♦✐ ❞✐❛❣r❛♠✱ ♦r ▲❛❣✉❡rr❡ ❞✐❛❣r❛♠✳■❢ ❛❧❧ r❛❞✐✐ ❛r❡ ❡q✉❛❧✱ t❤❡♥ t❤❡ ♣♦✇❡r

t❡st r❡❞✉❝❡s t♦ t❡st✐♥❣ ✇❤❡t❤❡r ❛ ♣♦✐♥t ❧✐❡s✐♥s✐❞❡✱ ♦✉ts✐❞❡✱ ♦r ♦♥ t❤❡ ❝✐r❝❧❡ ♣❛ss✐♥❣t❤r♦✉❣❤ t❤r❡❡ ♣♦✐♥ts❀ t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉✲❧❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡s ✐s t❤❡ ❉❡❧❛✉♥❛② tr✐❛♥❣✉✲❧❛t✐♦♥ DT ♦❢ t❤❡✐r ❝❡♥t❡rs✳

▼♦r❡ ❜❛❝❦❣r♦✉♥❞ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥❬❆✉r✽✼❪✳ ❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ st❛♥❞❛r❞t❡①t❜♦♦❦s ❢♦r ❛❧❣♦r✐t❤♠s ❝♦♠♣✉t✐♥❣ ❉❡❧❛✉✲♥❛② ❛♥❞ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥s ❬❞❇✈❑❖❙✵✵✱❇❨✾✽❪✳

❚❤✐s ❞❡✜♥✐t✐♦♥ ❣❡♥❡r❛❧✐③❡s ✐♥ ❛ ♥❛t✉r❛❧ ♠❛♥♥❡r t♦ t❤❡ ❝❛s❡ ♦❢ ❝✐r❝❧❡s ❧②✐♥❣♦♥ ❛ s♣❤❡r❡ S ✐♥ R

3✿ ❆♥❣❧❡s ❜❡t✇❡❡♥ ❝✐r❝❧❡s ❛r❡ ♠❡❛s✉r❡❞ ♦♥ t❤❡ s♣❤❡r❡✱tr✐❛♥❣❧❡s ❛r❡ ❞r❛✇♥ ♦♥ t❤❡ s♣❤❡r❡✱ t❤❡✐r ❡❞❣❡s ❜❡✐♥❣ ❛r❝s ♦❢ ❣r❡❛t ❝✐r❝❧❡s✳❆s ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ t❤❡ s♣❛❝❡ ♦❢ ❝✐r❝❧❡s ♣r♦✈✐❞❡s ❛ ❣❡♦♠❡tr✐❝♣r❡s❡♥t❛t✐♦♥ s❤♦✇✐♥❣ ✇✐t❤♦✉t ❛♥② ❝♦♠♣✉t❛t✐♦♥ t❤❛t t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥♦♥ S ✐s ❛ ❝♦♥✈❡① ❤✉❧❧ ✐♥ R

3 ❬❙✉❣✵✷❪✳■♥ t❤❡ s❡q✉❡❧✱ ✇❡ ❛ss✉♠❡ t❤❛t S ✐s ❣✐✈❡♥ ❜② ✐ts ❝❡♥t❡r✱ ❤❛✈✐♥❣ r❛t✐♦♥❛❧

❝♦♦r❞✐♥❛t❡s ✭✇❡ t❛❦❡ t❤❡ ♦r✐❣✐♥ O ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✮✱ ❛♥❞ ❛ r❛t✐♦♥❛❧sq✉❛r❡❞ r❛❞✐✉s R2✳ ❚❤✐s ✐s ❛❧s♦ ❤♦✇ s♣❤❡r❡s ❛r❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❝❣❛❧✳✶

✸ ❙♣❛❝❡ ♦❢ ❈✐r❝❧❡s

❈♦♠♣✉t❛t✐♦♥❛❧ ❣❡♦♠❡t❡rs ❛r❡ ❢❛♠✐❧✐❛r ✇✐t❤ t❤❡ ❝❧❛ss✐❝ ✐❞❡❛ ♦❢ ❧✐❢t✐♥❣ ✉♣ s✐t❡s❢r♦♠ t❤❡ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ♦♥t♦ t❤❡ ✉♥✐t ♣❛r❛❜♦❧♦✐❞ Π ✐♥ R

3 ❬❆✉r✾✶❪✳ ❲❡q✉✐❝❦❧② r❡❝❛❧❧ t❤❡ ♥♦t✐♦♥ ♦❢ s♣❛❝❡ ♦❢ ❝✐r❝❧❡s ❤❡r❡ ❛♥❞ r❡❢❡r t♦ t❤❡ ❧✐t❡r❛t✉r❡❢♦r ❛ ♠♦r❡ ❞❡t❛✐❧❡❞ ♣r❡s❡♥t❛t✐♦♥ ❬❉▼❚✾✷❪✳ ■♥ t❤✐s ❧✐❢t✐♥❣✱ ♣♦✐♥ts ♦❢ R

3 ❛r❡✈✐❡✇❡❞ ❛s ❝✐r❝❧❡s ♦❢ R

2 ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❝✐r❝❧❡s✿ ❆ ❝✐r❝❧❡ c = (p, r2) ✐♥ R2 ✐s

✶❲❡ ♠❡♥t✐♦♥ r❛t✐♦♥❛❧ ♥✉♠❜❡rs t♦ s✐♠♣❧✐❢② t❤❡ ♣r❡s❡♥t❛t✐♦♥✳ ❝❣❛❧ ❛❧❧♦✇s ♠♦r❡ ❣❡♥❡r❛❧♥✉♠❜❡r t②♣❡s t❤❛t ♣r♦✈✐❞❡ ✜❡❧❞ ♦♣❡r❛t✐♦♥s✿ +,−,×, /✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 9: Robust and Efficient Delaunay triangulations of points on ...

✻ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

♠❛♣♣❡❞ ❜② π t♦ t❤❡ ♣♦✐♥t π(c) = (xp, yp, x2p + y2

p − r2) ∈ R3✳ ❆ ♣♦✐♥t ♦❢

R3 ❧②✐♥❣ r❡s♣❡❝t✐✈❡❧② ♦✉ts✐❞❡✱ ✐♥s✐❞❡✱ ♦r ♦♥ t❤❡ ♣❛r❛❜♦❧♦✐❞ Π r❡♣r❡s❡♥ts ❛

❝✐r❝❧❡ ✇✐t❤ r❡s♣❡❝t✐✈❡❧② ♣♦s✐t✐✈❡✱ ✐♠❛❣✐♥❛r②✱ ♦r ♥✉❧❧ r❛❞✐✉s✳ ❚❤❡ ❝✐r❝❧❡ c ✐♥ R2

❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ ♣♦✐♥t π(c) ♦❢ R3 ♦✉ts✐❞❡ Π ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ♣r♦❥❡❝t✐♦♥

♦♥t♦ R2 ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ❜❡t✇❡❡♥ Π ❛♥❞ t❤❡ ❝♦♥❡ ❢♦r♠❡❞ ❜② ❧✐♥❡s t❤r♦✉❣❤

π(c) t❤❛t ❛r❡ t❛♥❣❡♥t t♦ Π❀ t❤✐s ✐♥t❡rs❡❝t✐♦♥ ✐s ❛❧s♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡♣♦❧❛r ♣❧❛♥❡ P (c) ♦❢ π(c) ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ q✉❛❞r✐❝ Π✳

P♦✐♥ts ❧②✐♥❣ r❡s♣❡❝t✐✈❡❧② ♦♥✱ ❛❜♦✈❡✱ ❜❡❧♦✇ P (c) ❝♦rr❡s♣♦♥❞ t♦ ❝✐r❝❧❡s ✐♥R

2 t❤❛t ❛r❡ r❡s♣❡❝t✐✈❡❧② ♦rt❤♦❣♦♥❛❧✱ s✉❜♦rt❤♦❣♦♥❛❧✱ s✉♣❡r♦rt❤♦❣♦♥❛❧ t♦ c✳❚❤❡ ♣r❡❞✐❝❛t❡ ♣♦✇(c, c′) ✐♥tr♦❞✉❝❡❞ ❛❜♦✈❡ ✐s t❤✉s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♦r✐❡♥✲t❛t✐♦♥ ♣r❡❞✐❝❛t❡ ✐♥ R

3 t❤❛t t❡sts ✇❤❡t❤❡r t❤❡ ♣♦✐♥t π(c′) ❧✐❡s ♦♥✱ ❛❜♦✈❡ ♦r❜❡❧♦✇ t❤❡ ♣❧❛♥❡ P (c)✳ ■❢ c ✐s t❤❡ ❝♦♠♠♦♥ ♦rt❤♦❣♦♥❛❧ ❝✐r❝❧❡ t♦ t❤r❡❡ ✐♥♣✉t❝✐r❝❧❡s c1, c2, ❛♥❞ c3 ✭✇❤❡r❡ ci = (pi, r

2i ) ❢♦r ❡❛❝❤ i✮✱ t❤❡♥ ♣♦✇(c, c′) ✐s t❤❡

♦r✐❡♥t❛t✐♦♥ ♣r❡❞✐❝❛t❡ ♦❢ t❤❡ ❢♦✉r ♣♦✐♥ts π(c1), π(c2), π(c3), π(c′) ♦❢ R3✳ ■t ❝❛♥

❜❡ ❡①♣r❡ss❡❞ ❛s

s✐❣♥

˛

˛

˛

˛

˛

˛

˛

˛

1 1 1 1

xp1xp2

xp3xp′

yp1yp2

yp3yp′

zp1zp2

zp3zp′

˛

˛

˛

˛

˛

˛

˛

˛

, ✭✶✮

✇❤❡r❡ zpi= x2

pi+ y2

pi− r2

i ❢♦r ❡❛❝❤ i ❛♥❞ z2p′ = x2

p′ + y2p′ − r′2✳ ■t ❛❧❧♦✇s

t♦ r❡❧❛t❡ ❉❡❧❛✉♥❛② ♦r r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥s ✐♥ R2 ❛♥❞ ❝♦♥✈❡① ❤✉❧❧s ✐♥ R

3

❬❆✉r✾✶❪✱ ✇❤✐❧❡ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠s ✐♥ R2 ❛r❡ r❡❧❛t❡❞ t♦ ✉♣♣❡r ❡♥✈❡❧♦♣❡s ♦❢

♣❧❛♥❡s ✐♥ R3✳

❯♣ t♦ ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ❛ s♣❤❡r❡ ✐♥ R3 ❝❛♥ ❜❡ ✉s❡❞ ❢♦r t❤❡

❧✐❢t✐♥❣ ✐♥st❡❛❞ ♦❢ t❤❡ ✉s✉❛❧ ♣❛r❛❜♦❧♦✐❞ ❬❇❡r✽✼❪✳ ■♥ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥✱ ❝♦♠✲♠♦♥ ✐♥ ❣❡♦♠❡tr②✱ t❤❡ s♣❤❡r❡ ❤❛s ❛ ♣♦❧❡✷ ❛♥❞ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ t♦ t❤❡ ❊✉✲❝❧✐❞❡❛♥ ♣❧❛♥❡ R

