More Object Representations Parametric instancing Hierarchical modeling Constructive Solid Geometry Sweep Objects Octrees Blobs and Metaballs and other such things Production rules

Parametric Instancing Many things, called primitives, are conveniently described by a label and a few parameters Cylinder: Radius, length, does it have end-caps, … Bolts: length, diameter, thread pitch, … Almost anything you buy from a catalogue This is a modeling format: Provide something that knows how to draw the object given the parameters Depends on the rendering style Can be an exact representation

Rendering Instances Generally, provide a routine that takes the parameters and produces a polygonal representation May include texture maps, normal vectors, colors, etc OpenGL utility library (GLu) defines routines for cubes, cylinders, disks, and other common shapes Renderman does similar things, so does POVray, … The procedure may be dynamic Adjust the polygon resolution according to distance from the viewer

OpenGL Support OpenGL defines display lists for encapsulating commands that are executed frequently list_id = glGenLists(1); glNewList(list_id, GL_COMPILE); glBegin(GL_TRIANGLES); draw some stuff glEnd(); glEndList(); And later glCallList(list_id);

Display Lists (2) The list can be: GL_COMPILE: things don’t get drawn, just stored GL_COMPILE_AND_EXECUTE: things are drawn, and also stored The list can contain almost anything: Useful for encapsulating lighting set-up commands The list cannot be modified after it is compiled The whole point is that the list may be internally optimized, which precludes modification For example, sequences of transformations may be composited into one transformation

Display Lists (3) When should you use display lists: Advantages: When you do the same thing over and over again Advantages: Can’t be much slower than the original way Can be much much faster Disadvantages: Doesn’t support real parameterized instancing, because you can’t have any parameters! Can’t use various commands that would offer other speedups For example, can’t use glVertexPointer()

Hierarchical Modeling A hierarchical model unites several parametric instances into one object For example: a desk is made up of many cubes Generally represented as a tree, with transformations and instances at any node Rendered by traversing the tree, applying the transformations, and rendering the instances Particularly useful for rigid-body animation Human is a hierarchy of body, head, upper arm, lower arm, etc… Animate by changing the transformations at the nodes

Hierarchical Model Example body Vitally Important Point: Every node has its own local coordinate system. This makes specifying transformations much much easier. xarm left arm rotate about shoulder upper arm l translate (l,0,0) rotate about (0,0,0) lower arm

OpenGL Support OpenGL defines glPushMatrix() and glPopMatrix() Takes the current matrix and pushes it onto a stack, or pops the matrix off the top of the stack and makes it the current matrix Note: Pushing does not change the current matrix Rendering a hierarchy (recursive): RenderNode(tree) glPushMatrix() Apply node transformation Draw node contents RenderNode(children) glPopMatrix()

Regularized Set Operations Hierarchical modeling is not good enough if objects in the hierarchy intersect each other Transparency will reveal internal surfaces that should not exist Computing properties like mass counts the same volume twice Solution is to define regularized set operations: Just a fancy name Every object must be a closed volume (mathematically closed) Define mathematical set operations (union, intersection, difference, complement) to the sets of points within the volume

Constructive Solid Geometry (CSG) Based on a tree structure, like hierarchical modeling, but now: The internal nodes are set operations: union, intersection or difference (sometimes complement) The edges of the tree have transformations associated with them The leaves contain only geometry Motivated by computer aided design and manufacture Difference in particular is like drilling or milling Other operations reduce the number of primitives required A common format in CAD products

CSG Example Fill it in! - - cube   cylinder cylinder scale translate

Rendering CSG Normals and texture coordinates typically come from underlying primitives (not a big deal with CAD) Some rendering algorithms can render CSG directly Raytracing (later in the course) Scan-line with an A-buffer Can do 2D with tesselators in OpenGL For OpenGL and other polygon renderers, must convert CSG to polygonal representation Must remove redundant faces, and chop faces up Basic algorithm: Split polygons until they are inside, outside, or on boundary. Then choose appropriate set for final answer. Generally difficult, messy and slow Numerical imprecision is the major problem

CSG Summary Advantages: Disadvantages: Good for describing many things, particularly machined objects Better if the primitive set is rich Early systems used quadratic surfaces Moderately intuitive and easy to understand Disadvantages: Not a good match for polygon renderers Some objects may be very hard to describe, if at all Geometric computations are sometimes easy, sometimes hard A volume representation (hence solid in the name) Boundary (surface representation) can also work

Sweep Objects Define a polygon by its edges Sweep it along a path The path taken by the edges form a surface - the sweep surface Special cases Surface of revolution: Rotate edges about an axis Extrusion: Sweep along a straight line

General Sweeps The path maybe any curve The polygon that is swept may be transformed as it is moved along the path Scale, rotate with respect to path orientation, … One common way to specify is: Give a poly-line (sequence of line segments) as the path Give a poly-line as the shape to sweep Give a transformation to apply at the vertex of each path segment Difficult to avoid self-intersection

Rendering Sweeps Convert to polygons Break path into short segments Create a copy of the sweep polygon at each segment Join the corresponding vertices between the polygons May need things like end-caps on surfaces of revolution and extrusions Normals come from sweep polygon and path orientation Sweep polygon defines one texture parameter, sweep path defines the other

Spatial Enumeration Basic idea: Describe something by the space it occupies For example, break the volume of interest into lots of tiny cubes, and say which cubes are inside the object Works well for things like medical data The process itself, like MRI or CAT scans, enumerates the volume Data is associated with each voxel (volume element) Problem to overcome: For anything other than small volumes or low resolutions, the number of voxels explodes Note that the number of voxels grows with the cube of linear dimension

Octrees (and Quadtrees) Build a tree where successive levels represent better resolution (smaller voxels) Large uniform spaces result in shallow trees Quadtree is for 2D (four children for each node) Octree is for 3D (eight children for each node)

Quadtree Example top left top right bot left bot right Octree principle is the same, but there are 8 children

Rendering Octrees Volume rendering renders octrees and associated data directly A special area of graphics, visualization, not covered in this class Can convert to polygons by a few methods: Just take faces of voxels that are on the boundary Find iso-surfaces within the volume and render those Typically do some interpolation (smoothing) to get rid of the artifacts from the voxelization Typically render with colors that indicate something about the data, but other methods exist

Spatial Data Structures Octrees are an example of a spatial data structure A data structure specifically designed for storing information of a spatial nature eg Storing the location of fire hydrants in a city In graphics, octrees are frequently used to store information about where polygons, or other primitives, are located in a scene Speeds up many computations by making it fast to determine when something is relevant or not Just like BSP trees speed up visibility Other spatial data structures include BSP trees, KD-Trees, Interval trees, …