1. Scene Graphs In 3D Graphics programming the structure used is a scene graph which is special tree structure designed to store information about a scene. Typical elements include –geometries –positional information –lights –fog

2. Recap 3D scene Camera 2D picture Content of 2D picture will depend on: camera parameters (position, direction, field of view,...), properties of scene objects, illumination,... Camera paradigm for 3D viewing 3D viewing is similar to taking picture with camera: –2D view of 3D scene

3. xwxw zwzw ywyw World coordinates Viewing Pipeline Coordinate transformations: –generation of 3D view involves sequence (pipeline) of coordinate transformations Camera Modelling coordinates 3D object 2D picture Device coordinates xmxm zmzm ymym xmxm zmzm ymym xmxm zmzm ymym xvxv zvzv yvyv Viewing coordinates

4. A scene graph is a data structure used to hold the elements that make up a scene. It may be either a tree or a Directed Acyclic Graph (DAG). The tree and DAG are similar, except that in the DAG the branches may possibly grow back together.

5. Trees Start at the root and move outward towards the leaves. Normally shown with the root at the top and the branches at the bottom. A node is a part of this tree that may have other nodes or leaves underneath it, a leaf cannot have other nodes or leaves under it. There is only one path from a leaf to the root of the tree. There are no "cycles", if you move outward along the branches, you never loop back around to end up at the root again.

6. Directed acyclic graph (DAG) Directed means that the parent-child relationship is one-way, Acyclic means that there can’t be loops, i.e. child cant be one of its own ancestors Like a tree, except maybe the branches grow back together sometimes, so that following a different sequence of branches outwards from the root might lead you to the exact same leaf. Branches never grow in a loop, though, so as long as you keep moving outwards, you always end up at a leaf eventually:

7. Nodes The scene graph contains 'nodes' such as shape, light, camera, etc. The tree structure is important because it allows the scope of influence of scene parameters to be clear and unambiguous. Nodes which have an unspecified number of children below them are known as Group nodes. One of the most important type of nodes is a Transform Group, this modifies all of the shapes below it by transforming them via a 4x4 matrix.

8. Simple scene graph Fog node Light node Root node Group node Xform node Geom node

9. Scene Graph Nodes Content Nodes –contain basic elements of a scene geometry light position fog Group Nodes –no content –link the hierarchy –allow grouping of nodes sharing a common state Parent Child #1 Parent Child #2

10. Example Hierarchy Geom Lampost Xform T2 Geom Dog Xform T1 Group “Dog” Group “Lampost” LightRoot

11. The Scene Graph The scene graph captures transformations and object-object relationships in a suitable structure: Robot BodyHead ArmTrunkLegEyeMouth Objects Instancing (i.e, a matrix) Legend World

12. Traversing the Scene Graph Traverse the scene graph in depth-first order, concatenating and undoing transforms: –For example, to render a robot Apply robot -to-head matrix Apply head -to-mouth matrix –Render mouth Un-apply head-to-mouth matrix Apply head-to-left eye matrix –Render eye Un-apply head-to-left eye matrix Apply head-to-right eye matrix –Render eye Un-apply head-to-right eye matrix Un-apply robot-to-head matrix Apply robot-to-body matrix

13. The Scene Graph in OpenGL OpenGL maintains a matrix stack of modeling and viewing transformations: ArmTrunk Leg EyeMouth HeadBody Robot Foot Matrix Stack Visited Unvisited Active

14. OpenGL: The Matrix Stack The user can save the current transformation matrix by pushing it onto the stack with glPushMatrix() The user can later restore the most recently pushed matrix with glPopMatrix() These commands really only make sense when in GL_MODELVIEW matrix mode

15. OpenGL: Matrix Stack Example glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(…); // save translation matrix: glPushMatrix(); glRotatef(…); // render something translated & rotated: glCallList(…); // restore pushed matrix, undoing rotation: glPopMatrix(); // render something else, no rotation: glCallList(…);

16. Data Structures Let’s have a look at the data structures employed in more detail Selection of data structures for computer graphics often driven by need for efficiency –storage –computation Trade-off between storage and computational efficiency often applied

17. Data Structures Data structures are required for: –scene specification object, polygon, point / vertex,... –mathematical manipulations vector, matrix, … –graphical display buffer,... Typical data structures: trees / scene graphs, linked lists, arrays

18. Data Structures Computer graphics often use hierarchical data structures, e.g. Scene ObjectsFacetsVertices Linked list of objects Linked lists of facets Linked lists of vertices Structures with x, y, z coordinates Note: other possible levels: object groups, facet groups (surfaces), edges vertex may also link back to facets which share vertex (for shading)

19. Data Structures Possible architecture Facet lists Facet M Facet 1 Vertex array Vertices Object list Object 1Object 2Object N Facet list and vertex array