2✳ ❲❤❛t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤✐s ♣❛♣❡r ✐s t❤❡ s♣❛❝❡ ♦❢ ❝✐r❝❧❡s❞r❛✇♥ ♦♥ t❤❡ s♣❤❡r❡ S ✐ts❡❧❢✱ ✇✐t❤♦✉t ❛♥② ♣♦❧❡✳ ❚❤✐s s♣❛❝❡ ♦❢ ❝✐r❝❧❡s ❤❛s❛ ♥✐❝❡ r❡❧❛t✐♦♥ t♦ t❤❡ ❞❡ ❙✐tt❡r s♣❛❝❡ ✐♥ ▼✐♥❦♦✇s❦✐❛♥ ❣❡♦♠❡tr② ❬❈♦①✹✸❪✳

S

cp = πS(c)

O

PS(p)

c1

p1 = πS(c1)

p2 = πS(c2)

c2

❋✐❣✉r❡ ✸✿ c1 ✐s s✉❜♦rt❤♦❣♦♥❛❧ t♦c✱ c2 ✐s s✉♣❡r♦rt❤♦❣♦♥❛❧ t♦ c✳

❲❡ ❝❛♥ st✐❧❧ ❝♦♥str✉❝t t❤❡ ❝✐r❝❧❡ c ♦♥ S t❤❛t ✐s❛ss♦❝✐❛t❡❞ t♦ ❛ ♣♦✐♥t p = πS(c) ♦❢ R

3 ❛s t❤❡✐♥t❡rs❡❝t✐♦♥ ❜❡t✇❡❡♥ S ❛♥❞ t❤❡ ♣♦❧❛r ♣❧❛♥❡PS(p) ♦❢ p ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ q✉❛❞r✐❝ S ✭❋✐❣✲✉r❡ ✸✮✳ ■ts ❝❡♥t❡r ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ p ♦♥t♦S ❛♥❞ ✐ts sq✉❛r❡❞ r❛❞✐✉s ✐s ‖Op‖2 − R2 ✭❛s❛❜♦✈❡✱ ✐♠❛❣✐♥❛r② r❛❞✐✐ ❛r❡ ♣♦ss✐❜❧❡✮✳✸ ❙♦✱ ✐♥t❤❡ ❞❡t❡r♠✐♥❛♥t ✐♥ ✭✶✮✱ xpi

, ypi, ❛♥❞ zpi

✭r❡✲s♣❡❝t✐✈❡❧② xp′ , yp′ , zp′✮ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts pi = πS(ci) ✭r❡s♣❡❝✲t✐✈❡❧② p′ = πS(p)✮✳ ❚❤✐s ✇✐❧❧ ❜❡ ❡①t❡♥s✐✈❡❧②✉s❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳ ❆❣❛✐♥✱ ✇❡ r❡♠❛r❦ t❤❛t❉❡❧❛✉♥❛② ❛♥❞ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥s ♦♥ S r❡❧❛t❡ t♦ ❝♦♥✈❡① ❤✉❧❧s ✐♥ ✸❉✳

✷❙❡❡ t❤❡ ♥✐❝❡ tr❡❛t♠❡♥t ♦❢ ✐♥✜♥✐t② ✐♥ ❬❇❡r✽✼❪✳✸❘❡♠❡♠❜❡r t❤❛t S ✐s ❝❡♥t❡r❡❞ ❛t O ❛♥❞ ❤❛s sq✉❛r❡❞ r❛❞✐✉s R2✳

■◆❘■❆

Page 10: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✼

■♥t❡r❡st✐♥❣❧②✱ r❛t❤❡r t❤❛♥ ✉s✐♥❣ ❛ ❝♦♥✈❡① ❤✉❧❧ ❛❧❣♦r✐t❤♠ t♦ ♦❜t❛✐♥ t❤❡❉❡❧❛✉♥❛② ♦r r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s✉r❢❛❝❡ ❛s ✉s✉❛❧❧② ❞♦♥❡ ❢♦r R

2

❬❆✉r✾✶❪✱ ✇❡ ✇✐❧❧ ❞♦ t❤❡ ❝♦♥✈❡rs❡ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

✹ ❆❧❣♦r✐t❤♠

❚❤❡ ✐♥❝r❡♠❡♥t❛❧ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦♠♣✉t✐♥❣ ❛ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❝✐r❝❧❡s ♦♥t❤❡ s♣❤❡r❡ S ✐s ❛ ❞✐r❡❝t ❛❞❛♣t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐♥ R

2 ❬❇♦✇✽✶❪✳ ❆ss✉♠❡t❤❛t RT i−1 = RT ({cj ∈ S, j = 1, . . . , i − 1}) ❤❛s ❜❡❡♥ ❝♦♠♣✉t❡❞✳✹ ❚❤❡✐♥s❡rt✐♦♥ ♦❢ ci = (pi, r

2i ) ✇♦r❦s ❛s ❢♦❧❧♦✇s✿

• ❧♦❝❛t❡ pi ✭✐✳❡✳ ✜♥❞ t❤❡ tr✐❛♥❣❧❡ t ❝♦♥t❛✐♥✐♥❣ pi✮✱• ✐❢ t ✐s ❤✐❞✐♥❣ pi ✭✐✳❡✳ ✐❢ ci ❛♥❞ t❤❡ ♦rt❤♦❣♦♥❛❧ ❝✐r❝❧❡ ♦❢ t ❛r❡ s✉❜♦rt❤♦❣♦♥❛❧✮t❤❡♥ st♦♣❀ pi ✐s ♥♦t ❛ ✈❡rt❡① ♦❢ RT i✳ ◆♦t❡ t❤❛t t❤✐s ❝❛s❡ ♥❡✈❡r ♦❝❝✉rs ❢♦r❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥s✳• ❡❧s❡ (i) ✜♥❞ ❛❧❧ tr✐❛♥❣❧❡s ✇❤♦s❡ ♦rt❤♦❣♦♥❛❧ ❝✐r❝❧❡s ❛r❡ s✉♣❡r♦rt❤♦❣♦♥❛❧t♦ ci ❛♥❞ r❡♠♦✈❡ t❤❡♠❀ t❤✐s ❢♦r♠s ❛ ♣♦❧②❣♦♥❛❧ r❡❣✐♦♥ t❤❛t ✐s st❛r✲s❤❛♣❡❞✇✐t❤ r❡s♣❡❝t t♦ pi❀✺ (ii) tr✐❛♥❣✉❧❛t❡ t❤❡ ♣♦❧②❣♦♥❛❧ r❡❣✐♦♥ ❥✉st ❝r❡❛t❡❞ ❜②❝♦♥str✉❝t✐♥❣ t❤❡ tr✐❛♥❣❧❡s ❢♦r♠❡❞ ❜② t❤❡ ❜♦✉♥❞❛r② ❡❞❣❡s ♦❢ t❤❡ r❡❣✐♦♥ ❛♥❞t❤❡ ♣♦✐♥t pi✳❙t❡♣s (i) ❛♥❞ (ii) ❝❛♥ ❛❧t❡r♥❛t✐✈❡❧② ❜❡ ❛❝❝♦♠♣❧✐s❤❡❞ ❜② ✢✐♣♣✐♥❣ ❬❊❙✾✻❪✱ ✇❤✐❝❤✐s t❤❡ ✈❡rs✐♦♥ ❝✉rr❡♥t❧② ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❝❣❛❧ ♣❛❝❦❛❣❡ ❬❨✈✐✵✾❪✳

❚✇♦ ♠❛✐♥ ♣r❡❞✐❝❛t❡s ❛r❡ ✉s❡❞ ❜② t❤✐s ❛❧❣♦r✐t❤♠✿❚❤❡ ♦r✐❡♥t❛t✐♦♥ ♣r❡❞✐❝❛t❡ ❛❧❧♦✇s t♦ ❝❤❡❝❦ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤r❡❡ ♣♦✐♥tsp, q, r ♦♥ t❤❡ s♣❤❡r❡ ✭t❤✐s ♣r❡❞✐❝❛t❡ ✐s ✉s❡❞ ✐♥ ♣❛rt✐❝✉❧❛r t♦ ❧♦❝❛t❡ ♥❡✇ ♣♦✐♥ts✮✳■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝♦♠♣✉t✐♥❣ t❤❡ s✐❞❡ ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② O, p, q ♦♥ ✇❤✐❝❤r ✐s ❧②✐♥❣✱ ✐✳❡✳ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ O, p, q, r ✐♥ R

3✳❚❤❡ ♣♦✇❡r t❡st ✐♥tr♦❞✉❝❡❞ ✐♥ ❙❡❝t✐♦♥ ✷ ✭t❤✐s ♣r❡❞✐❝❛t❡ ✐s ✉s❡❞ t♦ ✐❞❡♥t✐❢② t❤❡tr✐❛♥❣❧❡s ✇❤♦s❡ ♦rt❤♦❣♦♥❛❧ ❝✐r❝❧❡s ❛r❡ s✉♣❡r♦rt❤♦❣♦♥❛❧ t♦ ❡❛❝❤ ♥❡✇ ❝✐r❝❧❡✮❜♦✐❧s ❞♦✇♥ t♦ ❛♥ ♦r✐❡♥t❛t✐♦♥ ♣r❡❞✐❝❛t❡ ✐♥ R

3✱ ❛s s❡❡♥ ✐♥ ❙❡❝t✐♦♥ ✸✳❚❤❡ t✇♦ ❛♣♣r♦❛❝❤❡s q✉✐❝❦❧② ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ❢❛❧❧ ✐♥t♦ t❤❡

❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ♦❢ ❝♦♠♣✉t✐♥❣ t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❝✐r❝❧❡s ♦♥ t❤❡s♣❤❡r❡✳ ❚❤❡ ♥❡①t t✇♦ s❡❝t✐♦♥s s❤♦✇ ♠♦r❡ ♣r❡❝✐s❡❧② ❤♦✇ t❤❡s❡ ♣r❡❞✐❝❛t❡s ❛r❡❡✈❛❧✉❛t❡❞ ✐♥ ❡❛❝❤ ❛♣♣r♦❛❝❤✳

✹✳✶ ❋✐rst ❛♣♣r♦❛❝❤✿ ❯s✐♥❣ ♣♦✐♥ts ♦♥ t❤❡ s♣❤❡r❡

■♥ t❤✐s ❛♣♣r♦❛❝❤✱ ✐♥♣✉t ♣♦✐♥ts ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❛r❡ ❝❤♦s❡♥ t♦ ❜❡ t❤❡ ♣r♦✲❥❡❝t✐♦♥s ♦♥ S ♦❢ t❤❡ r♦✉♥❞❡❞ ♣♦✐♥ts ♦❢ t❤❡ ❞❛t❛✲s❡t ✇✐t❤ r❛t✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡s✳

✹❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❝❡♥t❡r O ♦❢ S ❧✐❡s ✐♥ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢t❤❡ ❞❛t❛✲s❡t✳ ❚❤✐s ✐s ❧✐❦❡❧② t♦ ❜❡ t❤❡ ❝❛s❡ ✐♥ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✳ ❙♦✱ ✇❡ ❥✉st ✐♥✐t✐❛❧✐③❡t❤❡ tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❢♦✉r ❞✉♠♠② ♣♦✐♥ts t❤❛t ❝♦♥t❛✐♥ O ✐♥ t❤❡✐r ❝♦♥✈❡① ❤✉❧❧ ❛♥❞ ❝❛♥♦♣t✐♦♥❛❧❧② ❜❡ r❡♠♦✈❡❞ ✐♥ t❤❡ ❡♥❞✳

✺❆s ♣r❡✈✐♦✉s❧② ♥♦t❡❞ ❢♦r t❤❡ ❡❞❣❡s ♦❢ tr✐❛♥❣❧❡s✱ ❛❧❧ ✉s✉❛❧ t❡r♠s r❡❢❡rr✐♥❣ t♦ s❡❣♠❡♥ts❛r❡ tr❛♥s♣♦s❡❞ t♦ ❛r❝s ♦❢ ❣r❡❛t ❝✐r❝❧❡s ♦♥ t❤❡ s♣❤❡r❡✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 11: Robust and Efficient Delaunay triangulations of points on ...

