sf3 RESTRICTED QUADTREE (VON HERZEN/BARR)

Transcription

1 SURFACE DATA

2 HIERARCHICAL TRIANGULAR DECOMPOSITION Appoximate suface S y plana tiangula patches whose vetices ae a suset of data points defining S Fo each patch, compute an appoximation eo. maximum eo of the data points with pojections on the x-y plane ovelapping the pojection of the patch on the x-y plane 2. sudivide futhe if a pedefined toleance is exceeded Types. tenay (De Floiani, Falcidieno, Nagy, and Pienovi) use intenal point with maximum deviation and decompose the tiangle into thee tiangles y joining it to the vetices suface is continuous at evey level tiangles ae thin and elongated 2. quatenay (Gomez and Guzman) select thee points on the sides of the tiangle and fom fou new tiangles y connecting them each tiangle can e adjacent to many tiangles intepolating suface is not continuous unless all tiangles ae split unifomly i.e., the esult is a complete quadtee good when points ae dawn fom a unifomly spaced gid sf

3 sf2 HIERARCHICAL RECTANGULAR DECOMPOSITION Simila to tiangula decomposition Good when data points ae the vetices of a ectangula gid Dawack is asence of continuity etween adjacent patches of unequal width (temed the alignment polem) Ovecoming the pesence of cacks. use the intepolated point instead of the tue point (Baea and Hinjosa) 2. tiangulate the squaes (Von Hezen and Ba) can split into 2, 4, o 8 tiangles depending on how many lines ae dawn though the midpoint if split into 2 tiangles, then cacks still emain no cacks if split into 4 o 8 tiangles

4 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 All 4-adjacent locks ae eithe of equal size o of atio 2: Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke)

5 RESTRICTED QUADTREE (VON HERZEN/BARR) 2 sf3 All 4-adjacent locks ae eithe of equal size o of atio 2: Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke)

6 RESTRICTED QUADTREE (VON HERZEN/BARR) 3 z 2 sf3 All 4-adjacent locks ae eithe of equal size o of atio 2: Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) 8-tiangle decomposition ule. decompose each lock into 8 tiangles (i.e., 2 tiangles pe edge) 2. unless the edge is shaed y a lage lock 3. in which case only tiangle is fomed

7 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 All 4-adjacent locks ae eithe of equal size o of atio 2: Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) 4 g 3 z 2 8-tiangle decomposition ule. decompose each lock into 8 tiangles (i.e., 2 tiangles pe edge) 2. unless the edge is shaed y a lage lock 3. in which case only tiangle is fomed 4-tiangle decomposition ule. decompose each lock into 4 tiangles (i.e., tiangle pe edge) 2. unless the edge is shaed y a smalle lock 3. in which case 2 tiangles ae fomed along the edge

8 RESTRICTED QUADTREE (VON HERZEN/BARR) sf3 All 4-adjacent locks ae eithe of equal size o of atio 2: Note: also used in finite element analysis to adptively efine an element as well as to achieve element compatiility (temed h-efinement y Kela, Peucchio, and Voelcke) 5 v 4 g 3 z 2 8-tiangle decomposition ule. decompose each lock into 8 tiangles (i.e., 2 tiangles pe edge) 2. unless the edge is shaed y a lage lock 3. in which case only tiangle is fomed 4-tiangle decomposition ule. decompose each lock into 4 tiangles (i.e., tiangle pe edge) 2. unless the edge is shaed y a smalle lock 3. in which case 2 tiangles ae fomed along the edge Pefe 8-tiangle ule as it is ette fo display applications (shading)

9 sf4 PROPERTY SPHERES (FEKETE) Appoximation of spheical data Uses icosahedon which is a Platonic solid. 20 faces each is a egula tiangle 2. lagest possile egula polyhedon

10 sf5 ALTERNATIVE SPHERICAL APPROXIMATIONS Could use othe Platonic solids. all have faces that ae egula polygons tetahedon: 4 equilateal tiangula faces hexahedon: 6 squae faces octahedon: 8 equilateal tiangula faces dodecahedon: 2 pentagonal faces 2. octahedon is nice fo modeling the gloe it can e aligned so that the poles ae at opposite vetices the pime meidian and the equato intesect at anothe vetex one sudivision line of each face is paallel to the equato Decompose on the asis of latitude and longitude values. not so good if want a patition into units of equal aea as geat polems aound the poles 2. poject sphee onto plane using Lamet s cylindical pojection which is locally aea peseving Instead of appoximating sphee with the solids, poject the faces of the solids on the sphee (Scott). all edges ecome su-acs of a geat cicle 2. use egula decomposition on tiangula, squae, o pentagonal spheical suface patches

11 sf6 2D PRISM TREE (PONCE/FAUGERAS) Hieachical epesentation of an appoximation used to model sufaces of 3d polyhedal ojects with no holes Based on polyhedal appoximation. initially model y a tiangle with vetices on the suface so suface is decomposed into thee segments 2. fo each segment of the suface pick the point M at a maximum distance fom the tiangle 3. if the distance fom M is geate than a pedefined theshold, then sudivide into two tiangles 4. epeat until all appoximations ae within theshold Can e epesented as a tee stuctue Appoximate each ounday segment y a quadilateal. easie to manipulate a ounday (e.g., detecting intesections etween oundaies of adjacent ojects) 2. esult is 2d pism tee H A G ABC H A G F D AB BC CA F D AD AF C B BD BE CE CF C B E AG DG FH AH E

12 sf7 3D PRISM TREE: CONSTRUCTION Use tetaheda instead of tiangles. stat with a tiangle that splits the suface into 2 segments 2. fo each segment pick a point on the suface at a maximum distance fom the tiangle esults in two tetaheda which is a tiangula ipyamid (= hexahedon)

