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Basic Modelling

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Lecture ObjectivesLecture Objectives

To understand how 3D scenes can be modelled - in terms of geometry, appearance and behaviour - and rendered on a display

To understand how to deliver interactive animated 3D graphics over the Internet

To be able to create interactive 3D graphics applications using industry standard software (OpenGL and VRML)

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The BasicsThe Basics

MODELLING– representing objects in 3D– transforming objects and composing scenes

VIEWING– projecting 3D scenes onto a 2D display

surface RENDERING

– illumination– shading– adding realism via textures, shadows

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Basic ModellingBasic Modelling

x

y

z

objects representedas set of faces - iepolygons- and facesas a set of points

scenes composedby scaling, rotating,translating objects tocreate a 3D world

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ViewingViewing

Clipping– selects a volume of interest– Projection– 3D scene is projected onto a 2D

plane

camera

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RenderingRendering

shading:how do we use ourknowledge of illuminationto shade surfaces in ourworld?

illumination:how is light reflectedfrom surfaces?

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RenderingRendering

texture

shadows

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InternetInternet

VRML– ISO standard for 3D graphics over

the Web– allows modelling of geometry,

appearance and behaviour

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Advanced RenderingAdvanced Rendering

ADVANCED RENDERING– direct versus global illumination

methods– ray tracing and radiosity

OTHER ADVANCED FEATURES– curve and surface modelling– image based rendering– non-photorealistic rendering

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Advanced RenderingAdvanced Rendering

Advanced Rendering - global illumination– ray tracing

– radiositybased on physics of radioactive heat

transfer between surfaces

light

eye

screen

objects

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Before we begin...mathematics!Before we begin...mathematics!

3D Co-ordinate Systems

LEFT RIGHT

x

yz

x

y

z

z points away z points toward

Align thumb with x, first finger with y, then second fingerof appropriate hand gives z direction. Common now touse a RIGHT HANDED system.

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Points and VectorsPoints and Vectors

We shall write points as column vectors

xyz

P =

Difference of two points gives a direction vector:D = P2 - P1

x

y

z

P2

P1

x

y

z

P

Note: If P1 and P2are on a plane, thenD lies in the plane

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Polygonal RepresentationPolygonal Representation

Any 3D object can be represented as a set of plane, polygonal surfaces

V1

V2V3

V4

V5V8

V7 V6

Note: each vertex part of severalpolygons

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Polygonal RepresentationPolygonal Representation

Objects with curved surfaces can be approximated by polygons - improved approximation by more polygons

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Scene OrganisationScene Organisation

Scene = list of objects Object = list of surfaces Surface = list of polygons Polygon = list of vertices

scene

objectsurfaces polygons

vertices

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Polygon Data StructurePolygon Data Structure

V1

V2V3

V4

V5V8

V7 V6

P1

P2

Object Table

Obj1P1, P2, P3,P4, P5, P6

Object Obj1

Vertex Table

V1X1, Y1, Z1

V2X2, Y2, Z2

. ...

Polygon Table

P1 V1, V2, V3, V4

P2 V1, V5, V6, V2

. ...

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Typical PrimitivesTypical Primitives

Graphics systems such as OpenGL typically support:– triangles, triangle strips and fans– quads, quad strips– polygons

Which way is front?– convention is that normal points

towards you if vertices are specified counter-clockwise

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Modelling Regular ObjectsModelling Regular Objects

Sweeping

Spinning

2D Profilesweep axis

spinning axis

R1 R2

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Sweeping a Circle to Generate a Cylinder as Polygons

Sweeping a Circle to Generate a Cylinder as Polygons

vertices at z=0

vertices at z=depthV1

V2

V3V4

V5

V6 V8

V7

V10

V9

V11

V12V13

V14

V15V16

V17

V18

V1[x] = R; V1[y] = 0; V1[z] = 0V2[x] = R cos ; V2[y] = R sin ; V2[z] = 0 (=/4)Vk[x] = R cos k; Vk[y] = R sin k; Vk[z] = 0wherek = 2 (k - 1 )/8, k=1,2,..8

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Complex PrimitivesComplex Primitives

Some systems such as VRML have cylinders, cones, etc as primitives– polygonal representation calculated

automatically OpenGL has a utility library (GLU)

which contains various high-level primitives– again converted to polygons

For conventional graphics hardware:– POLYGONS RULE!