✽ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

❚❤❡ t❤r❡❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛♥ ✐♥♣✉t ♣♦✐♥t ❛r❡ t❤✉s ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡rs ♦❢ ❞❡❣r❡❡t✇♦ ❧②✐♥❣ ✐♥ t❤❡ s❛♠❡ ❡①t❡♥s✐♦♥ ✜❡❧❞ ♦❢ t❤❡ r❛t✐♦♥❛❧s✳

■♥ t❤✐s ❛♣♣r♦❛❝❤ ✇❡✐❣❤ts✱ ♦r ❡q✉✐✈❛❧❡♥t❧② r❛❞✐✐ ✐❢ ❝✐r❝❧❡s✱ ❛r❡ ♥✉❧❧✳ ❚❤❡♣♦✇❡r t❡st ❝♦♥s✐sts ✐♥ t❤✐s ❝❛s❡ ✐♥ ❛♥s✇❡r✐♥❣ ✇❤❡t❤❡r ❛ ♣♦✐♥t s ❧✐❡s ✐♥s✐❞❡✱♦✉ts✐❞❡✱✻ ♦r ♦♥ t❤❡ ❝✐r❝❧❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ p, q, r ♦♥ t❤❡ s♣❤❡r❡✳ ❋♦❧❧♦✇✐♥❣❙❡❝t✐♦♥ ✸✱ t❤✐s ✐s ❣✐✈❡♥ ❜② t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ p, q, r ❛♥❞ s s✐♥❝❡ ♣♦✐♥ts ♦♥ t❤❡s♣❤❡r❡ ❛r❡ ♠❛♣♣❡❞ t♦ t❤❡♠s❡❧✈❡s ❜② πS✳

❚❤❡ ❞✐✣❝✉❧t② ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✐♥♣✉t ♣♦✐♥ts ❤❛✈❡ ❛❧❣❡❜r❛✐❝ ❝♦✲♦r❞✐♥❛t❡s✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t✇♦ ❞✐✛❡r❡♥t ✐♥♣✉t ♣♦✐♥ts ♦♥ t❤❡ s♣❤❡r❡ ❛r❡✐♥ ❣❡♥❡r❛❧ ❧②✐♥❣ ✐♥ ❞✐✛❡r❡♥t ❡①t❡♥s✐♦♥s✳ ❚❤❡♥ t❤❡ ✸❉ ♦r✐❡♥t❛t✐♦♥ ♣r❡❞✐❝❛t❡♦❢ p, q, r, s ❣✐✈❡♥ ❜② ✭✶✮ ✐s t❤❡ s✐❣♥ ♦❢ ❛♥ ❡①♣r❡ss✐♦♥ ❧②✐♥❣ ✐♥ ❛♥ ❛❧❣❡❜r❛✐❝❡①t❡♥s✐♦♥ ♦❢ ❞❡❣r❡❡ ✶✻ ♦✈❡r t❤❡ r❛t✐♦♥❛❧s✱ ♦❢ t❤❡ ❢♦r♠ a1

√α1 + a2

√α2 +

a3√

α3 + a4√

α4 ✇❤❡r❡ ❛❧❧ a✬s ❛♥❞ α✬s ❛r❡ r❛t✐♦♥❛❧✳ ❊✈❛❧✉❛t✐♥❣ t❤✐s s✐❣♥ ✐♥❛♥ ❡①❛❝t ✇❛② ❛❧❧♦✇s t♦ ❢♦❧❧♦✇ t❤❡ ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ❢r❛♠❡✇♦r❦ ❡♥s✉r✐♥❣ t❤❡r♦❜✉st♥❡ss ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳

❚❤♦✉❣❤ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡s ♦✛❡r ❡①❛❝t ♦♣❡r❛t✐♦♥s ♦♥ ❣❡♥❡r❛❧ ❛❧❣❡❜r❛✐❝♥✉♠❜❡rs ❬❝♦r✱ ❧❡❞❪✱ t❤❡② ❛r❡ ♠✉❝❤ s❧♦✇❡r t❤❛♥ ❝♦♠♣✉t✐♥❣ ✇✐t❤ r❛t✐♦♥❛❧ ♥✉♠✲❜❡rs✳ ❚❤❡ s✐❣♥ ♦❢ t❤❡ ❛❜♦✈❡ s✐♠♣❧❡ ❡①♣r❡ss✐♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿✕✶✕ ❡✈❛❧✉❛t❡ t❤❡ s✐❣♥s ♦❢ A1 = a1

√α1+a2

√α2 ❛♥❞ A2 = a3

√α3+a4

√α4✱ ❜②

❝♦♠♣❛r✐♥❣ ai√

αi ✇✐t❤ ai+1√

αi+1 ❢♦r i = 1, 3✱ ✇❤✐❝❤ r❡❞✉❝❡s ❛❢t❡r sq✉❛r✐♥❣t♦ ❝♦♠♣❛r✐♥❣ t✇♦ r❛t✐♦♥❛❧ ♥✉♠❜❡rs✱✕✷✕ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s ✐❢ A1 ❛♥❞ A2 ❤❛✈❡ ❞✐✛❡r❡♥t s✐❣♥s✱✕✸✕ ♦t❤❡r✇✐s❡✱ ❝♦♠♣❛r❡ A2

1 ✇✐t❤ A22✱ ✇❤✐❝❤ ✐s ❛♥ ❡❛s✐❡r ✐♥st❛♥❝❡ ♦❢ ✕✶✕✳

❚♦ s✉♠♠❛r✐③❡✱ t❤❡ ♣r❡❞✐❝❛t❡ ✐s ❣✐✈❡♥ ❜② t❤❡ s✐❣♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❡①♣r❡ss✐♦♥s ✐♥t❤❡ r❛t✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ r♦✉♥❞❡❞ ❞❛t❛ ♣♦✐♥ts✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞❡①❛❝t❧② ✉s✐♥❣ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ♦♥❧②✳

✹✳✷ ❙❡❝♦♥❞ ❛♣♣r♦❛❝❤✿ ❯s✐♥❣ ✇❡✐❣❤t❡❞ ♣♦✐♥ts

■♥ t❤✐s ❛♣♣r♦❛❝❤✱ t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ t❤❡ ✇❡✐❣❤t❡❞ ♣♦✐♥ts ✐s ❝♦♠♣✉t❡❞❛s ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❛♣♣r♦❛❝❤✱ ❜♦t❤ ♣r❡❞✐❝❛t❡s ✭♦r✐❡♥t❛t✐♦♥♦♥ t❤❡ s♣❤❡r❡ ❛♥❞ ♣♦✇❡r t❡st✮ r❡❞✉❝❡ t♦ ♦r✐❡♥t❛t✐♦♥ ♣r❡❞✐❝❛t❡s ♦♥ t❤❡ ❞❛t❛♣♦✐♥ts ✐♥ R

3✳ ◆♦t❡ t❤❛t ❙❡❝t✐♦♥ ✸ s❤♦✇❡❞ t❤❛t t❤❡ ✇❡✐❣❤t ‖Op‖2 − R2 ♦❢ ❛♣♦✐♥t p st❛②s ✐♠♣❧✐❝✐t✱ ✐t ♥❡❡❞ ♥♦t ❜❡ ❡①♣❧✐❝✐t❧② ❝♦♠♣✉t❡❞✳

❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✇❡✐❣❤ts✱ s♦♠❡ ♣♦✐♥ts ❝❛♥ ❜❡ ❤✐❞❞❡♥ ✐♥ ❛ r❡❣✉❧❛r tr✐❛♥✲❣✉❧❛t✐♦♥✳ ❲❡ ♣r♦✈❡ ♥♦✇ t❤❛t ✉♥❞❡r s♦♠❡ s❛♠♣❧✐♥❣ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ r♦✉♥❞❡❞❞❛t❛✱ t❤❡r❡ ✐s ❛❝t✉❛❧❧② ♥♦ ❤✐❞❞❡♥ ♣♦✐♥t✳

▲❡♠♠❛ ✹✳✶✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t ❛❧❧ ❞❛t❛ ♣♦✐♥ts ❧✐❡ ✇✐t❤✐♥ ❛ ❞✐st❛♥❝❡ δ ❢r♦♠

S✳ ■❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♣♦✐♥ts ✐s ❧❛r❣❡r t❤❛♥ 2√

Rδ✱ t❤❡♥ ♥♦ ♣♦✐♥t

✐s ❤✐❞❞❡♥✳

✻❖♥ S✱ t❤❡ ✐♥t❡r✐♦r ✭r❡s♣❡❝t✐✈❡❧② ❡①t❡r✐♦r✮ ♦❢ ❛ ❝✐r❝❧❡ c t❤❛t ✐s ♥♦t ❛ ❣r❡❛t ❝✐r❝❧❡ ♦❢ S

❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✐♥t❡r✐♦r ✭r❡s♣❡❝t✐✈❡❧② ❡①t❡r✐♦r✮ ♦❢ t❤❡ ❤❛❧❢✲❝♦♥❡ ✐♥ ✸❉ ✇❤♦s❡ ❛♣❡① ✐s t❤❡❝❡♥t❡r ♦❢ S ❛♥❞ t❤❛t ✐♥t❡rs❡❝ts S ❛❧♦♥❣ c✳

■◆❘■❆

Page 12: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✾

Pr♦♦❢✳ ❆ ♣♦✐♥t ✐s ❤✐❞❞❡♥ ✐✛ ✐t ✐s ❝♦♥t❛✐♥❡❞ ✐♥s✐❞❡ t❤❡ ✸❉ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡s❡t ♦❢ ❞❛t❛ ♣♦✐♥ts S✳ ▲❡t p ❜❡ ❛ ❞❛t❛ ♣♦✐♥t✱ ❛t ❞✐st❛♥❝❡ ρ ❢r♦♠ O✳ ❲❡❤❛✈❡ ρ ∈ [R − δ,R + δ]✳ ❉❡♥♦t❡ ❜② dp t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ p ❛♥❞t❤❡ ♦t❤❡r ♣♦✐♥ts✳ ■❢ dp >

