! Data mining. ! VLSI design. ! Computer vision. ! Mathematical models. ! Astronomical simulation. ! Geographic information systems.

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1 Geometri Algorithms Geometri Algorithms Applitions.! Dt mining.! VLSI design.! Computer vision.! Mthemtil models.! Astronomil simultion.! Geogrphi informtion systems. irflow round n irrft wing! Computer grphis (movies, gmes, virtul relity).! Models of physil world (mps, rhiteture, medil imging). Referene: Referene: Chpters 24-25, Algorithms in C, 2 nd Edition, Roert Sedgewik. History.! Anient mthemtil foundtions.! Most geometri lgorithms less thn 25 yers old. Roert Sedgewik nd Kevin Wyne Copyright Geometri Primitives Intuition Point: two numers (x, y). Line: two numers nd [x + y = 1] Line segment: two points. Polygon: sequene of points. ny line not through origin Wrning: intuition my e misleding.! Humns hve sptil intuition in 2D nd 3D.! Computers do not.! Neither hs good intuition in higher dimensions! Primitive opertions.! Is point inside polygon?! Compre slopes of two lines.! Distne etween two points.! Do two line segments interset?! Given three points p 1, p 2, p 3, is p 1 -p 2 -p 3 ounterlokwise turn? Other geometri shpes.! Tringle, retngle, irle, sphere, one,! 3D nd higher dimensions sometimes more omplited. Is given polygon simple? no rossings we think of this lgorithm sees this 3 4

2 Polygon Inside, Outside Polygon Inside, Outside: Crossing Numer Jordn urve theorem. [Velen 1905] Any ontinuous simple losed urve uts the plne in extly two piees: the inside nd the outside. Does line segment interset ry? Is point inside simple polygon? y = y i+1 " y i (x " x i ) + y i x i+1 " x i (x i+1, y i+1 ) where x i # x # x i+1 (x i, y i ) (x, y ) Applition. Drw filled polygon on the sreen. puli oolen ontins(doule x0, doule y0) { int rossings = 0; for (int i = 0; i < N; i++) { doule slope = (y[i+1] - y[i]) / (x[i+1] - x[i]); oolen ond1 = (x[i] <= x0) && (x0 < x[i+1]); oolen ond2 = (x[i+1] <= x0) && (x0 < x[i]); oolen ove = (y0 < slope * (x0 - x[i]) + y[i]); if ((ond1 ond2) && ove) rossings++; return (rossings % 2!= 0); 5 6 Implementing CCW Implementing CCW CCW. Given three point,, nd, is -- ounterlokwise turn?! Anlog of omprisons in sorting.! Ide: ompre slopes. CCW. Given three point,, nd, is -- ounterlokwise turn?! Determinnt gives twie re of tringle. x y 1 2 " Are(,, ) = x y 1 = ( x # x )( y # y ) # ( y # y )( x # x ) x y 1! If re > 0 then -- is ounterlokwise. yes no Yes (! slope)??? (olliner)??? (olliner)??? (olliner) If re < 0, then -- is lokwise. If re = 0, then -- re olliner. ( x, y ) ( x, y ) Lesson. Geometri primitives re triky to implement.! Deling with degenerte ses.! Coping with floting point preision. ( x, y ) > 0 ( x, y ) ( x, y ) < 0 ( x, y ) 7 8

3 Immutle Point ADT puli finl lss Point { puli finl int x; puli finl int y; Convex Hull puli Point(int x, int y) { this.x = x; this.y = y; puli doule distneto(point q) { return Mth.hypot(this.x - q.x, this.y - q.y); puli stti int w(point, Point, Point ) { doule re2 = (.x-.x)*(.y-.y) - (.y-.y)*(.x-.x); if else (re2 < 0) return -1; else if (re2 > 0) return +1; else if (re2 > 0 return 0; puli stti oolen olliner(point, Point, Point ) { return w(,, ) == 0; 9 10 Convex Hull Mehnil Solution A set of points is onvex if for ny two points p nd q in the set, the line segment pq is ompletely in the set. Mehnil lgorithm. Hmmer nils perpendiulr to plne; streth elsti ruer nd round points. Convex hull. Smllest onvex set ontining ll the points. p p q q onvex not onvex onvex hull Properties.! "Simplest" shpe tht pproximtes set of points.! Shortest (perimeter) fene surrounding the points.! Smllest (re) onvex polygon enlosing the points

