Mesh Generation. Quadtrees. Geometric Algorithms. Lecture 9: Quadtrees

Transcription

1 Lecture 9: Lecture 9:

2 VLSI Design To Lecture 9:

3 Finite Element Method To Lecture 9:

4 To Lecture 9:

5 To component not conforming doesn t respect input not well-shaped Lecture 9:

6 To Input: disjoint polygonal objects integer coordinates 0, 45, 90, 135 degree angles U 0 0 U Lecture 9:

7 To Input: disjoint polygonal objects integer coordinates 0, 45, 90, 135 degree angles Output: triangular mesh conforming: no vertices in the interior of edges respecting the input: input edges contained in mesh edges well-shaped: triangle angles at least 45 degrees not conforming component not well-shaped doesn t respect input Lecture 9:

8 Steiner points To without Steiner points angles in triangles can be very small Example: Triangulation that maximizes the minimum angle has a minimum angle of 5 degrees Lecture 9:

9 Uniform vs. Non-uniform To uniform non-uniform 512 triangles 52 triangles Lecture 9:

10 To Quadtree: a rooted tree in which every node has 4 children each node corresponds to a square in the plane each child of a node corresponds to a quadrant of the square of the node a quadtree induces a subdivision of the square of the root NE NW SW SE can be used for efficient storing of non-uniform meshes Lecture 9:

11 Quadtree Examples To in particular google-maps-coordinates-tile-bounds-projection/ Lecture 9:

12 Constructing a Quadtree To can store different types of data; here: point sets Principle: split squares until each square contains 1 point Recursive Construction: start with square containing all points; if there is more than 1 point in the square then split square into 4 quadrants assign points to squares recur on each of the quadrants Lecture 9:

13 Constructing a Quadtree To can store different types of data; here: point sets Principle: split squares until each square contains 1 point Recursive Construction: start with square containing all points; if there is more than 1 point in the square then split square into 4 quadrants assign points to squares recur on each of the quadrants σ NW σ SW x mid σ NE σ SE y mid Lecture 9:

14 Unbalanced To quadtrees become unbalanced when many points lie close together Lecture 9:

15 Side lengths To side lengths of squares in a quadtree halve with increasing depth side length = s/8 depth = 3 s Lecture 9:

16 Properties To Lemma: Let c be the smallest distance between any two points in a point set P, and let s be the side length of the initial (biggest) square. Then the depth of a quadtree for P is at most log(s/c) + 3/2. Lemma: A quadtree of depth d storing n points has O((d + 1)n) nodes and can be constructed in O((d + 1)n) time. Lecture 9:

17 Finding a Neighbor To quadtree is a subdivision into regions typical operation: moving around among regions Neighbor finding: Given a node v (a square) and a direction (north, south, west, or east), which node (square) is adjacent to v in the given direction? Lecture 9:

18 Algorithm for Finding a Neighbor To Input: a node v in a quadtree T Output: north neighbor of v if v is a SW- or SE-child, then its north neighbor is a sibling otherwise, climb T until a node w is reached that is a SW- or SE-child if this does not exist, return nil north-neighbor of parent(ν) σ(ν)... and descend down into the north neighbor of w finding the SW or SE nod at the same depth as v. this also works for south, west, east running time: O(d + 1) time in a quadtree of depth d Lecture 9:

19 Algorithm for Finding a Neighbor To Input: a node v in a quadtree T Output: north neighbor of v if v is a SW- or SE-child, then its north neighbor is a sibling otherwise, climb T until a node w is reached that is a SW- or SE-child if this does not exist, return nil north-neighbor of parent(ν) σ(ν)... and descend down into the north neighbor of w finding the SW or SE nod at the same depth as v. this also works for south, west, east running time: O(d + 1) time in a quadtree of depth d Lecture 9:

20 Algorithm for Finding a Neighbor To Input: a node v in a quadtree T Output: north neighbor of v if v is a SW- or SE-child, then its north neighbor is a sibling otherwise, climb T until a node w is reached that is a SW- or SE-child if this does not exist, return nil north-neighbor of parent(ν) σ(ν)... and descend down into the north neighbor of w finding the SW or SE nod at the same depth as v. this also works for south, west, east running time: O(d + 1) time in a quadtree of depth d Lecture 9:

21 Balanced To a quadtree is balanced if any two neighboring nodes differ at most 1 in depth an unbalanced quadtree subdivision Lecture 9:

22 Balancing a Quadtree To add nodes! balancing Lecture 9:

23 To Algorithm for Balancing a Quadtree Input: quadtree T Output: balanced version of T 1. insert all leaves of T into a linear list L 2. while L is not empty 3. do remove a leave µ from L 4. if σ(µ) has to be split 5. then make µ an internal node with 4 children; if µ stores a point, store it in the correct leave 6. insert the 4 new leaves into L 7. check if σ(µ) had neighbors that now need to be split, and if so, insert them into L 8. return T Lecture 9:

24 To Complexity of a balanced Quadtree Theorem: Let T be a quadtree with m nodes. Then the balanced version of T has O(m) nodes and can be constructed in O((d + 1)m) time. Lecture 9:

25 From to To Input: grid with 2 j 2 j squares and a set of disjoint polygonal objects inside (integer coord.; 0, 45, 90, 135 degree angles) U Output: triangular mesh of the inside and outside of the objects inside the square that is conforming, well-shaped, 0 non-uniform, and respects the input. 0 U Lecture 9:

26 From to To build a quadtree: split until no object intersects a square or square has size 1 1 this gives small squares at objects, (and larger ones further away) an object completely in the square and even a single point of intersection count as intersections! Lecture 9:

27 From to To an edge of an object can only cross a square as a diagonal make the quadtree balanced add a Steiner point in the middle and produce triangles where needed triangular mesh is stored as doubly-connected edge list Lecture 9:

28 From to To Theorem: Let S be a set of disjoint polygonal objects with vertices on a (integer) grid [0, U] [0, U]. Then there exists a non-uniform triangular mesh for S that is conforming, well-shaped and respects the input the number of triangles is O(p(S) log U), where p(s) is the sum of (lengths of) perimeters of the objects the mesh can be constructed in O(p(S) log 2 U) time Lecture 9: