Planar convex hulls (I)
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- Merry Mitchell
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1 Covx Hu Covxty Gv st P o ots 2D, tr ovx u s t sst ovx oyo tt ots ots o P A oyo P s ovx or y, P, t st s try P. Pr ovx us (I) Coutto Gotry [s 3250] Lur To Bowo Co ovx o-ovx Covx Hu Covx Hu Covx Hu Gv st P o ots 2D, tr ovx u s t sst ovx oyo tt ots ots o P T ro: Gv st P o ots 2D, sr ort to out tr ovx u O o t rst ros stu CG My soutos s, t, tutv, xos tus Lots o tos roots t rtto ros s roto srto ros outut: st o ots o t CH ( oury orr) 4 5 6
2 Atos Pt : (sortst) oso-r t ro strt to Atos Pt : (sortst) oso-r t ro strt to S yss, t, roto roxt ots y tr CH Atos ost ost strt strt It sow tt t t oows CH(ost) sortst t s to t s t sortr o t ur t owr t Prtto ros Atos os tr xst srt two ots? Prtto ros Atos os tr xst srt two ots? Atos F t two ots P tt r rtst wy YES NO
![Atos Out Covxty: r vw F t two ots P tt r rtst wy Prorts o CH St = st o ots o t or 1+2, wt 1,2 [0,1], 1+2=1 Aorts or out t CH (P) Brut-or 3 Gt wr (or: Jrvz r) Quu Gr s Arw s ooto 1 + (1-1) 1 2](/docs-images/87/97106825/images/3-0.jpg "11+22+(1-1-2)3 Irt Dv--our C w o ttr?")
3 Atos Out Covxty: r vw F t two ots P tt r rtst wy Prorts o CH St = st o ots o t or 1+2, wt 1,2 [0,1], 1+2=1 Aorts or out t CH (P) Brut-or 3 Gt wr (or: Jrvz r) Quu Gr s Arw s ooto 1 + (1-1) (1-1-2)3 Irt Dv--our C w o ttr? Lowr ou A ovx oto o ots 1, 2, s ot o t or , wt [0,1], =1 Ex: tr ossts o ovx otos o ts 3 vrts Wt ts otto, t ovx u CH(P) = ovx otos o ots P Extr ots A ot s xtr tr xsts trou, su tt t otr ots o P r o t s s o (or o ) Extr ots A ot s xtr tr xsts trou, su tt t otr ots o P r o t s s o ( ot o ) xtr Covx Hu Prorts
4 Extr ots A ot s xtr tr xsts trou, su tt t otr ots o P r o t s s o ( ot o ) Extr ots A ot s xtr tr xsts trou, su tt t otr ots o P r o t s s o ( ot o ) Extr ots A ot s xtr tr xsts trou, su tt t otr ots o P r o t s s o ( ot o ) xtr xtr xtr rt ss rt ss: osr o CH or ot Extr ots C: I ot s o t CH oy () t s xtr. Extr ots C: I ot s o t CH oy () t s xtr. CH Vrts Svr tys o ovx u outut r ov ots o t ovx u rtrry orr ots o t ovx u oury orr oy o-or ots rtrry orr oy o-or ots oury orr < xu or ots It y s tt out oury orr s rr w s tt ty t xtr ots s O( ) so sort s ot ot
5 Itror ots A ot s ot o t CH oy s ot t tror o tr or y tr otr ots o P (or tror o st or y two ots). Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) xtr ot xtr Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) C: A r o ots (, ) or o t CH (, ) s xtr. Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) C: A r o ots (, ) or o t CH (, ) s xtr. Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) C: A r o ots (, ) or o t CH (, ) s xtr. xtr xtr xtr
6 Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) C: A r o ots (, ) or o t CH (, ) s xtr. Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) C: A r o ots (, ) or o t CH (, ) s xtr. Extr s A (, ) s xtr t otr ots o P r o o s o t (or o) C: A r o ots (, ) or o t CH (, ) s xtr. xtr xtr xtr Brut or Gt wr (1970) CH y xtr s Aort (ut P) or stt rs (, ) (,) s xtr r Osrvtos CH ossts o xtr s srs vrtx wt xt I: us to t xt o Ayss? How to xtr to strt ro? Gv xtr, ow to t xt o?
7 Gt wr (1970) Gt wr (1970) Gt wr (1970) r C ot wt u x-oort s xtr ot wt xu x-oort s xtr ot wt u y-oort s xtr ot wt xu y-oort s xtr Proo C you t o so ots tt r urt to CH? Gt wr (1970) Gt wr (1970) Gt wr (1970) Strt ro ot wt sst x-oort or t o, rt ost Strt ro ot wt sst x-oort or t o, rt ost // rst. HOW? Strt ro ot wt sst x-oort or t o, rt ost // rst. HOW?