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Automatic Generation of Polygonal Objects

Automatic Generation of Polygonal Objects

3D scanners - or laser rangers - are able to generate computer representations of objects– object sits on rotating table– contour outline generated for a

given height– scanner moves up a level and next

contour created– successive contours stitched

together to give polygonal representation

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A PuzzleA Puzzle

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Modelling Objects and Creating Worlds

Modelling Objects and Creating Worlds

We have seen how boundary boundary representations representations of simple objects can be created

Typically each object is created in its own co-ordinate systemco-ordinate system

To create a world, we need to understand how to transform objects so as to place them in the right place - translationtranslation, at the right size - scalingscaling, in the right orientation- rotationrotation

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TransformationsTransformations

The basic linear transformations are:– translation: P = P + T, where T is

translation vector– scaling: P’ = S P, where S is a scaling

matrix– rotation: P’ = R P, where R is a rotation

matrix As in 2D graphics, we use

homogeneoushomogeneous co-ordinates in order to express all transformations as matrices and allow them to be combined easily

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Homogeneous Co-ordinatesHomogeneous Co-ordinates

In homogeneous coordinates, a 3D point P = (x,y,z)T

is represented as:P = (x,y,z,1)T

That is, a point in 4D space, with its ‘extra’ co-ordinate equal to 1

NoteNote: in homogeneous co-ordinates, multiplication by a constant leaves point unchanged– ie (x, y, z, 1)T = (wx, wy, wz, w)T

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TranslationTranslation

Suppose we want to translate P (x,y,z)T by a distance (Tx, Ty, Tz)T

We express P as (x, y, z, 1)T and form a translation matrix T as below

The translated point is P’

T P

x’y’z’1

P’ =

1 0 0 Tx0 1 0 Ty0 0 1 Tz0 0 0 1

xyz1

= x + Txy + Tyz + Tz1

=

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ScalingScaling

Scaling by Sx, Sy, Sz relative to relative to the originthe origin:

x’y’z’1

Sx 0 0 00 Sy 0 00 0 Sz 00 0 0 1

xyz1

P’ = S P

= = Sx . xSy . ySz . z1

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RotationRotation

Rotation is specified with with respect to an axis respect to an axis - easiest to start with co-ordinate axes

To rotate about the x-axis:

a positive angle corresponds to counter-clockwise direction lookingat origin from positive position on axis

x’y’z’1

= 1 0 0 00 cos -sin 00 sin cos 00 0 0 1

xyz1

P’ = Rz () P

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Composite TransformationsComposite Transformations

The attraction of homogeneous co-ordinates is that a sequence of transformations may be encapsulated as a single matrix

For example, scaling with respect to a with respect to a fixed position (a,b,c) fixed position (a,b,c) can be achieved by:– translate fixed point to origin- say, T(-a,-b,-c)– scale- S– translate fixed point back to its starting

position- T(a,b,c) Thus: P’ = T(a,b,c) S T(-a,-b,-c) P = M P

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Rotation about a Specified AxisRotation about a Specified Axis

It is useful to be able to rotate about any axis in 3D space

This is achieved by composing 7 elementary transformations

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Rotation through about Specified Axis

Rotation through about Specified Axis

x

y

z

x

y

zrotate throughrequ’d angle,

x

y

z

x

y

z

P2

P1x

y

z

P2

P1x

y

z

initial positiontranslate P1to origin

rotate so that P2 lies on z-axis(2 rotations)

rotate axisto orig orientation

translate back

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Inverse TransformationsInverse Transformations

As in this example, it is often useful to calculate the inverse of a transformation– ie the transformation that returns to

original state Translation: T-1 (a, b, c) = T (-a, -

b, -c) Scaling: S-1 ( Sx, Sy, Sz ) =

S ............ Rotation: R-1

z () = Rz (-)

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Viewing - Projections

SEN 992 Computer Graphics ISEN 992 Computer Graphics I

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ViewingViewing

Graphics display devices are 2D rectangular screens

Hence we need to understand how to transform our 3D world to a 2D surface

This involves:– selecting the observer position observer position (or

camera position)– selecting the view plane view plane (or camera film

plane)– selecting the type of projectionprojection

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Perspective ProjectionsPerspective Projections

There are two types of projection: perspectiveperspective and parallelparallel

In a perspectiveperspective projection, object positions are projected onto the view plane along lines which converge at the observerP1

P2

P1’

P2’

view plane

camera

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Parallel ProjectionParallel Projection

In a parallel projection, the observer position is at an infinite distance, so the projection lines are parallelP1

P2

view plane

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Perspective and Parallel Projection

Perspective and Parallel Projection

Parallel projection preserves the relative proportions of objects, but does not give a realistic view

Perspective projection gives realistic views, but does not preserve proportions– Projections of distant objects are

smaller than projections of objects of the same size which are closer to the view plane

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Perspective and Parallel Projection

Perspective and Parallel Projection

perspective parallel

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Viewing Coordinate SystemViewing Coordinate System

Viewing is easier if we work in a viewing co-ordinate systemviewing co-ordinate system, where the observer or camera position is on the z-axis, looking along the negative z-direction

xV

yV

zV

Camera is positioned at:(0 , 0, zC)

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View PlaneView Plane

We assume the view plane is perpendicular to the viewing direction

The view planeis positioned at:(0, 0, zVP)

Let d = zC - zVP be thedistance between thecamera and the plane

xv

yv

zv

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Perspective Projection CalculationPerspective Projection Calculation

xv

yv

zv

zVview plane

Q

camerayV

zCzQ zVP

looking along x-axis

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Perspective Projection CalculationPerspective Projection Calculation

zVview plane

Q

camerayV

P

By similar triangles, yP / yQ = (zC - zVP) / (zC - zQ)and soyP = yQ * (zC - zVP) / (zC - zQ)oryP = yQ * d / (zC - zQ)

zCzQ zVP

xP likewise

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Transformation Matrix for Perspective

Transformation Matrix for Perspective

1 0 0 0

0 1 0 0

0 0 -zVP/d zVPzC/d

0 0 -1/d zC/d

xQ

yQ

zQ

1

xH

yH

zH

wH

=

Then xP = xH / wH

iexP = xH / ( (zC - zQ) / d )iexP = xQ / ( (zC - zQ) / d )

yP likewise

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Note for LaterNote for Later

The original z co-ordinate of points is retained – we need relative depth in the scene

in order to sort out which faces are visible to the camera

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Vanishing PointsVanishing Points

When a 3D object is projected onto a view plane using perspective, parallel lines in object NOT parallel to the view plane converge to a vanishing vanishing pointpoint

view plane

vanishing point

one-pointperspectiveprojectionof cube

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One- and Two-Point Perspective Drawing

One- and Two-Point Perspective Drawing

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One-point PerspectiveOne-point Perspective

Said to be the firstpainting in perspective

This is:Trinity with the Virgin,St John and Donors,by Mastaccio in 1427

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Two-point PerspectiveTwo-point Perspective

EdwardHopperLighthouseat Two Lights

-seewww.postershop.com

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Parallel Projection - Two typesParallel Projection - Two types

OrthographicOrthographic parallel projection has view plane perpendicular to direction of projection

ObliqueOblique parallel projection has view plane at an oblique angle to direction of projection

P1

P2

view plane

P1

P2

view plane

We shall only consider orthographic projectionorthographic projection

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Parallel Projection Calculation

Parallel Projection Calculation

xv

yv

zv

zVview plane

Q

yV

zQ zVP

looking along x-axis

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Parallel Projection CalculationParallel Projection Calculation

zVview plane

Q

yV

P

yP = yQ

and similarly xP = xQ

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Parallel Projection CalculationParallel Projection Calculation

So this is much easier than perspective!– xP = xQ

– yP = yQ

– zP = zVP

The transformation matrix is simply1 0 0 0

0 1 0 00 0 zVP/zQ 00 0 0 1

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View Volumes - View WindowView Volumes - View Window

Type of lens in a camera is one factor which determines how much of the view is captured– wide angle lens captures more than

regular lens Analogy in computer graphics is the

view windowview window, a rectangle in the view plane

xv

yv

zv

view window

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View Volume - Front and Back Planes

View Volume - Front and Back Planes

We will also typically want to limit the view in the zV direction

We define two planes, each parallel to the view plane, to achieve this– front plane (or near plane)– back plane (or far plane)

front planeback plane

zV

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View Frustum - Perspective Projection

View Frustum - Perspective Projection

view window

backplane

frontplane

camera

view frustum

zV

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View Volume - Parallel ProjectionView Volume - Parallel Projection

view window

backplane

frontplane

zV

view volume

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View VolumeView Volume

The front and back planes act as important clipping planesclipping planes

Can be used to select part of a scene we want to view

Front plane Front plane important in perspective to remove near objects which will swamp picture

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VRML - An Introduction

SEN 992 Computer Graphics ISEN 992 Computer Graphics I

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Good Ideas are SimpleGood Ideas are Simple

Realization 1:– Hypertext + Internet = World Wide Web

Realization 2:– Adding images makes pages more interesting

Realization 3:– Images are pictures taken by the publisher -

why not send 3D scenes and allow the user to take the picture!

VRML: Virtual Reality Modelling Language– a language to describe 3D worlds - for the

Web

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World Wide Web PagesWorld Wide Web Pages

AUTHORINGPROCESS HTML

file

server

INTERNET

client

WEBBROWSER

text/html

.htmlMIMEtype

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VRMLVRML

AUTHORINGPROCESS VRML

file

server

INTERNET

client

VRMLBROWSER

x-world/x-vrml

.wrl

Major browser:CosmoPlayer

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Advantages of VRMLAdvantages of VRML

Transferring a 3D model - rather than a 2D image - to the browser has great advantages– viewer can choose how to look at

the model - or world– viewer can navigate the world– file size can often be much less

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How Did It All Begin?How Did It All Begin?

Original brainwave– Mark Pesce and Tony Parisi - early 1994– vision of a VR interface to the Web

VRML 1.0– practical realisation based on Open

Inventor (Silicon Graphics toolkit - similar to OpenGL but higher level)

– Open Inventor file format PLUS hyperlinks

– static, non-interactive worlds– Gavin Bell (SGI) - late 1994

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Where are we now?Where are we now?

VRML97– adds object behaviours and

interaction to allow creation of dynamicdynamic worlds

– ISO standard - December 1997– Gavin Bell, Rikk Carey (ex-SGI) and

Chris Marrin (SGI) 2000

– being reworked as XML (X3D)– many new developments

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A VRML FileA VRML File

VRML file consists of:

header nodes

– objects such as cylinders and spheres

– operations such as transformations

fields– parameters of

nodes

#VRML V2.0 utf8Shape {

geometry Cylinder {

radius3height 6

}}

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As Seen By A BrowserAs Seen By A Browser

#VRML V2.0 utf8

Shape {

geometry Cylinder {

radius2

height 4

}

}

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Adding Color to the SceneAdding Color to the Scene