(R + δ)2 − ρ2✱ t❤❡ s❡t B(O,R + δ) \ B(p, dp)✐s ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ❤❛❧❢✲s♣❛❝❡ H+ = {q : 〈q − p, O − p〉 > 0}✳ ❯♥❞❡r t❤❡s❡❝♦♥❞✐t✐♦♥s✱ ❛❧❧ ♦t❤❡r ♣♦✐♥ts ❜❡❧♦♥❣ t♦ H+ ❛♥❞ p ✐s ♥♦t ✐♥s✐❞❡ t❤❡ ❝♦♥✈❡① ❤✉❧❧♦❢ t❤❡ ♦t❤❡r ♣♦✐♥ts✳ ■t ❢♦❧❧♦✇s t❤❛t ✐❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ❞❛t❛♣♦✐♥ts ✐s ❧❛r❣❡r t❤❛♥ supρ

(R + δ)2 − ρ2 = 2√

Rδ✱ ♥♦ ♣♦✐♥t ✐s ❤✐❞❞❡♥✳

▲❡t ✉s ♥♦✇ ❛ss✉♠❡ ✇❡ ✉s❡ ❞♦✉❜❧❡ ♣r❡❝✐s✐♦♥ ✢♦❛t✐♥❣ ♣♦✐♥t ♥✉♠❜❡rs ❛s❞❡✜♥❡❞ ✐♥ t❤❡ ■❊❊❊ st❛♥❞❛r❞ ✼✺✹ ❬✐❡❡✵✽✱ ●♦❧✾✶❪✳ ❚❤❡ ♠❛♥t✐ss❛ ✐s ❡♥❝♦❞❡❞✉s✐♥❣ ✺✷ ❜✐ts✳ ▲❡t γ ❞❡♥♦t❡ t❤❡ ✇♦rst ❡rr♦r✱ ❢♦r ❡❛❝❤ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡✱❞♦♥❡ ✇❤✐❧❡ r♦✉♥❞✐♥❣ ❛ ♣♦✐♥t ♦♥ S t♦ t❤❡ ♥❡❛r❡st ♣♦✐♥t ✇❤♦s❡ ❝♦♦r❞✐♥❛t❡s❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❞♦✉❜❧❡ ♣r❡❝✐s✐♦♥ ✢♦❛t✐♥❣ ♣♦✐♥t ♥✉♠❜❡rs✳ ▲❡t ✉s ✉s❡t❤❡ st❛♥❞❛r❞ t❡r♠ ✉❧♣(x) ❞❡♥♦t✐♥❣ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✢♦❛t✐♥❣✲♣♦✐♥t♥✉♠❜❡rs ❝❧♦s❡st t♦ t❤❡ r❡❛❧ ✈❛❧✉❡ x ❬▼✉❧✵✺❪✳ ❆ss✉♠✐♥❣ ❛❣❛✐♥ t❤❛t t❤❡ ❝❡♥t❡r♦❢ S ✐s O✱ ♦♥❡ ❤❛s γ ≤ ✉❧♣(R) = 2−52+⌊log

2(R)⌋ ≤ 2−52R✳ ❚❤❡♥✱ δ ✐♥ t❤❡

♣r❡✈✐♦✉s ❧❡♠♠❛ ✐s s✉❝❤ t❤❛t δ ≤√

3/4γ < 2−52R✳ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ t❤❡❧❡♠♠❛✱ ♥♦ ♣♦✐♥t ✐s ❤✐❞❞❡♥ ✐♥ t❤❡ r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥ ❛s s♦♦♥ ❛s t❤❡ ❞✐st❛♥❝❡❜❡t✇❡❡♥ ❛♥② t✇♦ ♣♦✐♥ts ✐s ❣r❡❛t❡r t❤❛♥ 2−25R✱ ✇❤✐❝❤ ✐s ❤✐❣❤❧② ♣r♦❜❛❜❧❡ ✐♥♣r❛❝t✐❝❡✳

✺ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❛♥❞ ❊①♣❡r✐♠❡♥ts

❚❤❡ t✇♦ ♠❡t❤♦❞s ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹ ❤❛✈❡ ❜❡❡♥ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❈✰✰✱❜❛s❡❞ ♦♥ t❤❡ ❝❣❛❧ ♣❛❝❦❛❣❡ t❤❛t ❝♦♠♣✉t❡s tr✐❛♥❣✉❧❛t✐♦♥s ✐♥ R

2✳ ❚❤❡ ♣❛❝❦✲❛❣❡ ✐♥tr♦❞✉❝❡s ❛♥ ✐♥✜♥✐t❡ ✈❡rt❡① ✐♥ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥ t♦ ❝♦♠♣❛❝t✐❢② R

2✳ ❚❤✉st❤❡ ✉♥❞❡r❧②✐♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧ tr✐❛♥❣✉❧❛t✐♦♥ ✐s ❛ tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ t❤❡ t♦♣♦❧♦❣✲✐❝❛❧ s♣❤❡r❡✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ r❡✉s❡ t❤❡ ✇❤♦❧❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣❛rt ♦❢ ❝❣❛❧✷❉ tr✐❛♥❣✉❧❛t✐♦♥s ❬P❨✵✾❪ ✇✐t❤♦✉t ❛♥② ♠♦❞✐✜❝❛t✐♦♥✳ ❍♦✇❡✈❡r✱ t❤❡ ❣❡♦♠❡t✲r✐❝ ❡♠❜❡❞❞✐♥❣ ✐ts❡❧❢ ❬❨✈✐✵✾❪✱ ❜♦✉♥❞ t♦ R

2✱ ♠✉st ❜❡ ♠♦❞✐✜❡❞ ✐♥ ❛♥ ❡❛s②❜✉t ♥♦♥✲♥❡❣❧✐❣✐❜❧❡ ✇❛②✱ ❜② r❡♠♦✈✐♥❣ ❛♥② r❡❢❡r❡♥❝❡ t♦ t❤❡ ✐♥✜♥✐t❡ ✈❡rt❡①✳❆ s✐♠✐❧❛r ✇♦r❦ ✇❛s ❞♦♥❡ t♦ ❝♦♠♣✉t❡ tr✐❛♥❣✉❧❛t✐♦♥s ✐♥ t❤❡ ✸❉ ✢❛t t♦r✉s❬❈❚✵✾❜✱ ❈❚✵✾❛❪✱ r❡✉s✐♥❣ t❤❡ ❝❣❛❧ ✸❉ tr✐❛♥❣✉❧❛t✐♦♥ ♣❛❝❦❛❣❡ ❬P❚✵✾❛✱ P❚✵✾❜❪❛s ♠✉❝❤ ❛s ♣♦ss✐❜❧❡✳

❆❧s♦✱ t❤❡ ❣❡♥❡r✐❝✐t② ♦✛❡r❡❞ ✐♥ ❝❣❛❧ ❜② t❤❡ ♠❡❝❤❛♥✐s♠ ♦❢ tr❛✐ts ❝❧❛ss❡s✱t❤❛t ❡♥❝❛♣s✉❧❛t❡ t❤❡ ❣❡♦♠❡tr✐❝ ♣r❡❞✐❝❛t❡s ♥❡❡❞❡❞ ❜② t❤❡ ❛❧❣♦r✐t❤♠s✱ ❛❧❧♦✇s✉s t♦ ❡❛s✐❧② ✉s❡ ❡①❛❝t❧② t❤❡ s❛♠❡ ❛❧❣♦r✐t❤♠ ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t tr❛✐ts ❝❧❛ss❡s❢♦r ♦✉r t✇♦ ❛♣♣r♦❛❝❤❡s✳

❚♦ ❞✐s♣❧❛② t❤❡ tr✐❛♥❣✉❧❛t✐♦♥ ❛♥❞ ✐ts ❞✉❛❧✱ t❤❡ ❝♦❞❡ ✐s ✐♥t❡r❢❛❝❡❞ ✇✐t❤t❤❡ ❝❣❛❧ ✸❉ s♣❤❡r✐❝❛❧ ❦❡r♥❡❧ ❬❞❈❚✵✾✱ ❞❈❈▲❚✵✾❪✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s ♣r✐♠✐t✐✈❡s♦♥ ❝✐r❝✉❧❛r ❛r❝s ✐♥ ✸❉✳ ❚❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥s s❤♦✇♥ ❛r❡ t❤❡♣r♦❥❡❝t✐♦♥s ♦♥ t❤❡ s♣❤❡r❡ ♦❢ t❤❡ r♦✉♥❞❡❞ ❞❛t❛ ♣♦✐♥ts✳ ❚❤❡ ❝✐r❝✉❧❛r ❛r❝s ❛r❡❞r❛✇♥ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ s♣❤❡r❡ ✭s❡❡ ❋✐❣✉r❡s ✹✱ ✻ ❛♥❞ ✼✮✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 13: Robust and Efficient Delaunay triangulations of points on ...

✶✵ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

❋✐❣✉r❡ ✹✿ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ✭❧❡❢t✮ ❛♥❞ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ✭r✐❣❤t✮ ♦❢ t❤❡✾✵✸✶ ♣♦st✲♦✣❝❡s ✐♥ ❋r❛♥❝❡ ✭✐♥❝❧✉❞✐♥❣ ❖✈❡rs❡❛s ❉❡♣❛rt♠❡♥ts ❛♥❞ ❚❡rr✐t♦r✐❡s✮✳

❲❡ ❝♦♠♣❛r❡ t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ ♦✉r ♠❡t❤♦❞s ✇✐t❤ s❡✈❡r❛❧ ❛✈❛✐❧❛❜❧❡ s♦❢t✲✇❛r❡ ♣❛❝❦❛❣❡s✼✳ ❖♥ ❋✐❣✉r❡ ✺✱ ✇❡ ❝♦♥s✐❞❡r ❧❛r❣❡ s❡ts ♦❢ r❛♥❞♦♠ ❞❛t❛ ♣♦✐♥ts✽

✭✉♣ t♦ 223✮ ♦♥ t❤❡ s♣❤❡r❡✱ r♦✉♥❞❡❞ t♦ ❞♦✉❜❧❡ ❝♦♦r❞✐♥❛t❡s✳ ❋✐❣✉r❡ ✻ q✉✐❝❦❧②♠❡♥t✐♦♥s r✉♥♥✐♥❣ t✐♠❡s ♦♥ s♦♠❡ r❡❛❧✲❧✐❢❡ ❞❛t❛✳