8 Gt wr (1970) Gt wr (1970) Gt wr (1970) Strt ro ot wt sst x-oort or t o, rt ost // rst. HOW? Strt ro ot wt sst x-oort or t o, rt ost // rst or ot (!= ) out w- o wrt Strt ro ot wt sst x-oort or t o, rt ost // rst or ot (!= ) out w- o wrt t = ot wt sst outut (, ) s rst t = ot wt sst outut (, ) s rst //wt xt? rt ro u so Gt wr (1970) Gt wr (1970) Gt wr (1970) 0 = ot wt sst x-oort ( or t o, rt ost) 0 = ot wt sst x-oort ( or t o, rt ost) 0 = ot wt sst x-oort ( or t o, rt ost) = 0 = 0 = 0 rt or ot (!= ) rt or ot (!= ) rt or ot (!= ) out w- o wrt out w- o wrt out w- o wrt t = ot wt sst outut (, ) s CH t = ot wt sst outut (, ) s CH t = ot wt sst outut (, ) s CH = = = ut = 0 //ut t sovrs rst ot ut = 0 //ut t sovrs rst ot 0 ut = 0 //ut t sovrs rst ot
9 Gt wr (1970) Gt wr (1970) Gt wr (1970) 0 = ot wt sst x-oort ( or t o, rt ost) = 0 //t t ot rt or ot (!= ) out w- o wrt 0 = ot wt sst x-oort ( or t o, rt ost) = 0 rt or ot (!= ) out w- o wrt Dos t ort wor o rt ss? Gt wr rus O() t, wr s t sz o t CH(P) t = ot wt sst outut (, ) s CH t = ot wt sst outut (, ) s CH How s/r or st o ots? = ut = 0 //ut t sovrs rst ot 0 = ut = 0 0 Sow xs tt trr st/worst ss Dsuss w t-wr s oo o Ayss Ayss or vrtx o t CH, t ts O() ovr O(), wr s t sz o t CH(P) Sury ur u Gt wr Rus O() t, wr s t sz o t CH(P) ts s t s s or = O(), t wr ts O( 2 ) rt ost ot Fstr orts r ow Gt wr xts sy to 3D or y yrs ws t rry ort or 3D Quu t ost ot owr u
10 Quu (t 1970s) Quu (t 1970s) Quu (t 1970s) Sr to Qusort ( so wy) I: strt wt 2 xtr ots Quu (t 1970s) Quu (t 1970s) Quu (t 1970s) CH ossts o ur u (CH o P1) owr u (CH o P2) CH ossts o ur u (CH o P1) owr u (CH o P2) CH ossts o ur u (CH o P1) owr u (CH o P2) P1 P1 P1 P2 P2 P
11 Quu (t 1970s) Quu (t 1970s) Quu (t 1970s) W t CH o P1 CH o P2 srty Frst t s ous o P1 For ots P1: out st(, ) P1 P1 P Quu (t 1970s) Quu (t 1970s) Quu (t 1970s) F t ot wt rst st (.. urtst wy ro ) C: ust xtr ot ( tus o t CH o P1) C: ust xtr ot ( tus o t CH) Proo:? Proo:? t s or or ots or ow
12 Quu (t 1970s) Quu (t 1970s) Quu (t 1970s) Dsr ots s tr Dsr ots s tr Rurs o t ots t o rt o Quu (t 1970s) Quu (t 1970s) Quu (t 1970s) Cout CH o P2 sry Quu (P), rtto P to P1, P2 rtur Quu(,, P1) + Quu(,,P2) Quu(,,P) //vrt: P s st o ots o t t o P ty => rtur tyst or ot P: out ts st to t = ot wt x st t P1 = ots to t t o t P2 = ots to t t o rtur Quu(,,P1) + + Quu(,,P2)
13 Quu (t 1970s) Gr s (t 1960s) I t 60s to t B Ls rur t u o 10,000 ots, or w urt ort ws too sow Gr vo s s ort w rus O( ) o sort us r ss Ayss: Gr s Wrt rurr rto or ts ru t Wt s t st s ru t? Sow x Wt s t worst s ru t? Sow x Mor xrss: Aru tt Quu s vr oxty s O() o ots tt r uory strut Gr s (t 1960s) Gr s (t 1960s) I: strt ro ot tror to t u ots o t oury r r orr rou
14 Gr s (t 1960s) Gr s (t 1960s) Gr s (t 1960s) I: strt ro ot tror to t u I: strt ro ot tror to t u I: strt ro ot tror to t u orr ots y tr w wrt orr ots y tr w wrt orr ots y tr w wrt = t (.y -.y)/(.x -.x) Gr s (t 1960s) Gr s (t 1960s) Gr s (t 1960s) I: trvrs t ots ts orr,,,,,,, I: trvrs t ots ts orr,,,,,,, I: trvrs t ots ts orr,,,,,,, ty w ut, S ty w ut, S S = (, ) Ivrt: w t S s t CH o t ots trvrs so r S = (, ) Ivrt: w t S s t CH o t ots trvrs so r