4 Brute Fore Pkge Wrp (Jrvis Mrh) Oservtion 1. Edges of onvex hull of P onnet pirs of points in P. Oservtion 2. Edge pq is on onvex hull if ll other points re ounterlokwise of pq. q p Pkge wrp.! Strt with point with smllest y-oordinte.! Rotte sweep line round urrent point in w diretion.! First point hit is on the hull.! Repet. O(N 3 ) lgorithm. For ll points p nd q in P, hek whether pq is n edge of onvex hull. eh hek requires O(N) w lultions, where N is the numer of points in P Pkge Wrp (Jrvis Mrh) How Mny Points on the Hull? Implementtion.! Compute ngle etween urrent point nd ll remining points.! Pik smllest ngle lrger thn urrent ngle.! "(N) per itertion. Prmeters.! N = numer of points.! h = numer of points on the hull. Pkge wrp running time. "(N h) per itertion. How mny points on hull?! Worst se: h = N.! Averge se: diffiult prolems in stohsti geometry. in dis: h = N 1/3. in onvex polygon with O(1) edges: h = log N

5 Grhm Sn: Exmple Grhm Sn: Exmple Grhm sn.! Choose point p with smllest y-oordinte.! Sort points y polr ngle with p to get simple polygon.! Consider points in order, nd disrd those tht would rete lokwise turn. p Implementtion.! Input: p[1], p[2],, p[n] re points.! Output: M nd rerrngement so tht p[1],..., p[m] is onvex hull. // preproess so tht p[1] hs smllest y-oordinte // sort y ngle with p[1] points[0] = points[n]; // sentinel int M = 2; for (int i = 3; i <= N; i++) { while (Point.w(p[M-1], p[m], p[i]) <= 0) { M--; disrd points tht would rete lokwise turn M++; swp(points, M, i); dd i to puttive hull Running time. O() for sort nd O(N) for rest. 17 why? 18 Quik Elimintion Convex Hull Algorithms Costs Summry Quik elimintion.! Choose qudrilterl Q or retngle R with 4 points s orners.! If point is inside, n eliminte. 4 w tests for qudrilterl 4 omprisons for retngle Q R Algorithm Pkge wrp Grhm sn Running Time N h Three-phse lgorithm! Pss through ll points to ompute R.! Eliminte points inside R.! Find onvex hull of remining points. Quikhull Mergehull Sweep line Quik elimintion N t Best in theory N log h output sensitive running time Prtie. Cn eliminte lmost ll points in liner time. these points eliminted symptoti ost to find h-point hull in N-point set t ssumes "resonle" point distriution 19 20

6 Convex Hull: Lower Bound Models of omputtion.! Comprison sed: ompre oordintes. (impossile to ompute onvex hull in this model of omputtion) Closest Pir of Points (.x <.x) ((.x ==.x) && (.y <.y)))! Qudrti deision tree model: ompute ny qudrti funtion of the oordintes nd ompre ginst 0. (.x*.y -.y*.x +.y*.x -.x*.y +.x*.y -.x*.y) < 0 Theorem. [Andy Yo, 1981] In qudrti deision tree model, ny onvex hull lgorithm requires #() ops. higher degree polynomil tests don't help either [Ben-Or, 1983] even if hull points re not required to e output in ounterlokwise order Closest Pir of Points Closest Pir of Points Closest pir. Given N points in the plne, find pir with smllest Euliden distne etween them. Fundmentl geometri primitive.! Grphis, omputer vision, geogrphi informtion systems, moleulr modeling, ir trffi ontrol.! Speil se of nerest neighor, Euliden MST, Voronoi. fst losest pir inspired fst lgorithms for these prolems Algorithm.! Divide: drw vertil line L so tht roughly!n points on eh side.! Conquer: find losest pir in eh side reursively.! Comine: find losest pir with one point in eh side.! Return est of 3 solutions. L seems like "(N 2 ) Brute fore. Chek ll pirs of points p nd q with "(N 2 ) distne lultions D version. O() esy if points re on line. Assumption. No two points hve sme x oordinte. 12 to mke presenttion lener 23 26

7 Closest Pir of Points Closest Pir of Points Find losest pir with one point in eh side, ssuming tht distne < $.! Oservtion: only need to onsider points within $ of line L. Def. Let s i e the point in the 2$-strip, with the i th smllest y-oordinte.! Sort points in 2$-strip y their y oordinte.! Only hek distnes of those within 11 positions in sorted list! Clim. If i j % 12, then the distne etween s i nd s j is t lest $. Pf j! No two points lie in sme!$-y-!$ ox. 7 6 L! Two points t lest 2 rows prt hve distne % 2(!$).! 2 rows 29 30!$!$ Ft. Still true if we reple 12 with 7. i 27 28!$ 12 3 $ = min(12, 21) $ 30 $ $ 31 Closest Pir Algorithm Closest Pir of Points: Anlysis Closest-Pir(p 1,, p n ) { Compute seprtion line L suh tht hlf the points re on one side nd hlf on the other side. $ 1 = Closest-Pir(left hlf) $ 2 = Closest-Pir(right hlf) $ = min($ 1, $ 2 ) Delete ll points further thn $ from seprtion line L Sort remining points y y-oordinte. O() 2T(N / 2) O(N) O() Running time. T(N) " 2T ( N /2) + O() # T(N) = O(N log 2 N) Upper ound. Cn e improved to O(). void sorting y y-oordinte from srth Sn points in y-order nd ompre distne etween eh point nd next 11 neighors. If ny of these distnes is less thn $, updte $. O(N) Lower ound. In qudrti deision tree model, ny lgorithm for losest pir requires #() steps. return $