20. Data Structures Possible structure for 3D point or vertex /* 3D point or vertex with integer coordinates */ typedef struct structTag3DiPoint { intxCoordinate, /* x coordinate */ intxCoordinate, /* x coordinate */ yCoordinate, /* y coordinate */ yCoordinate, /* y coordinate */ zCoordinate; /* z coordinate */ zCoordinate; /* z coordinate */ } int3DPoint, /* 3D point */ } int3DPoint, /* 3D point */ * pInt3DPoint, /* pointer to a 3D point */ * pInt3DPoint, /* pointer to a 3D point */ int3DVertex, /* 3D vertex */ int3DVertex, /* 3D vertex */ * pInt3DVertex; /* pointer to a 3D vertex */ * pInt3DVertex; /* pointer to a 3D vertex */

21. Data Structures Possible structure for polygon /* Polygon in 3D space */ typedef struct structTag3DiPolygon { int3DVertexi3SidedPoly[3]; int3DVertexi3SidedPoly[3]; intcolour, intcolour,visibilityFlag; floatmagNormal; floatmagNormal; struct structTag3DiPolygon * link2NextPolygon; struct structTag3DiPolygon * link2NextPolygon; /* Other attributes can go here */ /* Other attributes can go here */ }int3DPolygon, /* 3D Polygon */ }int3DPolygon, /* 3D Polygon */ * pInt3DPolygon, /* pointer to a 3D Polygon */ * pInt3DPolygon, /* pointer to a 3D Polygon */ int3DFacet, /* 3D facet */ int3DFacet, /* 3D facet */ * pInt3DFacet; /* pointer to a 3D facet */ * pInt3DFacet; /* pointer to a 3D facet */

22. Data Structures Possible structure for 3D object /* Object in 3D space */ typedef struct structTag3DiObject { pInt3DFacetpFacetList; pInt3DFacetpFacetList; pInt3DVertexpVertexArray; pInt3DVertexpVertexArray; intnumVertices; intnumVertices; int3DPointworldPosition; int3DPointworldPosition; struct structTag3DiObject * link2NextObject; struct structTag3DiObject * link2NextObject; /* Other attributes can go here */ /* Other attributes can go here */ } int3DObject, /* 3D Object */ } int3DObject, /* 3D Object */ * pInt3DObject; /* pointer to a 3D Object */ * pInt3DObject; /* pointer to a 3D Object */

23. Data Structures To synthesise copies of an object –master / instance architecture master defines generic attributes of object instance defines attribute values of particular copy Master Instances

24. Data Structures Possible architecture Masters Instances Object 1 tmatt Object 2 tm att Master 1 car Facet list Edge list Vertex list Object N tm att tm: transf. matrix att: attributes Master M ball Facet list Edge list Vertex list

25. Background: linear algebra Quick review of important concepts Point: location (x, y, z) Vector: direction and magnitude

26. Vectors Magnitude of a vector: |v| Direction of a vector, unit vector: v Affine sum: P = a Q + (1-a) R ^

27. Dot Product Def: u v = u x v x + u y v y + u z v z u v = |u| |v| cos θ Uses: –Angle between two vectors? –Are two vectors perpendicular? –Do two vectors form acute or obtuse angle? –Is a face visible? (backface culling)

28. Cross Product u  v = Direction: normal to plane containing u, v (using right-hand rule in right-handed coordinate system) Magnitude: |u||v| sin θ Uses: –Face outward normal? –Angle between vectors? –Do two line segments intersect?

29. Face outward normals How to find the outward normal of a face? –Assume that vertices are listed in a standard order when viewed from the outside -- counter- clockwise –Cross product of the first two edges is outward normal vector –Note that first corner must be convex

30. Surface Normals For a polygon, a surface normal can be calculated as the vector cross product of two (non- parallel) edges of the polygon. For a plane given by the equation ax + by + cz = d, the vector (a,b,c) is a normal. For a plane given by the equation r = a + αb + βc, where a is a vector to get onto the plane and b and c are non-parallel vectors lying on the plane, the normal to the plane defined is given by b × c (the cross product of the vectors lying on the plane).

31. Coordinate systems and frames Hierarchical modeling May deal with many coordinate systems: viewer, model, world, viewport Frame: origin + basis vectors (axes) Need to transform between frames E.g. reading in a world description with several objects…

32. Transformations Changes in coordinate systems usually involve –Translation –Rotation –Scale Rotation and scale can be represented as 3x3 matrices, but not translation We're also interested in a perspective transformation We use 4D "Homogeneous coordinates"

33. Homogeneous Coordinates A point: (x, y, z, w) where w is a "scale factor" Converting a 3D point to homogeneous coordinates: (x, y, z)  (x, y, z, 1) Transforming back to 3-space: divide by w –(x, y, z, w)  (x/w, y/w, z/w) (3, 2, 1): same as (3, 2, 1, 1) = (6, 4, 2, 2) = (1.5, 1, 0.5, 0.5) Where is the point (3, 2, 1, 0)? –Point at infinity or "pure direction." –Used for vectors (vs. points)

34. Homogeneous transformations Most important reason for using homogeneous coordinates: –All affine transformations (line-preserving: translation, rotation, scale, perspective, skew) can be represented as a matrix multiplication –You can concatenate several such transformations by multiplying the matrices together. Just as fast as a single transform! –Modern graphics cards implement homogeneous transformations in hardware