#VRML V2.0 utf8

Shape {

geometry Cylinder {

radius 2

height 4

}

appearance Appearance {

material Material {

diffuseColor 1 0 0

specularColor 1 1 1 }

}

}

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Co-ordinate SystemsCo-ordinate Systems

Each node is drawn within its own local co-ordinate system..

x

z

y

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Modelling TransformationsModelling Transformations

...and can be subjected to a modelling transformation to scale, rotate, translate

Transform is a VRML node - treated just like an object

It applies to a group of children nodes

#VRML V2.0 utf8

Transform{

scale 5.0 0.5 1.0

children [

Shape {

geometry Cylinder {

radius 2

height 4

}

appearance Appearance {

material Material {

diffuseColor 1 0 0

specularColor 1 1 1 }

} } ] }

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Hierarchical StructureHierarchical Structure

#VRML V2.0 utf8

Transform{

scale 5.0 0.5 1.0

children [

Shape {

geometry Cylinder {

radius 2

height 4

}

appearance Appearance {

material Material {

diffuseColor 1 0 0

specularColor 1 1 1 }

} } ] }

TRANSFORM

SHAPE

parent

child

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Hierarchical StructureHierarchical Structure

This generalises to allow nodes to appear in a hierarchy

This is known as the VRML scene scene graphgraph

TRANSFORM

SHAPE TRANSFORM

SHAPE SHAPE

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TransformationsTransformations

Here we see Cylinder, Sphere and Cone nodes

Each has been subjected to a modelling transformation in order to position them

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InstancesInstances

A node can be given a ‘name’ and then used multiple times - with different transformations applied to each

DEF kwb Shape {...} gives it a name

USE kwb allows it to be included at other points in the scene graph

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AnchorsAnchors

A piece of geometry can act as a link to another URL

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TexturesTextures

Texture mapping: Images can be mapped to geometry to provide texturing

VRML looks like:Shape{

geometry Sphere { }

appearance Appearance{

texture ImageTexture{

url “http://..../kwb.gif”}

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Polygonal SurfacesPolygonal Surfaces

The general primitive for drawing polygonal surfaces is:

– IndexedFaceSet– coord field lists

the points– coordIndex

describes the polygons

Shape{geometry IndexedFaceSet {

coord Coordinate {point [ 17.5 11.2 -

1.2,17.5 15.0 -

1.2,… ]}

coordIndex [ 0 1 2 3 -1,4 1 0 5 -1,…]

}}

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LightsLights

VRML includes:– DirectionalLight– PointLight– SpotLight

Note # sign is a comment

Example:

PointLight{

on TRUE

intensity 0.75

color 1 0 0 #red

location 0 0 0

radius 100

}

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Viewing Viewing

Specified via the Viewpoint node

VRML file can have a number of Viewpoint nodes - browsers allow a user to jump from one viewpoint to another

Orientation = axis + angle of rotation

Example:

Viewpoint {

position 0 0 5

orientation 0 0 1 0.3

description“frontview”

}

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Other Basic FeaturesOther Basic Features

Inline node– incorporate VRML code from

another url– this allows us to composecompose large

worlds from many smaller ones Level of Detail

– different geometric representations depending on distance

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Information About VRMLInformation About VRML

Web3D Consortium– http://www.web3d.org– links to specifications and resources

Web3D Information– http://web3d.about.com/compute/web3d

Leeds University VRML/Java3D Information Centre – http://www.scs.leeds.ac.uk/vrmljava3d

VRML Repository– http://www.sdsc.edu/vrml/

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Range of books now available– The VRML 2.0 Handbook (Addison Wesley)

Jed Hartman and Josie Wernecke

http://vrml.sgi.com/handbook– The Annotated VRML Reference Book (Addison

Wesley)

Rikk Carey and Gavin Bell

http://www.best.com/~rikk/Book/ Tutorials include:

– Floppy’s guide: www.vapourtech.com/vrmlguide

Books and TutorialsBooks and Tutorials