8 1854 Choler Outrek, Golden Squre, London Nerest Neighor Nerest Neighor Voronoi Digrm Input. N Euliden points. Nerest neighor prolem. Given query point p, whih one of originl N points is losest to p? Voronoi region. Set of ll points losest to given point. Voronoi digrm. Plnr sudivision delineting Voronoi regions. Ft. Voronoi edges re perpendiulr isetor segments. Algorithm Brute Gol Preproess 1 Query N log N Voronoi of 2 points (perpendiulr isetor) Voronoi of 3 points (psses through irumenter) 36 37

9 Voronoi Digrm Voronoi Digrm: Applitions Voronoi region. Set of ll points losest to given point. Voronoi digrm. Plnr sudivision delineting Voronoi regions. Ft. Voronoi edges re perpendiulr isetor segments. Toxi wste dump prolem. N homes in region. Where to lote nuler power plnt so tht it is fr wy from ny home s possile? looking for lrgest empty irle (enter must lie on Voronoi digrm) Pth plnning. Cirulr root must nvigte through environment with N ostle points. How to minimize risk of umping into ostle? root should sty on Voronoi digrm of ostles Referene: J. O'Rourke. Computtionl Geometry. Quintessentil nerest neighor dt struture Voronoi Digrm: More Applitions Sientifi Redisoveries Anthropology. Identify influene of lns nd hiefdoms on geogrphi regions. Astronomy. Identify lusters of strs nd lusters of glxies. Biology, Eology, Forestry. Model nd nlyze plnt ompetition. Crtogrphy. Piee together stellite photogrphs into lrge "mosi" mps. Crystllogrphy. Study Wigner-Setiz regions of metlli sodium. Dt visuliztion. Nerest neighor interpoltion of 2D dt. Finite elements. Generting finite element meshes whih void smll ngles. Fluid dynmis. Vortex methods for invisid inompressile 2D fluid flow. Geology. Estimtion of ore reserves in deposit using info from ore holes. Geo-sientifi modeling. Reonstrut 3D geometri figures from points. Mrketing. Model mrket of US metro re t individul retil store level. Metllurgy. Modeling "grin growth" in metl films. Physiology. Anlysis of pillry distriution in ross-setions of musle tissue. Rootis. Pth plnning for root to minimize risk of ollision. Typogrphy. Chrter reognition, eveled nd rved lettering. Zoology. Model nd nlyze the territories of nimls. Referenes: Yer 1644 Disoverer Desrtes Disipline Astronomy Nme "Hevens" 1850 Dirihlet Mth Dirihlet tesseltion Voronoi Boldyrev Mth Geology Voronoi digrm re of influene polygons Thiessen Niggli Meteorology Crystllogrphy Thiessen polygons domins of tion Wigner-Seitz Frnk-Csper Physis Physis Wigner-Seitz regions tom domins Brown Med Eology Eology re of potentilly ville plnt polygons 1985 Hoofd et l. Antomy pillry domins Referene: Kenneth E. Hoff III 40 41

10 Fortune's Algorithm Fortune's Algorithm Fortune's lgorithm. Sweep-line lgorithm n e implemented in O() time. ut very triky to get right due to degenery nd floting point! Algorithm Brute Fortune Preproess 1 Query N log N Disretized Voronoi Hoff's Algorithm Disretized Voronoi. Solve nerest neighor prolem on n N-y-N grid. Hoff's lgorithm. Align pex of right irulr one with sites.! Minimum envelope of one intersetions projeted onto plne is the Voronoi digrm.! View ones in different olors & render Voronoi. Brute fore. For eh grid ell, mintin losest point. When dding new point to Voronoi, updte N 2 ells. Implementtion. Drw ones using stndrd grphis hrdwre!

11 Deluny Tringultion Summry Deluny tringultion. Tringultion of N points suh tht no point is inside irumirle of ny other tringle. Summry. Mny fundmentl geometri prolems require ingenuity to solve lrge instnes. Ft 0. It exists nd is unique (ssuming no degenery). Ft 1. Dul of Voronoi (onnet djent points in Voronoi digrm). Ft 2. No edges ross & O(N) edges. Ft 3. Mximizes the minimum ngle for ll tringulr elements. Ft 4. Boundry of Deluny tringultion is onvex hull. Ft 5. Shortest Deluny edge onnets losest pir of points. Prolem onvex hull losest pir Brute Cleverness N 2 N 2 furthest pir N 2 Deluny tringultion N 4 polygon tringultion N 2 N Deluny Voronoi symptoti time to solve 2D prolem with N points 46 47