15 Gr s (t 1960s) Gr s (t 1960s) Gr s (t 1960s) Now w r ot : wt o w o wt t? Now w r ot : wt o w o wt t? Now w r ot : wt o w o wt t? Is (,, ) ovx? Is (,, ) ovx? YES! S = (, ) Ivrt: w t S s t CH o t ots trvrs so r S = (, ) Ivrt: w t S s t CH o t ots trvrs so r S = (, ) Ivrt: w t S s t CH o t ots trvrs so r Gr s (t 1960s) Gr s s t o? Gr s (t 1960s) Now w r ot : (+S) stys ovx: to S Now w r ot : (+S) stys ovx : to S Now w r ot : Is (,, ) ovx? YES! Is (,, ) ovx? YES! S = (,, ) Ivrt: w t S s t CH o t ots trvrs so r S = (,, ) Ivrt: w t S s t CH o t ots trvrs so r S = (,, ) Ivrt: w t S s t CH o t ots trvrs so r
16 Gr s (t 1960s) Gr s (t 1960s) Gr s (t 1960s) Now w r ot : s t o? NO Now w r ot : s t o? NO Now w r ot : s t o? NO // t, us (,,,) ot ovx S = (,, ) Ivrt: w t S s t CH o t ots trvrs so r Ivrt: w t S s t CH o t ots trvrs so r S = (,,, ) S = (,, ) Ivrt: w t S s t CH o t ots trvrs so r Gr s (t 1960s) Gr s (t 1960s) Gr s (t 1960s) Now w r ot : s t o? NO Now w r ot : s t o? NO Now w r ot : s t o? NO o ; s t o? o ; s t o? o ; s t o? YES ==> srt S S = (, ) Ivrt: w t S s t CH o t ots trvrs so r S = (, ) Ivrt: w t S s t CH o t ots trvrs so r S = (,, ) Ivrt: w t S s t CH o t ots trvrs so r
17 Gr s (t 1960s) Gr s (t 1960s) Gr s (t 1960s) I r I r.. Cs os W r xt ot : W r xt ot : t = (S), = xt() t = (S), = xt() s t o : to S s rt o : o ; rt ut s t o, t to S us o us S = (,,.) S = (,,,.) S = (,,.) S = (,,.) Gr s (t 1960s) Gr s (t 1960s) Gr s: ANALYSIS F tror ot 0 Sort otr ots w rou 0, t 1, 2, 3, -1 ts orr Itz st S = (2, 1) or =3 to -1 o s t o (so(s),rst(s)): s us o S o o S w s rt o (so(s), rst(s)) us o S Coos st o trst ots o trou t ort Drt ss: os t ort rt ss? I ot, ow o you x t? How to tror ot? Ayss: How o os t t? F tror ot 0 Sort otr ots w rou 0.. Itz st S = (2, 1) or =3 to -1 o s t o (so(s),rst(s)): us o S s o o S w s rt o (so(s), rst(s)) us o S
18 Gr s: ANALYSIS Gr s: ANALYSIS Gr s: ANALYSIS F tror ot 0 O() (w t o t tr) F tror ot 0 O() (w t o t tr) F tror ot 0 O() (w t o t tr) Sort otr ots w rou 0.. Sort otr ots w rou 0.. O( ) Sort otr ots w rou 0.. O( ) Itz st S = (2, 1) Itz st S = (2, 1) Itz st S = (2, 1) or =3 to -1 o or =3 to -1 o or =3 to -1 o s t o (so(s),rst(s)): s t o (so(s),rst(s)): s t o (so(s),rst(s)): us o S us o S us o S s s s How o os ts t? o o o o S o S o S w s rt o (so(s), rst(s)) w s rt o (so(s), rst(s)) w s rt o (so(s), rst(s)) us o S us o S us o S Gr s: ANALYSIS Gr s: Dts Gr s: Dts F tror ot 0 O() (w t o t tr) How to tror ot? How to tror ot? Sort otr ots w rou 0.. O( ) A sto s to 0 s t owst ot Itz st S = (2, 1) or =3 to -1 o s t o (so(s),rst(s)): us o S s How o os ts t? o o S w s rt o (so(s), rst(s)) us o S O() vry ot s us o o t ost o
19 Gr s: Dts Gr s: Dts S u Gr s How to tror ot? A sto s to 0 s t owst ot tz st S = (1, 0) //ot r o CH S w wys ot t st 2 ots H or-ts Wt s w you ru o ts ut? How you x t? S u Gr s S u Gr s S u Gr s
20 S u Gr s 115
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[ppto ot] AC QUIT Y U P L C / E L S / U V: ONE M E T HO DO LOGY FO R F FA, FAM E AND ANA LYSIS O F BIO D I E S E L Ptr J. L A J. D Go Wtrs Corporto, Mor, MA U.S. INT RODUCTION Bos, tr o or wt oss s u,