Mesh Data Structures. Geometry processing. In this course. Mesh gallery. Mesh data

Transcription

1 Gomtry procssing Msh Data Structurs Msh data Gomtry Connctivity Data structur slction dpnds on Msh typ Algorithm rquirmnts 2 Msh gallry In this cours Only orintabl, triangular, manifold mshs Singl componnt, closd, triangular, orintabl manifold Multipl componnts With boundaris Not orintabl Singl componnt, closd, triangular, orintabl manifold Not only triangls Multipl componnts Non manifold With boundaris Not orintabl Not only triangls Non manifold 3 4

2 Algorithms rquirmnts Storing msh data Data structur should b fficint for Rndring How good is a data structur? Tim to construct - prprocssing Gomtry quris What ar th vrtics of #3? Which s ar adjacnt to #7? Tim to answr a qury Tim to updat Gomtry oprations Rmov/add a / Vrtx split, dg collaps Spac complxity Rdundancy 5 6 List of s List of s - quris List of vrtics - 3D List of s - triplts of pointrs (,, (x, y, z f f 4 (x, y, z vrtics (ccw f (,, (,, 7 f f 4 What ar th vrtics of #3? O( - third triplt from list Rquirs a full pass ovr all s vrtics (ccw f (,, (,, 8

3 List of s pros and cons Adjacncy matrix Simpl Low mmory rquirmnts Can rprsnt polygon soups O( F for quris lik Vrtics adjacncy Facs nar List of vrtics - 3D List of s - triplts of pointrs (,, Adjacncy matrix A of siz V * V A(i,j = iff v i and v j ar adjacnt f f 4 (x, y, z vrtics (ccw f (,, (,, 9 0 Adjacncy matrix - quris Adjacncy matrix pros and cons What ar th vrtics of #3? O( third triplt from list O(: A(i,j ==? Which s ar adjacnt to j? Rquirs a full pass on all s (x, y, z vrtics (ccw f (,, (,, Information on vrtics adjacncy Can rprsnt non-manifold mshs No connction btwn a and its adjacnt s O( F for s nar qury 2

4 DCEL (Doubly-Connctd Edg List Rcord for ach, dg and : Vrtx rcord: Coordinats Pointr to an outgoing half-dg DCEL Fac rcord: Pointr to an adjacnt half-dg Gomtric information Topological information Attribut information v f aka Half-Edg Structur 3 4 DCEL Half-dg rcord: Pointr to origin origin( Pointr to twin half-dg twin( prv( DCEL nxt( IncFac( Pointr to adjacnt IncidntFac( lft of from origin to dstination twin( Nxt and prvious dg of IncidntFac( nxt( prv( 5 origin( 6

5 DCEL (cont. DCEL - xampl Oprations supportd: Walk around boundary of givn Visit all dgs incidnt to v Quris: Most quris ar O( (x, y, z coordinat adjacnt dg f, 5, outgoing dg 2, 5,, v, 4, 2, 3, 5, f Half-dg origin twin Incidnt Fac nxt prv 3, 3, f, 5, 2, 4, 7 8 DCEL a largr xampl DCEL a largr xampl 6, 6,, 2, 4, 7, 7,2 9, f f 4 3, 5, 8,, 2, f f 4 3, 5, 4, 7, 7,2 9, 8, coordinat (x,y,z,y 2,z 2,y 3,z 3 (x 4,y 4,z 4 (x 5,y 5,z 5 (x 6,y 6,z 6 IncidntEdg 2, 5,, 7, 9, 7,2 dg f, 5, f 4 8, 9 Half-dg origin 3, 4, twin 3, 4, IncidntFac f nxt, 5, 7, prv 2, 4, 5, 6, 20

6 DCEL pros and cons Cornr tabl All quris in O( tim All oprations ar O( (usually Rprsnts only manifold mshs Cornr : Coupling of with on of its incidnt triangls. Cornr c contains: c.l.v c.l=c.p.o Triangl c.t Vrtx c.v Nxt cornr in c.t (ccw - c.n Prvious cornr - c.p c.n c.n.v (== c.n.n Cornr opposit c - c.o Edg E opposit c not incidnt on c.v Triangl T adjacnt to c.t across E c.o.v of T that is not incidnt on E c.v c c.t Edg E c.o c.o.v c.r=c.n.o c.p c.p.v c.r.v 2 Right cornr - c.r - cornr opposit c.n (== c.n.o. Lft cornr - c.l (== c.p.o == c.n.n.o 22 Cornr tabl - xampl c c 3 9 c4 c 7 c 2 Cornr tabl pros and cons c f c f 6 c 0 4 c2 c5 c c 8 All quris in O( tim All oprations ar O( (usually cornr c.v c.t c.n c.p c.o c.r c.l c f c 2 c 3 c 2 f c 3 c NULL NULL c 3 f c c 2 NULL c 4 c 5 NULL c 7 c c 5 c 4 c 7 c c 4 c 5 c NULL c 7 NULL NULL NULL NULL 23 Only triangular, manifold mshs High rdundancy (but not too high 24

7 Cornr tabl Additional dtails For ach a list of all its cornrs Cornrs numbr j*3, j*3+ and j*3+2 match numbr j Cornr numbr = EdgID of opposit dg Exampls of quris What ar th vrtics of #3? Chck c.v of cornrs 9, 0, Scan all cornrs of i, chck if c.p.v or c.n.v ar j Which s ar adjacnt to j? Chck c.t of all cornrs of j 25 26