●r❛♣❤ ✶st ♦❢ ❋✐❣✉r❡ ✺ s❤♦✇s t❤❡ r❡s✉❧ts ♦❢ ♦✉r ✜rst ❛♣♣r♦❛❝❤✳ ❲❡ ❝♦❞❡❞❛ tr❛✐ts ❝❧❛ss ✐♠♣❧❡♠❡♥t✐♥❣ t❤❡ ❡①❛❝t ♣r❡❞✐❝❛t❡s ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✶✱t♦❣❡t❤❡r ✇✐t❤ s❡♠✐✲st❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ✜❧t❡r✐♥❣ ❬▲P❨✵✺❪✳ ❚❤❡ ♥♦♥✲❧✐♥❡❛r✐t②♦❢ t❤❡ ❜❡❤❛✈✐♦r ✐s ❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t s❡♠✐✲st❛t✐❝ ✜❧t❡rs ❤❛r❞❧② ❡✈❡r ❢❛✐❧ ❢♦r❧❡ss t❤❛♥ 213 ♣♦✐♥ts✱ ❛♥❞ ❛❧♠♦st ❛❧✇❛②s ❢❛✐❧ ❢♦r ♠♦r❡ t❤❛♥ 218 ♣♦✐♥ts✳

●r❛♣❤ ✷♥❞ s❤♦✇s t❤❡ r❡s✉❧ts ♦❢ t❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤✳ ❖♥❡ ♦❢ t❤❡ ♣r❡✲❞❡✜♥❡❞ ❦❡r♥❡❧s✾ ♦❢ ❝❣❛❧ ♣r♦✈✐❞❡s ✉s ❞✐r❡❝t❧② ✇✐t❤ ❛♥ ❡①❛❝t ✐♠♣❧❡♠❡♥t❛t✐♦♥♦❢ t❤❡ ♣r❡❞✐❝❛t❡s✱ ✜❧t❡r❡❞ ❜♦t❤ s❡♠✐✲st❛t✐❝❛❧❧② ❛♥❞ ❞②♥❛♠✐❝❛❧❧②✳ ❲❡ ♦❜s❡r✈❡t❤❛t ♥♦ ♣♦✐♥t ✐s ❤✐❞❞❡♥ ✇✐t❤ s✉❝❤ ❞✐str✐❜✉t✐♦♥s✱ ❡✈❡♥ ✇❤❡♥ t❤❡ ❞❛t❛✲s❡t ✐s❧❛r❣❡✱ ✇❤✐❝❤ ❝♦♥✜r♠s ✐♥ ♣r❛❝t✐❝❡ t❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ ❙❡❝t✐♦♥ ✹✳✷✳

❚❤❡ ❝❣❛❧ ✸❉ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ✭❣r❛♣❤ ❉❚✸✮ ❬P❚✵✾❜❪ ❛❧s♦ ♣r♦✈✐❞❡st❤✐s ❝♦♥✈❡① ❤✉❧❧ ❛s ❛ ❜②✲♣r♦❞✉❝t✳ ❲❡ ✉s❡ t❤❡ s❛♠❡ ❝❣❛❧ ❦❡r♥❡❧ ❛♥❞ ✐♥s❡rt t❤❡❝❡♥t❡r ♦❢ t❤❡ s♣❤❡r❡ t♦ ❛✈♦✐❞ ♣❡♥❛❧✐③✐♥❣ t❤✐s ❝♦❞❡ ✇✐t❤ t♦♦ ♠❛♥② ❞❡❣❡♥❡r❛t❡❝❛s❡s t❤❛t ✇♦✉❧❞ ❛❧✇❛②s ❝❛✉s❡ ✜❧t❡rs t♦ ❢❛✐❧✳❋♦r t❤❡s❡ t❤r❡❡ ❛♣♣r♦❛❝❤❡s✱ ✸❉ s♣❛t✐❛❧ s♦rt✐♥❣ r❡❞✉❝❡s t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢t❤❡ ❧♦❝❛t✐♦♥ st❡♣ ♦❢ ♣♦✐♥t ✐♥s❡rt✐♦♥ ❬❉❡❧✵✾✱ ❇✉❝✵✾❪✳

✼❚✐♠❡s r❡♣♦rt❡❞ ❛r❡ ✐♥ s❡❝♦♥❞s✳ ❖❢ ❝♦✉rs❡✱ t❤❡ t✐♠❡ ♦❢ ✐♥♣✉t✴♦✉♣✉t ✐s ♥♦t ❝♦✉♥t❡❞✳❈♦♠♣✉t❛t✐♦♥s ♦♥ ❛♥ ▼❛❝ ❇♦♦❦ Pr♦ ✸✱✶ ▼❆❈ ❖❙ ❳ ✈❡rs✐♦♥ ✶✵✳✺✳✼✱ Pr♦❝❡ss♦r ✷✳✻ ●❍③■♥t❡❧ ❈♦r❡ ✷ ❉✉♦✱ ▼❡♠♦r② ✷ ●♦ ✻✻✼ ▼❍③ ❉❉❘✷ ❙❉❘❆▼✳ ❈♦♠♣✐❧❡r ❣✰✰ ✹✳✸✳✷ ✇✐t❤ ✲❖✸❛♥❞ ✲❉◆❉❊❇❯● ✢❛❣s✳ ❈♦♠♣✐❧❡r ❣✼✼ ✸✳✹✳✸ ✇✐t❤ ✲❖✸ ❢♦r ❋♦rtr❛♥✳

✽❣❡♥❡r❛t❡❞ ❜② ❈●❆▲✿✿❘❛♥❞♦♠❴♣♦✐♥ts❴♦♥❴s♣❤❡r❡❴✸✾♣r❡❝✐s❡❧② ❈●❆▲✿✿❊①❛❝t❴♣r❡❞✐❝❛t❡s❴✐♥❡①❛❝t❴❝♦♥str✉❝t✐♦♥s❴❦❡r♥❡❧

■◆❘■❆

Page 14: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✶✶

10 12 14 16 18 20 22 24

log #vertices

0.001

0.01

0.1

1

10

100

1000

seconds

QHULL

SUG

HULL

STRIPACK

DT3

2nd

1st

2

❋✐❣✉r❡ ✺✿ ❈♦♠♣❛r❛t✐✈❡ ❜❡♥❝❤♠❛r❦s✳ ❚❤❡ ♣r♦❣r❛♠s ✇❡r❡ ❛❜♦rt❡❞ ✇❤❡♥ t❤❡✐rr✉♥♥✐♥❣ t✐♠❡ ✇❛s ❛❜♦✈❡ ✶✵ ♠✐♥✉t❡s✳

◆♦ ❣r❛♣❤ ✐s s❤♦✇♥ ❢♦r t❤❡ ❝♦❞❡ ♦❢ ❋♦❣❡❧ ❛♥❞ ❙❡tt❡r ❬❋❙✱ ❋❙❍✵✽❪✿ ■t ✐s♣r❡❧✐♠✐♥❛r② ❛♥❞ ❤❛s ♥♦t ❜❡❡♥ ❢✉❧❧②✲♦♣t✐♠✐③❡❞ ②❡t❀ ✐ts r✉♥♥✐♥❣ t✐♠❡ ✐s ❝❧♦s❡ t♦✻✵✵ s❡❝♦♥❞s ❢♦r 212 ♣♦✐♥ts✳ ◆♦t❡ t❤❛t ✐t ✐s r❡str✐❝t❡❞ t♦ ♣♦✐♥ts ✇✐t❤ r❛t✐♦♥❛❧❝♦♦r❞✐♥❛t❡s ❧②✐♥❣ ♣r❡❝✐s❡❧② ♦♥ ❛ s♣❤❡r❡✳

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡s ❝♦♠♣✉t✐♥❣ ❛ ❝♦♥✈❡① ❤✉❧❧✐♥ ✸❉✱✶✵ ❢♦r ✇❤✐❝❤ t❤❡ ❞❛t❛ ♣♦✐♥ts ❛r❡ ✜rst r♦✉♥❞❡❞ t♦ ♣♦✐♥ts ✇✐t❤ ✐♥t❡❣❡r❝♦♦r❞✐♥❛t❡s✳ Pr❡❞✐❝❛t❡s ❛r❡ ❡✈❛❧✉❛t❡❞ ❡①❛❝t❧② ✉s✐♥❣ s✐♥❣❧❡ ♣r❡❝✐s✐♦♥ ❝♦♠♣✉✲t❛t✐♦♥s✳●r❛♣❤ ❍❯▲▲ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❝♦❞❡ ❬❤✉❧❪ ♦❢ ❈❧❛r❦s♦♥✱ ✇❤♦ ✉s❡s ❛ r❛♥❞♦♠✲✐③❡❞ ✐♥❝r❡♠❡♥t❛❧ ❝♦♥str✉❝t✐♦♥ ❬❈▼❙✾✸❪ ✇✐t❤ ❛♥ ❡①❛❝t ❛r✐t❤♠❡t✐❝ ♦♥ ✐♥t❡❣❡rs❬❈❧❛✾✷❪✳●r❛♣❤ ❙❯● s❤♦✇s t❤❡ r✉♥♥✐♥❣ t✐♠❡s ♦❢ ❙✉❣✐❤❛r❛✬s ❝♦❞❡ ✐♥ ❋♦rtr❛♥ ❬s✉❣✱❙✉❣✵✷❪✳

●r❛♣❤ ◗❍❯▲▲ s❤♦✇s t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❢❛♠♦✉s ◗❤✉❧❧ ♣❛❝❦❛❣❡ ♦❢❇❛r❜❡r ❡t ❛❧✳ ❬q❤✉❪ ✇❤❡♥ ❝♦♠♣✉t✐♥❣ t❤❡ ✸❉ ❝♦♥✈❡① ❤✉❧❧ ♦❢ t❤❡ ♣♦✐♥ts✳ ❚❤❡♦♣t✐♦♥ ✇❡ ✉s❡ ❤❛♥❞❧❡s r♦✉♥❞✲♦✛ ❡rr♦rs ❢r♦♠ ✢♦❛t✐♥❣ ♣♦✐♥t ❛r✐t❤♠❡t✐❝ ❜②♠❡r❣✐♥❣ ❢❛❝❡ts ♦❢ t❤❡ ❝♦♥✈❡① ❤✉❧❧ ✇❤❡♥ ♥❡❝❡ss❛r②✳

❘❡♥❦❛ ❝♦♠♣✉t❡s t❤❡ tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧❣♦r✐t❤♠ s✐♠✐❧❛r t♦ ♦✉r ✜rst❛♣♣r♦❛❝❤✱ ❜✉t ❤✐s s♦❢t✇❛r❡ ❙❚❘■P❆❈❑✱ ✐♥ ❋♦rtr❛♥✱ ✉s❡s ❛♣♣r♦①✐♠❛t❡ ❝♦♠✲♣✉t❛t✐♦♥s ✐♥ ❞♦✉❜❧❡ ❬❘❡♥✾✼❪✳ ❈♦♥s❡q✉❡♥t❧②✱ ✐t ♣❡r❢♦r♠s q✉✐t❡ ✇❡❧❧ ♦♥ r❛♥❞♦♠

✶✵❚❤❡ ♣❧♦t ❞♦❡s ♥♦t s❤♦✇ t❤❡ r❡s✉❧ts ♦❢ t❤❡ ❝❣❛❧ ✸❉ ❝♦♥✈❡① ❤✉❧❧ ♣❛❝❦❛❣❡ ❬❍❙✵✾❪✱ ✇❤♦s❡✐♠♣❧❡♠❡♥t❛t✐♦♥ ❞❛t❡s ❜❛❝❦ t♦ s❡✈❡r❛❧ ②❡❛rs✱ ❜❡❝❛✉s❡ ✐t ✇❛s ♠✉❝❤ s❧♦✇❡r t❤❛♥ ❛❧❧ ♦t❤❡r♠❡t❤♦❞s ✭r♦✉❣❤❧② ✺✵✵ t✐♠❡s s❧♦✇❡r t❤❛♥ ◗❤✉❧❧✮✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 15: Robust and Efficient Delaunay triangulations of points on ...