35. Using Modelling Software Maya 3d Studio Max VRML generators

36. Features Primitives Groups Complex, irregular shapes Lines, Points, Facets Curved surfaces Texture mapped surfaces Lighting and Shading Interactions and Behaviours

37. Primitives Facets constructed from known geometric relationships Uses polygons to map to standard mesh structure Scaling, shearing, rotation and translation used to modify vertex information, vertex ordering remains same

38. Complex or Irregular objects Manual construction –Lines and vertices positioned by hand/ eye –Modification of primitives –Extrusions Curved surfaces –B-splines –Bezier curves –Parametric meshes –etc

39. Scene view Scene hierarchy required Must have mechanism to store results Output file structure must link to internal structure –Hierarchy –Relationship between hierarchical nodes –Vertex list –Vertex ordering list –Lighting information –Texture mapping –May also hold recent viewing parameters

40. 3DS File structure May be ASCII output Tags outline structures Must read between Tags Comma delimitation usually to separate vertex list and ordering information

41. The 3ds File Structure Sometimes more efficient to read binary output Consists of chunks of binary data

42. Chunks A chunk is composed of 4 fields: –Identifier: a hexadecimal number of two byte of length that identify the chunk. With this information we can immediately realise if the chunk is useful for our purpose. If we need the chunk we extrapolate the contained information in it and, if necessary, in its children, if instead the chunk is useless we jump it using the following information... –Length of the chunk: another number, this time of 4 byte, that is the sum of the chunk length and all the lengths of every contained sub-chunks. –Chunk data: this field has a variable length. The real data of the chunk are contained in this field. –Sub-Chunks:

43. Typical structure OffsetLength 02Chunk identifier 24 Chunk length: chunk data + sub- chunks(6+n+m) 6nData 6+nmSub-chunks

44. Example MAIN CHUNK 0x4D4D 3D EDITOR CHUNK 0x3D3D OBJECT BLOCK 0x4000 TRIANGULAR MESH 0x4100 VERTICES LIST 0x4110 FACES DESCRIPTION 0x4120 FACES MATERIAL 0x4130 MAPPING COORDINATES LIST 0x4140 SMOOTHING GROUP LIST 0x4150 LOCAL COORDINATES SYSTEM 0x4160 LIGHT 0x4600 SPOTLIGHT 0x4610 CAMERA 0x4700 MATERIAL BLOCK 0xAFFF

45. Traversal If we for example want to reach the chunk VERTICES LIST –we have to read the MAIN CHUNK, –the 3D EDITOR CHUNK, –the OBJECT BLOCK –and finally the TRIANGULAR MESH.

46. Chunk Identification MAIN CHUNK Identifier0x4D4D Length0 + sub-chunks length Chunk fatherNone Sub chunks3D EDITOR CHUNK DataNone 3D EDITOR CHUNK Identifier0x3D3D Length0 + sub-chunks length Chunk fatherMAIN CHUNK Sub chunksOBJECT BLOCK, MATERIAL BLOCK, KEYFRAMER CHUNK DataNone

47. OBJECT BLOCK Identifier0x4000 LengthObject name length + sub-chunks length Chunk father3D EDITOR CHUNK Sub chunksTRIANGULAR MESH, LIGHT, CAMERA DataObject name TRIANGULAR MESH Identifier0x4100 Length0 + sub-chunks length Chunk fatherOBJECT BLOCK Sub chunksVERTICES LIST, FACES DESCRIPTION, MAPPING COORDINATES LIST DataNone VERTICES LIST Identifier0x4110 Lengthvarying + sub-chunks length Chunk fatherTRIANGULAR MESH Sub chunksNone DataVertices number (unsigned short) Vertices list: x1,y1,z1,x2,y2,z2 etc. (for each vertex: 3*float)

48. FACES DESCRIPTION Identifier0x4120 Lengthvarying + sub-chunks length Chunk fatherTRIANGULAR MESH Sub chunksFACES MATERIAL DataPolygons number (unsigned short) Polygons list: a1,b1,c1,a2,b2,c2 etc. (for each point: 3*unsigned short) Face flag: face options, sides visibility etc. (unsigned short) MAPPING COORDINATES LIST Identifier0x4140 Lengthvarying + sub-chunks length Chunk fatherTRIANGULAR MESH Sub chunksSMOOTHING GROUP LIST DataVertices number (unsigned short) Mapping coordinates list: u1,v1,u2,v2 etc. (for each vertex: 2*float)

49. Load a 3DS object Implement a "while" loop that continues its execution until the end of file is reached. For each cycle read the chunk_id and the chunk_length. Through a switch analyse the content of the chunk_id –If the chunk is a section of the tree not needed to pass then jump the whole length of the chunk moving the file pointer to the position calculated using the length of the chunk added to the current position. In this way jump the chunk and all the contained sub-chunks. –Or if the chunk enables reach of another needed chunk, or maybe it contains data that is needed, then read its data, then read the next chunk.