고려대학교 컴퓨터학과 김 창 헌 10 three-dimensional object representations

고려대학교 컴퓨터학과김 창 헌

1010 Three-Dimensional Object Representations

3D Object Representation

Contents

Constructing a Solid - Sweep Representations

Constructive Solid-Geometry Methods

Octrees

BSP Trees

Fractal-Geometry Methods

Shape Grammars and Other Procedural Methods

Particle Systems

Physically Based Modeling

Visualization of Data Sets


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


Constructive Solid-Geometry Methods

CSGTo 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


CSG Methods (cont’)

Ray-Casting method to be used to imp

lement CSG to determine surface inte

rsections to sort the intersection po

ints

x

y

z

FiringPlane

Pixel Ray

x,y

z

FiringPlane

Pixel Ray

AB

CD

obj1 obj2Operation Surface Limits

Union A, DIntersection C, BDifference B, D(obj2-obj1)

Implementing CSG Operation Using Ray Casting


CSG Methods (cont’)

Ray-Casting (cont’)to be used to determine physical properties, such

as volume and mass

x

y

z

FiringPlane

Aij

zij

V A zij ij ij


Octrees

Hierarchical Tree Structure Advantage

spatial coherence to reduce storage requirements for 3D objects

storing information about object interiors

Quadtrees

10

3 2

0 1 2 3

0 1 2 3


Octrees (cont’)

Octreesfor a solid objectoperations

– union : to combine regions for each of the input objects

– intersection or difference : looking for regions of overlap

– rotation : transformation to the occupied octants

4 5

0 1

3 2

7

6

0 1 2 3 4 5 6 7


BSP Trees

Binary Space-Partitioning Treeto be similar to octreeexcept to divide space into 2 partitionsmore efficientto be useful for identifying visible surfaces and for s

pace partitioning


Fractal-Geometry Methods

Euclidean-Geometry MethodsObject shapes are described with equations.to be adequate for describing manufactured

objects

Fractal-Geometry MethodsNatural objects, such as mountains and clouds,

can be realistically described.

Fractal Object’s Basic Characteristicsinfinite detail at every pointcertain self-similarity


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



Ex) The Dragon Curve

Dragon Curve Gerneration 10

Dragon Curve and Paper folding

Dragon Fractal



Ex) The Branched Fractal



Fractal Image examples


Particle Systems

“Fluid-Like” Properties Good for Describing

objects to change over time by flowing, billowing, spattering, or expanding

clouds, smoke, fire, fireworks, waterfalls, water spray, and clumps of grass

Random Processes are Usedto generate objects within some defined region of

space to vary objects’ parameters over time


Particle Systems (cont’)

Particle Motionto be controlled by specified forces, such as a

gravity field

Particle system examples


Particle Systems (cont’)

A scene, entitled Road to Point Reyes, showing particle-system grass, fractal mountains, and texture-mapped surfaces


Physically Based Modeling

Nonrigid Objects Representationa rope, a piece of cloth, or a soft rubber ball

Hooke’s Law

x

Fx

k(unstretched position)

F F kxs x


Physically Based Modeling (cont’)

A two-dimensional spring network Modeling the flexible behavior of a banana peel with a spring network


Physically Based Modeling (cont’)

Modeling a Nonrigid Objectto set up the external forces acting on the objectto solve a set of simultaneous equations

– propagation of the forces throughout the network representing the object

Animationto 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


Visualization of Data Sets

Scientific Visualizationto 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


Visualization of Data Sets (cont’)

Visual Representation for Scalar Fieldsto use graphs or charts that the distribution of data

valuespseudo-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 10-124– an example of 3, overlapping, color-coded contour plots in xy

plane



Visual 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 10-127 )– 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 10-130 : volume visualization of medical data set



Visual Representations for Vector Fieldsto plot each data point as a small arrow

( page 400, figure 10-131 )– 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 10-135 )– for animation of fluid flow

the behavior of the vector field can be visualized by tracking particles along the flow direction



Visual Representation for Tensor Fieldsactually, 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 10-136

Visual Representations for Multivariate Data Fieldsto construct graphical objects(glyphs) with multiple

partsexample : page 403, figure 10-138

고려대학교 컴퓨터학과 김 창 헌 10 three-dimensional object representations

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