고려대학교 컴퓨터학과 김 창 헌 10 Three-Dimensional Object Representations 

3D Object Representation Contents oConstructing a Solid - Sweep Representations oConstructive Solid-Geometry Methods oOctrees oBSP Trees oFractal-Geometry Methods oShape Grammars and Other Procedural Methods oParticle Systems oPhysically Based Modeling oVisualization of Data Sets

3D Object Representation Constructing a Solid - Sweep Representations  A sweep that moves the 2D shape through a region of space –translational, rotational, or other symmetries –2D shape : circles, rectangles, closed spline-curves –translational sweep, rotational sweep Constructing a torus using rotational sweep

3D Object Representation Constructive Solid-Geometry Methods oCSG  To create a new volume by applying the union, intersection, or difference operation to 2 specified volumes  An object designed CSG is represented with a binary tree. CSG tree example

3D Object Representation CSG Methods (cont’) oRay-Casting  method to be used to implement CSG  to determine surface intersections  to sort the intersection points x y z Firing Plane Pixel Ray x,y z Firing Plane Pixel Ray A B C D obj 1 obj 2 OperationSurface Limits UnionA, D IntersectionC, B DifferenceB, D (obj 2 -obj 1 ) Implementing CSG Operation Using Ray Casting

3D Object Representation CSG Methods (cont’) oRay-Casting (cont’)  to be used to determine physical properties, such as volume and mass x y z Firing Plane A ij  z ij

3D Object Representation Octrees oHierarchical Tree Structure oAdvantage  spatial coherence to reduce storage requirements for 3D objects  storing information about object interiors oQuadtrees

3D Object Representation Octrees (cont’) oOctrees  for a solid object  operations –union : to combine regions for each of the input objects –intersection or difference : looking for regions of overlap –rotation : transformation to the occupied octants

3D Object Representation BSP Trees oBinary Space-Partitioning Tree  to be similar to octree  except to divide space into 2 partitions  more efficient  to be useful for identifying visible surfaces and for space partitioning

3D Object Representation Fractal-Geometry Methods oEuclidean-Geometry Methods  Object shapes are described with equations.  to be adequate for describing manufactured objects oFractal-Geometry Methods  Natural objects, such as mountains and clouds, can be realistically described. oFractal Object’s Basic Characteristics  infinite detail at every point  certain self-similarity

3D Object Representation Fractal-Geometry Methods (cont’) Initiator and generator for the Koch curve First three iteration in the generation of the Koch curve Ex) The Koch Curve

3D Object Representation Fractal-Geometry Methods (cont’) Ex) The Dragon Curve Dragon Curve Gerneration 10 Dragon Curve and Paper folding Dragon Fractal

3D Object Representation Fractal-Geometry Methods (cont’) Ex) The Branched Fractal

3D Object Representation Fractal-Geometry Methods (cont’) Fractal Image examples

3D Object Representation Particle Systems o“Fluid-Like” Properties oGood for Describing  objects to change over time by flowing, billowing, spattering, or expanding  clouds, smoke, fire, fireworks, waterfalls, water spray, and clumps of grass oRandom Processes are Used  to generate objects within some defined region of space  to vary objects’ parameters over time

3D Object Representation Particle Systems (cont’) oParticle Motion  to be controlled by specified forces, such as a gravity field Particle system examples

3D Object Representation Particle Systems (cont’) A scene, entitled Road to Point Reyes, showing particle- system grass, fractal mountains, and texture-mapped surfaces

3D Object Representation Physically Based Modeling oNonrigid Objects Representation  a rope, a piece of cloth, or a soft rubber ball oHooke’s Law x FxFx k (unstretched position)

3D Object Representation Physically Based Modeling (cont’) A two-dimensional spring network Modeling the flexible behavior of a banana peel with a spring network

3D Object Representation Physically Based Modeling (cont’) oModeling a Nonrigid Object  to set up the external forces acting on the object  to solve a set of simultaneous equations –propagation of the forces throughout the network representing the object oAnimation  to more accurately describe motion paths  in the past –using spline paths and kinematics, where motion parameters are based only on position and velocity  to describe motion using dynamical equations, involving forces and accelerations

3D Object Representation Visualization of Data Sets oScientific Visualization  to visually display, enhance, and manipulate information to allow better understanding of the data  example –dealing with the output of high-volume data sources  similar methods –other nonscientific areas : business visualization

3D Object Representation Visualization of Data Sets (cont’) oVisual Representation for Scalar Fields  to use graphs or charts that the distribution of data values  pseudo-color methods –to combine color-coding techniques with graph and chart methods  contour plots –to display isolines(lines of constant scalar value)  page 397, figure –an example of 3, overlapping, color-coded contour plots in xy plane

3D Object Representation Visualization of Data Sets (cont’) oVisual Representation for Scalar Fields (cont’)  3D scalar data fields –to take cross-sectional slice –to display the 2-D data distributions over the slices  isosurfaces ( page 398, figure ) –simply 3-D contour plots  volume rendering –somewhat like an X-ray picture –in medical applications  opacity factor : bone(opaque), tissue(low opaque)  to display the accumulated opacity value as pixel-intensity level –page 399, figure : volume visualization of medical data set

3D Object Representation Visualization of Data Sets (cont’) oVisual Representations for Vector Fields  to plot each data point as a small arrow ( page 400, figure ) –to be most used with cross-sectional slices  because it can be difficult to see the data trends in a 3-D region cluttered with overlapping arrows  to plot field lines or streamlines ( page 401, figure ) –for animation of fluid flow  the behavior of the vector field can be visualized by tracking particles along the flow direction

3D Object Representation Visualization of Data Sets (cont’) oVisual Representation for Tensor Fields  actually, to be used for a second-order tensor –tensor quantity in 3D space : 3 by 3 matrix  physical tensor –physical, second-order tensors are stress and strain –physical tensor quantities are symmetric  example : page 402, figure oVisual Representations for Multivariate Data Fields  to construct graphical objects(glyphs) with multiple parts  example : page 403, figure