Today

1. Today • Homework • Bounding Boxes • Quadtrees • Per-pixel collision • Drawing text (if we have time)

2. Secret Game • Two Parts • Part I: Debug • Syntax errors • Logical errors • Improvements • Comments • Part II: Complete Game • Score • Lives

3. Common Bounding Volumes • Most introductory game programming texts call AABBs simply “bounding boxes” Circle/Sphere Axis-Aligned Bounding Box(AABB) Oriented Bounding Box (OBB) Convex Hull Better bounds, better culling Faster test, less memory

4. Circle Bounding Box • Simple storage, easy intersection test • Rotationally invariant Compare Euclidean distance between circle centers against sum of circle radii. boolcircle_intersect(circle a, circle b) { Point d; // d = b.c – a.c d.x = a.c.x – b.c.x; d.y = a.c.y – b.c.y; int dist2 = d.x*d.x + d.y*d.y; // d dot d intradiusSum = a.r + b.r; if (dist2 <= radiusSum * radiusSum) { return true; } else { return false; } } r c struct circle { Point c; // center intr; // radius } struct Point { intx; inty; }

5. Axis-Aligned Bounding Boxes (AABBs) • Three common representations • Min-max • Min-widths • Center-radius min // min.x<=x<=max.x // min.y<=y<=max.y struct AABB { Point min; Point max; } // min.x<=x<=min.x+dx // min.y<=y<=min.y+dy struct AABB { Point min; intdx; // x width intdy; // y width } // | c.x-x | <= rx | c.y-y | <= ry struct AABB { Point c; intrx; // x radius intry; // y radius } max min dx dy ry rx c Can easily be extended to 3D

6. Axis Aligned Bounding Box Intersection (min-max) • Two AABBs intersect only if they overlap on both axes a.min.x=b.min.x a.min.x>b.max.x a.max.x<b.min.x a.max.y<b.min.y bool IntersectAABB(AABB a, AABB b) { { if (a.max.x < b.min.x || a.min.x < b.max.x) return false; if (a.max.y < b.min.y || a.min.y < b.max.y) return false; return true; } a.min.y=b.min.y a.min.y>b.max.y

7. Axis Aligned Bounding Box Intersection (min-width) • Two AABBs intersect only if they overlap on both axes (a.min.x-b.min.x)>b.dx -(a.min.x-b.min.x)>a.dx a.min.x=b.min.x boolIntersectAABB(AABB a, AABB b) { { intt; t=a.min.x-b.min.x; if (t>b.dx || -t>a.dx) return false; t=a.min.y-b.min.y; if (t>b.dy || -t>a.dy) return false; return true; } // Note: requires more operations than // min-max case (2 more subtractions, 2 more negations) -(a.min.y-b.min.y)>a.dy a.min.y=b.min.y (a.min.y-b.min.y)>b.dy

8. AABB Intersection (center-radius) • Two AABBs intersect only if they overlap on both axes a.c.x-b.c.x >a.rx+b.rx b.c.x-a.c.x >a.rx+b.ry a.c.x=b.c.x b.c.y-a.c.y > a.ry+b.ry bool IntersectAABB(AABB a, AABB b) { { if (Abs(a.c.x – b.c.x) > (a.r.dx + b.r.dx)) return false; if (Abs(a.c.y – b.c.y) > (a.r.dy + b.r.dy)) ) return false; return true; } // Note: Abs() typically single instruction on modern processors a.c.y=b.c.y a.c.y-b.c.y >a.ry+b.ry

9. Trees root 56 • A tree is a data structure • Places contents into nodes • Each node has • Contents • Pointers to 0 or more other levels • Children are represented as references a b 45 75 d c 24 51 // A simple tree node class Class Node { List<Node> children; // List of references to Nodes int contents; } A tree with 5 nodes (root, a, b, c, d).Root node has contents 56.Node a has contents 45, and two children, c and d.Node b has contents 75, and no children.Node c has contents 24, and no children.Node d has contents 51, and no children.

10. Tree Operations • Adding a child • Create new node • n = new Node(); • Add it to list of children of parent • parentNode.children.Add(n); • Removing a child • Find and remove child node from list of children • parentNode.children.Remove(childToRemove); • (childToRemove is reference to Node of child to be deleted)

11. Point Quadtree root 0 1 • A tree where each node has four children • Each level represents subdividing space into 4 quadrants • Each node holds information about one quadrant • A deeper tree indicates more subdivisions MX Quadtree Demo http://donar.umiacs.umd.edu/quadtree/points/mxquad.html Demo both points and rectangles 0 3 1 2 2 3 root 1.1 1.0 0 3 1 2 0 1 1.2 1.3 1.0 1.1 1.2 1.3 2 3

12. C# Node Representation class Node { Point min[4]; Point max[4]; int level; Node Quad[4]; // holds 4 quadrants List<IGameObject> objectList; // list of game objects in a Node } • quads is an array with four elements • Each element is a reference to a Node • Quad[0] – upper left, etc. • A recursive data structure • Nodes hold pointers to Nodes • Min[] is an array with four elements • Each element is upper left point of quadrant • Max[] holds lower right point of each quadrant • Level indicates how many levels down in the tree • Useful for bounding the depth of the tree Quad[0] Quad[1] Quad[2] Quad[3]

13. Adding object to tree Insert(Node n, IGameObject g) • Iterate through all 4 quadrants of current node (n) • If at maximum depth of the tree • Add IGameObject to objectList and return • If AABB (bounding box) of IGameObject lies fully within a quadrant • Create child node for that quadrant, if necessary • Then call insert on that node (recursion) • Otherwise, AABB does not lie in just one quadrant • Add to contents list at this level • First call is Insert(root, g) • That is, start at root node for insertions

14. Finding, removing object in tree Find(Node n, IGameObject g) • Iterate through all 4 quadrants of current node (n) • Check if AABB (bounding box) of IGameObject spans multiple quadrants • Yes: IGameObject is in this node. Return node. • At maximum level of tree? • Yes: IGameObject is at this level • If AABB (bounding box) of IGameObject lies fully within a quadrant • Call find on that node (recursion) • Removing object from tree • Found_node = Find(root, g) • Found_node.objectList.Remove(g)

15. Per-pixel collision

16. Drawing Text • Create a new SpriteFont in Content folder • Add a SpriteFontmyText variable to your game • In LoadContent(), use myText =Content.Load<SpriteFont>(“SpriteFont1”) • In Draw(), spriteBatch.Begin(); spriteBatch.DrawString(myText, … ); spriteBatch.End();