✶✷ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

♣♦✐♥ts ✭❜❡tt❡r t❤❛♥ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❢♦r s♠❛❧❧ r❛♥❞♦♠ ❞❛t❛✲s❡ts✮✱ ❜✉t ✐t❢❛✐❧s ♦♥ s♦♠❡ ❞❛t❛✲s❡ts ❧✐❦❡ t❤❡ s❡ts s❤♦✇♥ ✐♥ ❋✐❣✉r❡s ✻ ❛♥❞ ✼✳ ❖✉r ✐♠♣❧❡✲♠❡♥t❛t✐♦♥s ❤❛♥❞❧❡ ❡✈❡♥ ❤✐❣❤❧② ❞❡❣❡♥❡r❛t❡ ❝❛s❡s✳

❋✐❣✉r❡ ✻✿ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ✭t♦♣✮ ❛♥❞ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ✭❜♦tt♦♠✮ ♦❢✷✵✾✺✵ ✇❡❛t❤❡r st❛t✐♦♥s ❛❧❧ ❛r♦✉♥❞ t❤❡ ✇♦r❧❞✳ ❉❛t❛ ❛♥❞ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❝❛♥❜❡ ❢♦✉♥❞ ❛t ❤tt♣✿✴✴✇✇✇✳❧♦❝❛t✐♦♥✐❞❡♥t✐❢✐❡rs✳♦r❣✴✳ ❖✉r s❡❝♦♥❞ ❛♣♣r♦❛❝❤❝♦♠♣✉t❡s t❤❡ r❡s✉❧t ✐♥ ✵✳✶✹ s❡❝♦♥❞s✱ ✇❤✐❧❡ ◗❤✉❧❧ ♥❡❡❞s ✵✳✸✺ s❡❝♦♥❞s✱ ❛♥❞ t❤❡✜rst ❛♣♣r♦❛❝❤ ✵✳✺✼ s❡❝♦♥❞s✳ ❙❚❘■P❆❈❑ ❢❛✐❧s ♦♥ t❤✐s ❞❛t❛✲s❡t✳

❚♦ t❡st ❢♦r ❡①❛❝t♥❡ss ✇❡ ❞❡✈✐s❡❞ ❛ ♣♦✐♥t s❡t t❤❛t ✐s ❡s♣❡❝✐❛❧❧② ❤❛r❞ t♦ tr✐✲❛♥❣✉❧❛t❡ ❜❡❝❛✉s❡ ✐t ②✐❡❧❞s ♠❛♥② s❧✐✈❡r✲s❤❛♣❡❞ tr✐❛♥❣❧❡s ✐♥ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥✳❚❤✐s ♣♦✐♥t s❡t ✐s ❞❡✜♥❡❞ ❛s

Sn =

cos θ sin φsin θ sin φ

cos φ

θ ∈{

0,π

n, . . . ,

(n − 1)π

n, π

}

, φ =(θ2 + 1)

π2

100

,

010

,

001

,− 1√3

111

■♥ ❋✐❣✉r❡ ✼ ✇❡ s❤♦✇ t❤❡ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ S250 ❛♥❞ t❤❡ ❱♦r♦♥♦✐❞✐❛❣r❛♠ ♦❢ S100✳

❈♦♥❝❧✉s✐♦♥

❚❤❡ r❡s✉❧ts s❤♦✇ t❤❛t ♦✉r s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ②✐❡❧❞s ❜❡tt❡r t✐♠✐♥❣s t❤❛♥❛❧❧ t❤❡ ♦t❤❡r t❡st❡❞ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡s ❢♦r ❧❛r❣❡ ❞❛t❛✲s❡ts✱ ✇❤✐❧❡ ❜❡✐♥❣ ❢✉❧❧②r♦❜✉st✳ ◆♦t❡ t❤❛t ✐t ❝❛♥ ✐♥ ❢❛❝t ❜❡ ✉s❡❞ ❛s ✇❡❧❧ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦♥✈❡① ❤✉❧❧♦❢ ♣♦✐♥ts t❤❛t ❛r❡ ♥♦t ❝❧♦s❡ t♦ ❛ s♣❤❡r❡✿ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ s♣❤❡r❡ ❝❛♥ ❜❡

■◆❘■❆

Page 16: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✶✸

❋✐❣✉r❡ ✼✿ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ S250 ✭❧❡❢t✮✱ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ♦❢ S100

✭r✐❣❤t✮✳ ❙❚❘■P❆❈❑ ❢❛✐❧s ❢♦r ❡✳❣✳ n = 1, 500✳

❝❤♦s❡♥ ❛t ❛♥② ♣♦✐♥t ✐♥s✐❞❡ ❛ t❡tr❛❤❡❞r♦♥ ❢♦r♠❡❞ ❜② ❛♥② ❢♦✉r ♥♦♥✲❝♦♣❧❛♥❛r❞❛t❛ ♣♦✐♥ts✳

❚❤❡ ✜rst ❛♣♣r♦❛❝❤ ✐s s❧♦✇❡r ❜✉t st✐❧❧ ♦♥❡ ♦❢ t❤❡ ♠♦st s❝❛❧❛❜❧❡✳ ■t ✐st❤❡ ♦♥❧② ♦♥❡ t❤❛t ❡①❛❝t❧② ❝♦♠♣✉t❡s t❤❡ tr✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ❢♦r ✐♥♣✉t♣♦✐♥ts ✇✐t❤ ❛❧❣❡❜r❛✐❝ ❝♦♦r❞✐♥❛t❡s✱ ❛♥❞ t❤✉s ❡♥s✉r❡s t❤❛t ✐♥ ❛♥② ❝❛s❡ ❛❧❧ ♣♦✐♥ts✇✐❧❧ ❛♣♣❡❛r ✐♥ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥✳

❆❝❦♥♦✇❧❡❞❣♠❡♥ts

❲❡ ✇❛r♠❧② ❛❝❦♥♦✇❧❡❞❣❡ ❖♣❤✐r ❙❡tt❡r ❛♥❞ ❊✜ ❋♦❣❡❧ ✇❤♦ ❦✐♥❞❧② ✇♦r❦❡❞ ♦♥t❤❡✐r ❝♦❞❡ t♦ ❣✐✈❡ ✉s ❛❝❝❡ss t♦ ✐t✳ ❲❡ ✇✐s❤ t♦ t❤❛♥❦ ❏❡❛♥✲▼❛r❝ ❙❝❤❧❡♥❦❡r ❢♦r✈❡r② ✐♥t❡r❡st✐♥❣ ❞✐s❝✉ss✐♦♥s ♦♥ t❤❡ ❞❡ ❙✐tt❡r s♣❛❝❡✳

❘❡❢❡r❡♥❝❡s

❬❆✉r✽✼❪ ❋r❛♥③ ❆✉r❡♥❤❛♠♠❡r✳ P♦✇❡r ❞✐❛❣r❛♠s✿ ♣r♦♣❡rt✐❡s✱ ❛❧❣♦r✐t❤♠s❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t✐♥❣✱ ✶✻✿✼✽✕✾✻✱ ✶✾✽✼✳

❬❆✉r✾✶❪ ❋r❛♥③ ❆✉r❡♥❤❛♠♠❡r✳ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠s✿ ❆ s✉r✈❡② ♦❢ ❛ ❢✉♥✲❞❛♠❡♥t❛❧ ❣❡♦♠❡tr✐❝ ❞❛t❛ str✉❝t✉r❡✳ ❆❈▼ ❈♦♠♣✉t✐♥❣ ❙✉r✈❡②s✱✷✸✭✸✮✿✸✹✺✕✹✵✺✱ ❙❡♣t❡♠❜❡r ✶✾✾✶✳

❬❇❡r✽✼❪ ▼❛r❝❡❧ ❇❡r❣❡r✳ ❚❤❡ s♣❛❝❡ ♦❢ s♣❤❡r❡s✳ ■♥ ●❡♦♠❡tr② ✭✈♦❧s✳ ✶✲✷✮✱♣❛❣❡s ✸✹✾✕✸✻✶✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✶✾✽✼✳

❬❇❋❍+✵✾❛❪ ❊r✐❝ ❇❡r❜❡r✐❝❤✱ ❊✜ ❋♦❣❡❧✱ ❉❛♥ ❍❛❧♣❡r✐♥✱ ▼✐❝❤❛❡❧ ❑❡r❜❡r✱ ❛♥❞❖♣❤✐r ❙❡tt❡r✳ ❆rr❛♥❣❡♠❡♥ts ♦♥ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡s ■■✿ ❈♦♥✲❝r❡t✐③❛t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ✷✵✵✾✳ ❙✉❜♠✐tt❡❞✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 17: Robust and Efficient Delaunay triangulations of points on ...

✶✹ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

❬❇❋❍+✵✾❜❪ ❊r✐❝ ❇❡r❜❡r✐❝❤✱ ❊✜ ❋♦❣❡❧✱ ❉❛♥ ❍❛❧♣❡r✐♥✱ ❑✉rt ▼❡❤❧❤♦r♥✱ ❛♥❞❘♦♥ ❲❡✐♥✳ ❆rr❛♥❣❡♠❡♥ts ♦♥ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡s ■✿ ●❡♥❡r❛❧❢r❛♠❡✇♦r❦ ❛♥❞ ✐♥❢r❛str✉❝t✉r❡✱ ✷✵✵✾✳ ❙✉❜♠✐tt❡❞✳

❬❇♦✇✽✶❪ ❆✳ ❇♦✇②❡r✳ ❈♦♠♣✉t✐♥❣ ❉✐r✐❝❤❧❡t t❡ss❡❧❧❛t✐♦♥s✳ ❚❤❡ ❈♦♠♣✉t❡r❏♦✉r♥❛❧✱ ✷✹✭✷✮✿✶✻✷✕✶✻✻✱ ✶✾✽✶✳

❬❇r♦✽✵❪ ❑✳ ◗✳ ❇r♦✇♥✳ ●❡♦♠❡tr✐❝ tr❛♥s❢♦r♠s ❢♦r ❢❛st ❣❡♦♠❡tr✐❝ ❛❧❣♦✲

r✐t❤♠s✳ P❤✳❉✳ t❤❡s✐s✱ ❉❡♣t✳ ❈♦♠♣✉t✳ ❙❝✐✳✱ ❈❛r♥❡❣✐❡✲▼❡❧❧♦♥❯♥✐✈✳✱ P✐tts❜✉r❣❤✱ P❆✱ ✶✾✽✵✳ ❘❡♣♦rt ❈▼❯✲❈❙✲✽✵✲✶✵✶✳

❬❇✉❝✵✾❪ ❑❡✈✐♥ ❇✉❝❤✐♥✳ ❈♦♥str✉❝t✐♥❣ ❉❡❧❛✉♥❛② tr✐❛♥❣✉❧❛t✐♦♥s ❛❧♦♥❣s♣❛❝❡✲✜❧❧✐♥❣ ❝✉r✈❡s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ❊✉r♦♣❡❛♥ ❙②♠♣♦s✐✉♠ ♦♥

❆❧❣♦r✐t❤♠s✱ ✷✵✵✾✳ ❚♦ ❛♣♣❡❛r✳

❬❇❨✾✽❪ ❏❡❛♥✲❉❛♥✐❡❧ ❇♦✐ss♦♥♥❛t ❛♥❞ ▼❛r✐❡tt❡ ❨✈✐♥❡❝✳ ❆❧❣♦r✐t❤♠✐❝ ●❡✲♦♠❡tr②✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❯❑✱ ✶✾✾✽✳ ❚r❛♥s❧❛t❡❞ ❜②❍❡r✈é ❇rö♥♥✐♠❛♥♥✳

❬❝❣❛❪ ❈❣❛❧✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr② ❆❧❣♦r✐t❤♠s ▲✐❜r❛r②✳❤tt♣✿✴✴✇✇✇✳❝❣❛❧✳♦r❣✳

❬❈❧❛✾✷❪ ❑✳ ▲✳ ❈❧❛r❦s♦♥✳ ❙❛❢❡ ❛♥❞ ❡✛❡❝t✐✈❡ ❞❡t❡r♠✐♥❛♥t ❡✈❛❧✉❛t✐♦♥✳ ■♥Pr♦❝❡❡❞✐♥❣s ✸✸r❞ ❆♥♥✉❛❧ ■❊❊❊ ❙②♠♣♦s✐✉♠ ♦♥ ❋♦✉♥❞❛t✐♦♥s ♦❢

❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✽✼✕✸✾✺✱ ❖❝t♦❜❡r ✶✾✾✷✳

❬❈▼❙✾✸❪ ❑✳ ▲✳ ❈❧❛r❦s♦♥✱ ❑✳ ▼❡❤❧❤♦r♥✱ ❛♥❞ ❘✳ ❙❡✐❞❡❧✳ ❋♦✉r r❡s✉❧ts ♦♥r❛♥❞♦♠✐③❡❞ ✐♥❝r❡♠❡♥t❛❧ ❝♦♥str✉❝t✐♦♥s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠✲❡tr②✿ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✸✭✹✮✿✶✽✺✕✷✶✷✱ ✶✾✾✸✳

❬❝♦r❪ ❈♦r❡ ♥✉♠❜❡r ❧✐❜r❛r②✳❤tt♣✿✴✴❝s✳♥②✉✳❡❞✉✴❡①❛❝t✴❝♦r❡❴♣❛❣❡s✳

❬❈♦①✹✸❪ ❍✳ ❙✳ ▼✳ ❈♦①❡t❡r✳ ❆ ❣❡♦♠❡tr✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❢♦r ❞❡ ❙✐tt❡r✬s✇♦r❧❞✳ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥t❤❧②✱ ✺✵✿✷✶✼✕✷✷✽✱ ✶✾✹✸✳

❬❈❚✵✾❛❪ ▼❛♥✉❡❧ ❈❛r♦❧✐ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ✸❉ ♣❡r✐♦❞✐❝ tr✐❛♥❣✉✲❧❛t✐♦♥s✳ ■♥ ❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱ ❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞

❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧✳ ✸✳✺ ❡❞✐t✐♦♥✱ ✷✵✵✾✳ ❚♦ ❛♣♣❡❛r✳

❬❈❚✵✾❜❪ ▼❛♥✉❡❧ ❈❛r♦❧✐ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ❈♦♠♣✉t✐♥❣ ✸❉ ♣❡r✐♦❞✐❝tr✐❛♥❣✉❧❛t✐♦♥s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ✶✼t❤ ❊✉r♦♣❡❛♥ ❙②♠♣♦s✐✉♠ ♦♥

❆❧❣♦r✐t❤♠s✱ ✷✵✵✾✳ ❚♦ ❛♣♣❡❛r✳ ❋✉❧❧ ✈❡rs✐♦♥ ❛✈❛✐❧❛❜❧❡ ❛s ■◆❘■❆❘❡s❡r❝❤ ❘❡♣♦rt ◆♦ ✻✽✷✸✱ ❤tt♣✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴✐♥r✐❛✲✵✵✸✺✻✽✼✶✳

❬❞❇✈❑❖❙✵✵❪ ▼❛r❦ ❞❡ ❇❡r❣✱ ▼❛r❝ ✈❛♥ ❑r❡✈❡❧❞✱ ▼❛r❦ ❖✈❡r♠❛rs✱ ❛♥❞ ❖t✲❢r✐❡❞ ❙❝❤✇❛r③❦♦♣❢✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr②✿ ❆❧❣♦r✐t❤♠s ❛♥❞

❆♣♣❧✐❝❛t✐♦♥s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ●❡r♠❛♥②✱ ✷♥❞ ❡❞✐t✐♦♥✱✷✵✵✵✳

■◆❘■❆

Page 18: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✶✺

❬❞❈❈▲❚✵✾❪ P❡❞r♦ ▼✳ ▼✳ ❞❡ ❈❛str♦✱ ❋ré❞ér✐❝ ❈❛③❛❧s✱ ❙é❜❛st✐❡♥ ▲♦r✐♦t✱ ❛♥❞▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ❉❡s✐❣♥ ♦❢ t❤❡ ❈●❆▲ ✸❉ ❙♣❤❡r✐❝❛❧ ❑❡r♥❡❧❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❛rr❛♥❣❡♠❡♥ts ♦❢ ❝✐r❝❧❡s ♦♥ ❛ s♣❤❡r❡✳ ❈♦♠♣✉✲t❛t✐♦♥❛❧ ●❡♦♠❡tr② ✿ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✹✷✭✻✲✼✮✿✺✸✻✕✺✺✵✱✷✵✵✾✳

❬❞❈❚✵✾❪ P❡❞r♦ ▼✳ ▼✳ ❞❡ ❈❛str♦ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ✸❉ s♣❤❡r✐❝❛❧❣❡♦♠❡tr② ❦❡r♥❡❧✳ ■♥ ❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱ ❡❞✐t♦r✱ ❈●❆▲❯s❡r ❛♥❞ ❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧✳ ✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❬❉❡❧✵✾❪ ❈❤r✐st♦♣❤❡ ❉❡❧❛❣❡✳ ❙♣❛t✐❛❧ s♦rt✐♥❣✳ ■♥ ❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞ ❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧✳ ✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❬❉▼❚✾✷❪ ❖❧✐✈✐❡r ❉❡✈✐❧❧❡rs✱ ❙t❡❢❛♥ ▼❡✐s❡r✱ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ❚❤❡s♣❛❝❡ ♦❢ s♣❤❡r❡s✱ ❛ ❣❡♦♠❡tr✐❝ t♦♦❧ t♦ ✉♥✐❢② ❞✉❛❧✐t② r❡s✉❧ts♦♥ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ✹t❤ ❈❛♥❛❞✐❛♥ ❈♦♥✲

❢❡r❡♥❝❡ ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr②✱ ♣❛❣❡s ✷✻✸✕✷✻✽✱ ✶✾✾✷✳❋✉❧❧ ✈❡rs✐♦♥ ❛✈❛✐❧❛❜❧❡ ❛s ■◆❘■❆ ❘❡s❡❛r❝❤ ❘❡♣♦rt ◆♦ ✶✻✷✵✱❤tt♣✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴✐♥r✐❛✲✵✵✵✼✹✾✹✶✳

❬❊❙✾✻❪ ❍✳ ❊❞❡❧s❜r✉♥♥❡r ❛♥❞ ◆✳ ❘✳ ❙❤❛❤✳ ■♥❝r❡♠❡♥t❛❧ t♦♣♦❧♦❣✐❝❛❧ ✢✐♣✲♣✐♥❣ ✇♦r❦s ❢♦r r❡❣✉❧❛r tr✐❛♥❣✉❧❛t✐♦♥s✳ ❆❧❣♦r✐t❤♠✐❝❛✱ ✶✺✿✷✷✸✕✷✹✶✱ ✶✾✾✻✳

❬❋❙❪ ❊✜ ❋♦❣❡❧ ❛♥❞ ❖♣❤✐r ❙❡tt❡r✳ ❙♦❢t✇❛r❡ ❢♦r ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ♦♥❛ s♣❤❡r❡✳ P❡rs♦♥❛❧ ❝♦♠♠✉♥✐❝❛t✐♦♥✳

❬❋❙❍✵✽❪ ❊✜ ❋♦❣❡❧✱ ❖♣❤✐r ❙❡tt❡r✱ ❛♥❞ ❉❛♥ ❍❛❧♣❡r✐♥✳ ❊①❛❝t ✐♠♣❧❡♠❡♥✲t❛t✐♦♥ ♦❢ ❛rr❛♥❣❡♠❡♥ts ♦❢ ❣❡♦❞❡s✐❝ ❛r❝s ♦♥ t❤❡ s♣❤❡r❡ ✇✐t❤ ❛♣✲♣❧✐❝❛t✐♦♥s✳ ■♥ ❆❜str❛❝ts ♦❢ ✷✹t❤ ❊✉r♦♣❡❛♥ ❲♦r❦s❤♦♣ ♦♥ ❈♦♠✲

♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr②✱ ♣❛❣❡s ✽✸✕✽✻✱ ✷✵✵✽✳

❬❋❚✵✻❪ ❊✜ ❋♦❣❡❧ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ●❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❛♥❞t❤❡ ❈●❆▲ ❧✐❜r❛r②✳ ■♥ ❏❡❛♥✲❉❛♥✐❡❧ ❇♦✐ss♦♥♥❛t ❛♥❞ ▼♦♥✐q✉❡❚❡✐❧❧❛✉❞✱ ❡❞✐t♦rs✱ ❊✛❡❝t✐✈❡ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr② ❢♦r ❈✉r✈❡s❛♥❞ ❙✉r❢❛❝❡s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❱✐s✉❛❧✐③❛t✐♦♥✱✷✵✵✻✳

❬●♦❧✾✶❪ ❉✳ ●♦❧❞❜❡r❣✳ ❲❤❛t ❡✈❡r② ❝♦♠♣✉t❡r s❝✐❡♥t✐st s❤♦✉❧❞ ❦♥♦✇❛❜♦✉t ✢♦❛t✐♥❣✲♣♦✐♥t ❛r✐t❤♠❡t✐❝✳ ❆❈▼ ❈♦♠♣✉t✐♥❣ ❙✉r✈❡②s✱✷✸✭✶✮✿✺✕✹✽✱ ▼❛r❝❤ ✶✾✾✶✳

❬❍❙✵✾❪ ❙✉s❛♥ ❍❡rt ❛♥❞ ❙t❡❢❛♥ ❙❝❤✐rr❛✳ ✸❉ ❝♦♥✈❡① ❤✉❧❧s✳ ■♥ ❈●❆▲❊❞✐t♦r✐❛❧ ❇♦❛r❞✱ ❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞ ❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧✳✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❬❤✉❧❪ ❍✉❧❧✱ ❛ ♣r♦❣r❛♠ ❢♦r ❝♦♥✈❡① ❤✉❧❧s✳❤tt♣✿✴✴✇✇✇✳♥❡t❧✐❜✳♦r❣✴✈♦r♦♥♦✐✴❤✉❧❧✳❤t♠❧✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 19: Robust and Efficient Delaunay triangulations of points on ...

