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 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 xarm l left arm rotate about shoulder upper arm translate l 0 0 rotate about 0 0 0 lower arm Vitally Important Point Every node has its own local coordinate system This makes specifying transformations much much easier 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 scale translate cube scale translate scale translate cylinder cylinder 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 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
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