MKtree Generation and Simulations

The problem to represent very complex systems has been studied by several authors obtaining solu tions based on di erent data structures In this paper aK dimensional tree Multirresolution Kdtree MKtree is introduced The MKtree represents a hierarchical subdivision of the scene objects that guarantees a minimum space overlap between node regions MKtrees are useful for collision detec tion and for time critical rendering in very large environments requiring external memory storage Examples in ship design applications are described MKtrees Introduction In this paper we introduce a new bounding volume hierarchy the Multirresolution Kdtree MKtree to represent very complex systems This kind of tree is generated taking ad vantage of two partitioning criteria space partition and scene objects partition Thereby our MKtree can be considered as an hybrid between Kdtrees and R trees Our method precomputes and automatically stores di erent levels of detail of objects and groups of objects when constructing the tree Our MKtree incorporates the following features The MKtree represents a hierarchical subdivision of the scene objects that guarantees a minimum space overlap between node regions Levels of detail LOD with bounded tolerance are supported at every hierarchical level of the tree Bounding approximations of objects and groups of objects are used The method has been conceived to manage memory e ciently by developping exter nal memory based algorithms and it is therefore useful for collision detection in large environments and for time critical rendering for instance The particular structure of the MKtrees is specially well suited for collision and proximity detection using external memory in very large virtual environments MKtrees are also useful for frustum based collision detection on line collision detection during navigation through the virtual model Related De nitions Before introducing the MKtrees let us start with some related de nitions De nition The environment is de ned as a collections of D objects that are usu ally polyhedra even though they can be a set of more general models Thus S fo oNg De nition We name R S the axis aligned bounding box AABB of all the set S In general R Sn is the AABB of a subset of objects of S Sn Where Sn S De nition xOverlap R S R S is the length of the intersection of the one di mensional x projections of R S and R S say xProj R S and xProj R S In other words xOverlap R S R S length xProj R S xProj R S The functions yOverlap and zOverlap are de ned in a similar way Assumption There exists a boolean function InPage Sn that returns true if the com plete geometry of Sn ts in one disk block and can be retrieved to the core memory using a single disk paging operation and false otherwise Description Following a similar nomenclature of KHM He a MKtree is a tree MKT S that speci es a bounding volume hierarchy on S S is a set of objects Each node n of MKT S corresponds to a subset Sn S with the root being associated with the full set S Each internal node has two children Thus the MKtree is a binary tree The union of the two subsets associated to the children of n is equal to Sn Each node is also associated to the AABB box of Sn R Sn A basic issue that is directly related to the performance of a MKtree is the selection of the splitting rules when building the hierarchy The main goal is that during tree construction we would like to assign subsets Sn and Sn of objects to each child of a node n in such a way to minimize the probability that their R Sn and R Sn intersect Let us assume that we have a procedure xDivideList that divides the collection of objects belonging to Sn in two subsets Sn and Sn minimizing the xOverlap R Sn R Sn Let us assume also that we have similar procedures for the y and z dimensions yDivideList and zDivideList Now we can de ne a node n of a MKtree corresponding to a subset Sn If InPage Sn then n is a leaf node and stores the geometry of objects in Sn If no InPage Sn then n is an internal node with two pointers to n and n respec tively Sn and Sn are obtained by using wDivideList where w can be x y or z with w being such that wOverlap R Sn R Sn is equal to Min xOverlap R Sn R Sn yOverlap R Sn R Sn zOverlap R Sn R Sn Note The overlap band is the value of wOverlap R Sn R Sn In this way the tree construction procedure automatically generates a subdivision and a hierarchy of all objects of S with a minimum overlap Groups of objects are automatically generated like in WLML by using here a minimum overlap criterium As it will be seen in the result sections sec and when dealing with virtual ship environments in many cases overlap band is null In these cases our building tree method is at the same time an space and object partitioning method Thereby our MKtree can be considered as an intermediate model between Kdtrees DECM Sam and Rtrees SRF BKSS Gut in D Speci cation The bounding approximations of objects b are described in this section It is desirable that these class of shapes accomplish some properties The bounding approximation of an object oi must ful ll oi b oi k where k is the level of detail The bounding approximation b oi k must approximate objects depending on a de sired resolution LOD k The volume distance vDist ABA between the geometry of an object oi and its bounding approximation associated to one speci ed level of detail k b oi k has to satisfy to be less than a known geometric threshold k In other words p oi p b oi k j dist p p k and p b oi k p oi j dist p p k We have chosen as bounding volumes surrounding polyhedra with small number of faces And AAB Each object is modeled by an o set polyhedron generated by topology simpli cation ABA In fact each object oi and each group of objects Sn have a set of bounding approximations one for each level of detail k Ranging k from to M The approximation error k that restricts the volume distance between bounding approximation of an object and its geometry vDist b oi k oi is decreasing while k is increasing In other words b S k is more accurate than b S k Faces b S k is the number of faces of the bounding approximation of S at k level of resolution On the other hand the set of bounding approximations of objects oi oj that belong to a subset Sn will be represented by B Sn k Thus B Sn k fb oi k b oj k g Figure shows a simple example in D of b Sn k and B Sn k Observe that b Sn k corresponds to the bounding polyhedron of a pipe red color and B Sn k corresponds to the set of bounding approximations of the set of the pipe elements green color Figure presents an other example in D where the b Sn k is the bounding approximation of a whole set of pipes in a ship environment