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SI23Introduction to Computer

Graphics

SI23Introduction to Computer

Graphics

Lecture 10 – Introduction to 3D Graphics

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Course OutlineCourse Outline

ImageDisplay

URLGIMP

colour

2D vectorgraphics

URL SVGViewer

lines,areas

graphicsalgorithms

interaction

VRMLviewer

3DGraphics

URL

surfaces

viewing, shading

GraphicsProgramming

OpenGLAPI

animation

Graphics programming

– Using OpenGL with C, C++

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2D Graphics – Picture Description

2D Graphics – Picture Description

SVG is a vector graphics description language<?xml version="1.0" ?>

<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20010904//EN" "http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd">

<svg width = "300" height = "300">

<rect x = "100" y = "100" width = "100" height = "100" style = "fill:red" />

</svg>

SVG Viewer

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2D Graphics – Programming Approach

2D Graphics – Programming Approach

For programming graphics, there are a number of APIs, or Application Programming Interfaces

OpenGL is industry standard

– Both 2D and 3D

User Programcalling OpenGL

functions

OpenGL Library

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Objectives for This PartObjectives for This Part

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

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

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Lecture Outline - The Basics

Lecture Outline - The Basics

MODELLING and VIEWING– representing objects in 3D– transforming objects and composing scenes– specifying the camera viewpoint

PROJECTION– projecting 3D scenes onto a 2D display surface

RENDERING– illumination– shading– adding realism via textures, shadows

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Basic Modelling and Viewing

Basic Modelling and Viewing

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

camera

Camera positionspecified, togetherwith its direction of view

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ProjectionProjection

Projection– 3D scene is projected onto a 2D

plane

camera

viewplane

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A PuzzleA Puzzle

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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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Creating 3D GraphicsCreating 3D Graphics

OpenGL is an API that allows us to program 3D graphics

– As well as 2D

VRML is a language that allows us to describe 3D graphics scenes

– Cf SVG for 2D

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Applications - Computer Games

Applications - Computer Games

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Applications - Computer-Aided Design

Applications - Computer-Aided Design

This is Hubble Space Telescope modeled using the BRL-CAD system

Uses CSG modeling and ray tracing for rendering

http://ftp.arl.mil/brlcad

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Applications - Virtual Reality

Applications - Virtual Reality

Virtual oceanarium built for EXPO in Lisbon

Example taken from Fraunhofer Institute site

http://www.igd.fhg.de

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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 P2

are on a plane, thenD lies in the plane

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Magnitude of a VectorMagnitude of a Vector

The magnitude of a vector V = (v1,v2,v3)T is given by:

|V| = sqrt(v1*v1 + v2*v2 + v3*v3)

eg (1,2,3)T has magnitude sqrt(14) A unit vector has magnitude 1 A unit vector in the direction of

V is V / |V|

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Scalar or Dot ProductScalar or Dot Product

The scalar product, or dot product, of two vectors U and V is defined as:

U.V = u1*v1 + u2*v2 + u3*v3

It is important in computer graphics because we can show that also:

U.V = |U|*|V|*coswhere is the angle between U and V

This lets us calculate angle ascos = (u1*v1 + u2*v2 + u3*v3) / (|U|*|V|)

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Diffuse LightingDiffuse Lighting

Diffuse reflection depends on angle between light direction and surface normal:reflected intensity = light intensity *

cosine of angle between light direction and surface normal

light normal

scalar product letsus calculate cos

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Vector or Cross ProductVector or Cross Product

The vector or cross product is defined as:UxV = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)

We can also show that:UxV = N |U||V| sin where N is unit vector orthogonal to U and V

(forming a right handed system) and is angle between U and V

This allows us to find the normal to a plane

– cross-product of two directions lying in plane , eg (P3-P2), (P2-P1), where P1, P2, P3 are three points in the plane

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ExercisesExercises

Convince yourself that the x-axis is represented by the vector (1,0,0)

What is the unit normal in the direction (2,3,4)?

What is the angle between the vectors (1,1,0) and (1,0,0)?

Which vector is orthogonal to the vectors (1,0,0) and (0,1,0)?

What is the normal to the plane through the points (1,2,3), (3,4,5) and (0,0,0)?

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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 PrimitivesOrder of VerticesTypical PrimitivesOrder of Vertices

Graphics systems such as OpenGL typically support:

– triangles, triangle strips and fans– quads, quad strips– polygons

How are vertices ordered?– convention is that vertices are ordered

counter-clockwise when looking from outside an object

– allows us to distinguish outer and inner faces of a polygon

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Complex PrimitivesComplex Primitives

OpenGL has utility libraries (GLU and GLUT) which contain various high-level primitives– Sphere, cone, torus– Polygonal representation

constructed automatically Similarly for VRML For conventional graphics

hardware:– POLYGONS RULE!

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Automatic Generation of Polygonal Objects

Automatic Generation of Polygonal Objects

3D laser scanners are able to generate computer representations of objects

– for successive heights, 2d outline generated as object rotates

– contours stitched together into 3D polygonal representation

Cyberware Cyberscanner in Med Physics at LGI able to scan human faces

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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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Exercise and Further Reading

Exercise and Further Reading

Spinning:– Work out formulae to spin an

outline (in the xy plane) about the y-axis

READING:– Hearn and Baker, Chapter 10