✶✻ ❈❛r♦❧✐ ✫ ▼❛❝❤❛❞♦ ✫ ▲♦r✐♦t ✫ ❲♦r♠s❡r ✫ ❚❡✐❧❧❛✉❞

❬✐❡❡✵✽❪ ■❊❊❊ st❛♥❞❛r❞ ❢♦r ✢♦❛t✐♥❣✲♣♦✐♥t ❛r✐t❤♠❡t✐❝✳ ■❊❊❊ ❙t❞ ✼✺✹✲

✷✵✵✽✱ ♣❛❣❡s ✶✕✺✽✱ ❆✉❣✉st ✷✵✵✽✳

❬❑▼P+✵✽❪ ▲✉t③ ❑❡tt♥❡r✱ ❑✉rt ▼❡❤❧❤♦r♥✱ ❙②❧✈❛✐♥ P✐♦♥✱ ❙t❡❢❛♥ ❙❝❤✐rr❛✱❛♥❞ ❈❤❡❡ ❨❛♣✳ ❈❧❛ssr♦♦♠ ❡①❛♠♣❧❡s ♦❢ r♦❜✉st♥❡ss ♣r♦❜❧❡♠s✐♥ ❣❡♦♠❡tr✐❝ ❝♦♠♣✉t❛t✐♦♥s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr②✿ ❚❤❡♦r②❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✹✵✿✻✶✕✼✽✱ ✷✵✵✽✳

❬▲❛✇✼✼❪ ❈✳ ▲✳ ▲❛✇s♦♥✳ ❙♦❢t✇❛r❡ ❢♦r C1 s✉r❢❛❝❡ ✐♥t❡r♣♦❧❛t✐♦♥✳ ■♥ ❏✳ ❘✳❘✐❝❡✱ ❡❞✐t♦r✱ ▼❛t❤✳ ❙♦❢t✇❛r❡ ■■■✱ ♣❛❣❡s ✶✻✶✕✶✾✹✳ ❆❝❛❞❡♠✐❝Pr❡ss✱ ◆❡✇ ❨♦r❦✱ ◆❨✱ ✶✾✼✼✳

❬❧❡❞❪ ▲❡❞❛✱ ▲✐❜r❛r② ❢♦r ❡✣❝✐❡♥t ❞❛t❛ t②♣❡s ❛♥❞ ❛❧❣♦r✐t❤♠s✳❤tt♣✿✴✴✇✇✇✳❛❧❣♦r✐t❤♠✐❝✲s♦❧✉t✐♦♥s✳❝♦♠✴❡♥❧❡❞❛✳❤t♠✳

❬▲P❨✵✺❪ ❈✳ ▲✐✱ ❙✳ P✐♦♥✱ ❛♥❞ ❈✳ ❑✳ ❨❛♣✳ ❘❡❝❡♥t ♣r♦❣r❡ss ✐♥ ❡①❛❝t ❣❡♦♠❡t✲r✐❝ ❝♦♠♣✉t❛t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝ ❛♥❞ ❆❧❣❡❜r❛✐❝ Pr♦❣r❛♠♠✐♥❣✱✻✹✭✶✮✿✽✺✕✶✶✶✱ ✷✵✵✺✳

❬▼✉❧✵✺❪ ❏❡❛♥✲▼✐❝❤❡❧ ▼✉❧❧❡r✳ ❖♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✉❧♣(x)✳ ❘❡s❡❛r❝❤❘❡♣♦rt ✺✺✵✹✱ ■◆❘■❆✱ ❋❡❜r✉❛r② ✷✵✵✺✳ ❤tt♣✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴✐♥r✐❛✲✵✵✵✼✵✺✵✸✴✳

❬◆▲❈✵✷❪ ❍②❡♦♥✲❙✉❦ ◆❛✱ ❈❤✉♥❣✲◆✐♠ ▲❡❡✱ ❛♥❞ ❖t❢r✐❡❞ ❈❤❡♦♥❣✳ ❱♦r♦♥♦✐❞✐❛❣r❛♠s ♦♥ t❤❡ s♣❤❡r❡✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ●❡♦♠❡tr②✿ ❚❤❡♦r② ❛♥❞❆♣♣❧✐❝❛t✐♦♥s✱ ✷✸✿✶✽✸✕✶✾✹✱ ✷✵✵✷✳

❬P❚✵✾❛❪ ❙②❧✈❛✐♥ P✐♦♥ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ✸❉ tr✐❛♥❣✉❧❛t✐♦♥ ❞❛t❛str✉❝t✉r❡✳ ■♥ ❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱ ❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞

❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧✳ ✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❬P❚✵✾❜❪ ❙②❧✈❛✐♥ P✐♦♥ ❛♥❞ ▼♦♥✐q✉❡ ❚❡✐❧❧❛✉❞✳ ✸❉ tr✐❛♥❣✉❧❛t✐♦♥s✳ ■♥❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱ ❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞ ❘❡❢❡r❡♥❝❡

▼❛♥✉❛❧✳ ✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❬P❨✵✾❪ ❙②❧✈❛✐♥ P✐♦♥ ❛♥❞ ▼❛r✐❡tt❡ ❨✈✐♥❡❝✳ ✷❉ tr✐❛♥❣✉❧❛t✐♦♥ ❞❛t❛ str✉❝✲t✉r❡✳ ■♥ ❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱ ❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞ ❘❡❢✲

❡r❡♥❝❡ ▼❛♥✉❛❧✳ ✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❬q❤✉❪ ◗❤✉❧❧✳ ❤tt♣✿✴✴✇✇✇✳q❤✉❧❧✳♦r❣✴✳

❬❘❡♥✾✼❪ ❘♦❜❡rt ❏✳ ❘❡♥❦❛✳ ❆❧❣♦r✐t❤♠ ✼✼✷✿ ❙❚❘■P❆❈❑✿ ❉❡❧❛✉✲♥❛② tr✐❛♥❣✉❧❛t✐♦♥ ❛♥❞ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ♦♥ t❤❡ s✉r✲❢❛❝❡ ♦❢ ❛ s♣❤❡r❡✳ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ▼❛t❤❡♠❛t✐✲

❝❛❧ ❙♦❢t✇❛r❡✱ ✷✸✭✸✮✿✹✶✻✕✹✸✹✱ ✶✾✾✼✳ ❙♦❢t✇❛r❡ ❛✈❛✐❧❛❜❧❡ ❛t❤tt♣✿✴✴♦r✐♦♥✳♠❛t❤✳✐❛st❛t❡✳❡❞✉✴❜✉r❦❛r❞t✴❢❴sr❝✴str✐♣❛❝❦✴str✐♣❛❝❦✳❤t♠❧✳

❬s✉❣❪ ❚❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥✈❡① ❤✉❧❧s✳❤tt♣✿✴✴✇✇✇✳s✐♠♣❧❡①✳t✳✉✲t♦❦②♦✳❛❝✳❥♣✴∼s✉❣✐❤❛r❛✴♦♣❡♥s♦❢t✴♦♣❡♥s♦❢t❡✳❤t♠❧✳

■◆❘■❆

Page 20: Robust and Efficient Delaunay triangulations of points on ...

❚r✐❛♥❣✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡ ✶✼

❬❙✉❣✵✷❪ ❑♦❦✐❝❤✐ ❙✉❣✐❤❛r❛✳ ▲❛❣✉❡rr❡ ❱♦r♦♥♦✐ ❞✐❛❣r❛♠ ♦♥ t❤❡ s♣❤❡r❡✳❏♦✉r♥❛❧ ❢♦r ●❡♦♠❡tr② ❛♥❞ ●r❛♣❤✐❝s✱ ✻✭✶✮✿✻✾✕✽✶✱ ✷✵✵✷✳

❬❨❉✾✺❪ ❈✳ ❑✳ ❨❛♣ ❛♥❞ ❚✳ ❉✉❜é✳ ❚❤❡ ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥ ♣❛r❛❞✐❣♠✳ ■♥❉✳✲❩✳ ❉✉ ❛♥❞ ❋✳ ❑✳ ❍✇❛♥❣✱ ❡❞✐t♦rs✱ ❈♦♠♣✉t✐♥❣ ✐♥ ❊✉❝❧✐❞❡❛♥

●❡♦♠❡tr②✱ ✈♦❧✉♠❡ ✹ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ❙❡r✐❡s ♦♥ ❈♦♠♣✉t✐♥❣✱♣❛❣❡s ✹✺✷✕✹✾✷✳ ❲♦r❧❞ ❙❝✐❡♥t✐✜❝✱ ❙✐♥❣❛♣♦r❡✱ ✷♥❞ ❡❞✐t✐♦♥✱ ✶✾✾✺✳

❬❨✈✐✵✾❪ ▼❛r✐❡tt❡ ❨✈✐♥❡❝✳ ✷❉ tr✐❛♥❣✉❧❛t✐♦♥s✳ ■♥ ❈●❆▲ ❊❞✐t♦r✐❛❧ ❇♦❛r❞✱❡❞✐t♦r✱ ❈●❆▲ ❯s❡r ❛♥❞ ❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧✳ ✸✳✹ ❡❞✐t✐♦♥✱ ✷✵✵✾✳

❘❘ ♥➦ ✵✶✷✸✹✺✻✼✽✾

Page 21: Robust and Efficient Delaunay triangulations of points on ...

Centre de recherche INRIA Sophia Antipolis – Méditerranée2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence CedexCentre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier

Centre de recherche INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, avenue Halley - 59650 Villeneuve d’AscqCentre de recherche INRIA Nancy – Grand Est : LORIA, Technopôle de Nancy-Brabois - Campus scientifique

615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy CedexCentre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay CedexCentre de recherche INRIA Rennes – Bretagne Atlantique : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex

Centre de recherche INRIA Saclay – Île-de-France : Parc Orsay Université - ZAC des Vignes : 4, rue Jacques Monod - 91893 Orsay Cedex

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

❤tt♣✿✴✴✇✇✇✳✐♥r✐❛✳❢r

ISSN 0249-6399