13 3D PRISM TREE: CONSTRUCTION Use tetaheda instead of tiangles. stat with a tiangle that splits the suface into 2 segments 2. fo each segment pick a point on the suface at a maximum distance fom the tiangle esults in two tetaheda which is a tiangula ipyamid (= hexahedon) 3. decompose the tiangula faces into tetaheda 2 sf7

14 3D PRISM TREE: CONSTRUCTION Use tetaheda instead of tiangles. stat with a tiangle that splits the suface into 2 segments 2. fo each segment pick a point on the suface at a maximum distance fom the tiangle esults in two tetaheda which is a tiangula ipyamid (= hexahedon) 3. decompose the tiangula faces into tetaheda unless the common edge AB etween 2 adjacent tiangles T and T 2 is such that the appoximation of the suface fomed y them is not within a specified toleance A 3 z 2 sf7 M T T 2 M 2 B

15 4 3 2 sf7 g z 3D PRISM TREE: CONSTRUCTION Use tetaheda instead of tiangles. stat with a tiangle that splits the suface into 2 segments 2. fo each segment pick a point on the suface at a maximum distance fom the tiangle esults in two tetaheda which is a tiangula ipyamid (= hexahedon) 3. decompose the tiangula faces into tetaheda unless the common edge AB etween 2 adjacent tiangles T and T 2 is such that the appoximation of the suface fomed y them is not within a specified toleance M A T T 2 M 2 M M n A M 2 B in this case, emove the common edge etween them and eplace it and tiangles T and T 2 y 4 new tiangles with vetices that ae the emaining vetices of T and T 2 (i.e., M and M 2 ) and a new point Mn 4 new tiangles M A M n, M 2 A M n, M B M n, and M 2 B M n whee M n satisfies a. eing on the suface. it is on the plane that isects the 2 tiangles that ae eing eplaced c. of all points satisfying a and, it is the futhest point fom the emoved edge B

16 sf8 3D PRISM TREE: APPROXIMATION Can e epesented as a tee stuctue. 2 sons at initial level 2. emaining levels have 3 sons unless an edge was emoved in which case thee ae 2 sons Appoximate each ounday segment y a 5-sided polyhedon in the shape of a tuncated tetahedon (misnamed a pism). one side is tiangula ase T of tetahedon 2. one side is a plane paallel to T passing though the point on the suface at the maximum distance fom T 3. emaining 3 sides ae fomed y isecting planes of T and the tiangles adjacent at the edges of T

17 sf9 COMPARISON OF PRISM AND STRIP TREES Appoximation of ounday segments. stip tee uses ectangles while the pism tee uses tapezoids (tuncated tetaheda) 2. fo convex ojects the inteios of the tapezoids ae disjoint while adjacent ones have common edges (faces) 3. disjointness and common oundaies aely aise in a stip tee Shape of the appoximating tapezoid fo a ounday segment S. depends on the line (tiangle) that appoximates the segments adjacent to S as well as on the level of the tee at which they ae found 2. shape of ectangles used in the stip tee ae independent of the appoximations of the neighoing segments and of thei level

18 DISPLAYING CURVED SURFACES (CATMULL) sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones I

19 DISPLAYING CURVED SURFACES (CATMULL) 2 sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones A I B D C quadilateal ABCD appoximates patch ABCD

20 DISPLAYING CURVED SURFACES (CATMULL) 3 z 2 sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones A I B D C quadilateal ABCD appoximates patch ABCD

21 DISPLAYING CURVED SURFACES (CATMULL) 4 g 3 z 2 sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones A F E I B D H G C quadilateal ABCD appoximates patch ABCD decompose ABCD into fou quadilateal patches

22 DISPLAYING CURVED SURFACES (CATMULL) 5 4 g 3 z 2 sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones A F D E H I M J K B N L G C quadilateal ABCD appoximates patch ABCD decompose ABCD into fou quadilateal patches sudivide futhe ut no distinction etween supatches and thei quadilateal appoximations

23 DISPLAYING CURVED SURFACES (CATMULL) 6 g 5 4 g 3 z 2 sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones A F D E H I M J K B N L G C quadilateal ABCD appoximates patch ABCD decompose ABCD into fou quadilateal patches sudivide futhe ut no distinction etween supatches and thei quadilateal appoximations each supatch spans at most one pixel cente

24 DISPLAYING CURVED SURFACES (CATMULL) 7 v 6 g 5 4 g 3 z 2 sf0 Can epesent y a collection of paametic icuic suface patches Recusively decompse each patch into supatches until the supatches ae so small that they only span the cente of one pixel (o can e shown to liie outside the display egion) Test fo nume of pixel centes spanned y a patch is ased on an appoximation y a polygon connecting the patch s cones A F D E H I M J K B N L G C quadilateal ABCD appoximates patch ABCD decompose ABCD into fou quadilateal patches sudivide futhe ut no distinction etween supatches and thei quadilateal appoximations each supatch spans at most one pixel cente aste image esulting fom the decomposition

25 sf MECHANICS OF CURVED SURFACE DISPLAY Pocessing follows quadtee decomposition paadigm in paamete space does not geneate the quadtee explicitly Seveal patches can span the same pixel cente as they ae in thee-dimensional space and thus use z-uffe to keep tack of patch that is closest to the viewpoint Can use with othe patch epesentations. chaacteistic polyheda (Bezie and B-spline patches) 2. allow appoximation y convex hull enclosing its contol points moe accuate than Catmull s method which is ased only on the fou cones of the patch