Solid Modeling Techniques

Transcription

1 ML70 LTUR Solid Modeling Techniques onstructive Solid eometry (S) omputational Solid eometry Primitive based: It is based on the notion that a physical object can be divided into a set of primitives basic elements or shapes that can be combined in a certain order following a set of rules ( oolean operations) to create the object. Primitives themselves are considered valid S models. ach primitives is bounded by closed and orientable surfaces. S model is fundamentally and topologically different from a -rep model in that it does not store faces, edges and vertices explicitly. Instead it evaluates them whenever needed by algorithms. S representation is of considerable importance for manufacturing

2 Primitive based Vs Half-space based S Schemes There are two types of S schemes Primitive ased S: It is based on bounded valid solid primitives, r-sets. It is the most popular S scheme. Half space ased S :This scheme uses unbounded half spaces (non r-sets). ounded solid primitives are considered composite half spaces and the boundaries of these are the surfaces of the component half spaces. - + Primitive based S Half space based S ata Structure of S Like in -rep, the database stores topology and geometry. The validity checking in S scheme occurs indirectly. The each primitive that is combined using a oolean operations (r-sets) to build the S model is checked for its validity. The common data structures used for S are graphs and trees

3 S ata Structure raph: graph is defined as a set of nodes connected by a set of branches or lines. Path: ach node in a tree belongs to a path.g. to ycle: If starting and ending nodes are the same the path is called a cycle Root node raph Leaf nodes igraph S ata Structure yclic raph: If a graph contains a cycle it is called cyclic otherwise it is acyclic. Path: ach node in a tree belongs to a path.g. to ycle: If starting and ending nodes are the same the path is called a cycle Indegree = Outdegree = raph igraph

4 Tree and inary Tree Tree: tree is an acyclic digraph in which a single node called root node has a zero indegree and every other nodes has an indegree of one. inary Tree: In a tree if the descendent of each node are in order (from left to right) and each node except the leaf node has two decedents (left and right), then the tree is called a binary tree. Tree inary Tree inary Tree and Subtree Subtree: ny binary tree can be thought of as joining together of subtrees. left subtree and a right subtree rooted at two successor nodes Left subtree n L = n R alanced binary tree Right subtree n L n R Unbalanced binary tree

5 Inverted inary Tree for S Model Inverted inary Tree: If the direction of each connector (arrow) are reversed then we get an inverted binary tree wherein each node has one outdegree except the root node. Root node: node with outdegree = 0. ny node that does not have descendent. Leaf node: ny node that does not have a predecessor or indegree = 0. Interior node: ny node with outdegree > 0 is an interior node. Inverted inary Tree Inverted inary Tree as S Tree S Tree: The inverted binary tree is very convenient to represent the S operations. Here each of the leaf nodes is a valid solid primitive. The intermediate nodes are the transition states of the solid modeling (r-set) operations. The root node is the resulting solid from the set operations. Here n primitives require (n-) oolean operations to complete the construction of the object. S=UU S Primitives = U U Operations= Nodes=n-=7 S Tree

6 Tree Traversal Tree Traversal: Visiting the nodes in sequential or orderly and efficient way. Many of the sorting and searching algorithms need to do the tree traversal Traversal methods: Two broad methods are:. epth first. readth first The depth first is further divided into the following types depending on the order in which the root node is visited a) Preorder b) Inorder c)postorder Preorder Inorder Postorder Tree Traversal Preorder: We have the following recursive algorithm lgorithm. Visit the root. Traverse the left subtree in preorder. Traverse the right subtree 9 Preorder Traversal

7 Tree Traversal Reverse Preorder: We have the following recursive algorithm lgorithm. Traverse the right subtree. Traverse the left subtree in preorder. Visit the root 7 Reverse Preorder Traversal Tree Traversal Inorder: We have the following recursive algorithm lgorithm. Traverse the left subtree in preorder. Visit the root. Traverse the right subtree 6 0 Inorder Traversal 7 9 7

8 Tree Traversal Reverse Inorder: We have the following recursive algorithm lgorithm. Traverse the right subtree. Visit the root. Traverse the left subtree in preorder 0 6 Reverse Inorder Traversal 9 7 Tree Traversal Postorder: We have the following recursive algorithm lgorithm. Traverse the left subtree in preorder. Traverse the right subtree. Visit the root 7 Postorder Traversal 6 9 0

9 Tree Traversal Reverse Postorder: We have the following recursive algorithm lgorithm. Visit the root. Traverse the right subtree. Traverse the left subtree in preorder 9 Reverse Postorder Traversal asic lements Primitives: ounded solid primitives are the basic building blocks of S. These (parametric) solids have two sets of geometric data:. onfiguration Parameters (size information). Rigid motion Parameters (orientation information) Y L Y Z P Y H LOK X Z R YLINR Y X W SPHR Z H P X Z R X 9

10 uilding Operations The main building operations are regularised set operatoins like union (U*), intersection ( *) and difference (-*). Hence the S models are known as set-theoretic, oolean or combinatorial models. In contrast to uler operations, the oolean operations are not based on any equation or law. They are based on the set theory and the closure property. These operations are considered higher-level operations than uler operations. Some implementations of solid modelers provide derived types of operations like SSML and LU Main algorithms in S Operations. dge / Solid intersection algorithm. omputing set membership classification a) ivide and conquer: It is like ray tracing. Instead of a ray an edge is used as a reference b) Neighborhood: It deals with in, on and out decisions When a point is in the interior of solid face then it is called face neighborhood dge neighborhood occurs when the point lies on the solid edge When a point is a vertex, vertex neighborhood occurs. This is a complex case becouse the point is shared between three solid faces. P P P 0

11 Summary of a S algorithm The following steps describe a general S algorithm based on divide and conquer approach:. enerate a sufficient number of t-faces, set of faces of participating primitives, say and.. lassify self edges of w.r.t including neighborhood.. lassify self edges of w.r.t using & paradigm. If or is not primitive then this step is followed recursively.. ombine the classifications in step and via oolean operations.. Regularize the on segment that result from step discarding the segments that belong to only one face of S. 6. Store the final on segments that result from step as part of the boundary of S. Steps to 6 is performed for each of t-edge of a given t-face of. 7. Utilize the surface/surface intersection to find cross edges that result from intersecting faces of (one at a time) with the same t-face mentioned in step 6.. lassify each cross edge w.r.t S by repeating steps to with the next self edge of. 9. Repeat steps and 6 for each cross edge 0. Repeat steps to 9 for each t-face of.. Repeat stpes to 6 for each t-face of. S xample reate the S model of the following solid S. y d x b c d z c + - a eometry of the primitives LOK : xl = a d, yl = d, zl = c, P ( x, y, z) = P ( d,0, c) LOK : xl = d, yl = b, zl = c, P ( x, y, z) = P (0,0, c) YLINR : R = R, H = d, P ( x, y, z) = P ( d + a /, d, c / ) S U*-* U* lassify ombine on out on S Null on S M(,) M(,) U* * -* M(,) M(,) on out in lassify U* * -* on S on S on S ombine