Graphics6-ViewingIn3D

7/29/2019 Graphics6-ViewingIn3D

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23New Lecture And Lab Information

Lectures: Thursday 12:00 13:00 (room tba)

Friday 15:00 16:00 (A28)

Labs: Wednesday 10:00 11:00 (A305)

Wednesday 17:00 18:00 (Aungier St. 1-005)

Sorry for all of the messing around!


2/23

Course Website:http://www.comp.dit.ie/bmacnamee

Computer Graphics 7:

Viewing in 3-D
http://www.comp.dit.ie/bmacnameehttp://www.comp.dit.ie/bmacnamee


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3

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23Contents

In todays lecture we are going to have alook at:

Transformations in 3-D

How do transformations in 3-D work? 3-D homogeneous coordinates and matrix based

transformations

Projections

History

Geometrical Constructions

Types of Projection

Projection in Computer Graphics


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Ima

gestakenfromHearn&B

aker,ComputerGraphics

withOpenGL(2004)

3-D Coordinate Spaces

Remember what we mean by a 3-Dcoordinate space

x axis

y axis

z axis

P

y

z

x

Right-Hand

Reference System


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23Translations In 3-D

To translate a point in three dimensions bydx, dy and dzsimply calculate the newpoints as follows:

x = x + dx y = y + dy z = z + dz

(x, y, z)

(x, y, z)

Translated PositionImagestakenfromHearn&B


withOpenGL(2004)


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23Scaling In 3-D

To sale a point in three dimensions bysx,syandszsimply calculate the new points asfollows:

x = sx*x y = sy*y z = sz*z

(x, y, z)

Scaled Position

(x, y, z)

Ima

gestakenfromHearn&B


withOpenGL(2004)


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23Rotations In 3-D

When we performed rotations in twodimensions we only had the choice of

rotating about thezaxis

In the case of three dimensions we havemore options

Rotate aboutx pitch

Rotate abouty yaw

Rotate aboutz- roll


8/23

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23

Ima

gestakenfromHearn&B


withOpenGL(2004)

Rotations In 3-D (cont)

x = xcos -ysin

y = xsin +ycos

z = z

x = x

y = ycos -zsin

z = ysin +zcos

x = zsin +xcos

y = y

z = zcos -xsin

The equations for the three kinds ofrotations in 3-D are as follows:


9/23

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23Homogeneous Coordinates In 3-D

Similar to the 2-D situation we can usehomogeneous coordinates for 3-D

transformations - 4 coordinate

column vectorAll transformations can

then be represented

as matrices

1

z

y

x

x axis

y axis

z axis

P

y

z

xP(x, y, z) =


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233D Transformation Matrices

1000

100

010

001

dz

dy

dx

1000

000

000

000

z

y

x

s

s

s

1000

0cos0sin

0010

0sin0cos

Translation by

dx, dy, dzScaling by

sx, sy, sz

1000

0cossin0

0sincos0

0001

Rotate About X-Axis

1000

0100

00cossin

00sincos

Rotate About Y-Axis Rotate About Z-Axis


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23Remember The Big Idea

ImagestakenfromHearn&B

aker,ComputerGraphicswithOpenGL(2004)


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23What Are Projections?

Our 3-D scenes are all specified in 3-Dworld coordinates

To display these we need to generate a 2-D

image -projectobjects onto apicture plane

So how do we figure out these projections?

Picture Plane

Objects inWorld Space


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23Converting From 3-D To 2-D

Projection is just one part of the process ofconverting from 3-D world coordinates to a

2-D image

Clip against

view volume

Project onto

projection

plane

Transform to

2-D device

coordinates

3-D world

coordinate

output

primitives

2-D device

coordinates


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23Types Of Projections

There are two broad classes of projection: Parallel: Typically used for architectural and

engineering drawings

Perspective: Realistic looking and used incomputer graphics

Perspective ProjectionParallel Projection


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23Types Of Projections (cont)

For anyone who did engineering or technicaldrawing


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23Parallel Projections

Some examples of parallel projections

Orthographic Projection

Isometric Projection


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23Isometric Projections

Isometric projections have been used incomputer games from the very early days of

the industry up to today

Q*Bert Sim City Virtual Magic Kingdom


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23Perspective Projections

Perspective projections are much morerealistic than parallel projections


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23Perspective Projections

There are a number of different kinds ofperspective views

The most common are one-point and two

point perspectives

ImagestakenfromHearn&B

aker,ComputerGraphicswithOpenGL(2004)

One Point Perspective

Projection

Two-Point

Perspective

Projection


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23Elements Of A Perspective Projection

Virtual

Camera


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23The Up And Look Vectors

The look vectorindicates the direction in

which the camera is

pointingThe up vector

determines how the

camera is rotatedFor example, is the camera held vertically or

horizontally

Up vectorLook vector

Position

Projection of

up vector


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23Summary

In todays lecture we looked at: Transformations in 3-D

Very similar to those in 2-D

Projections 3-D scenes must be projected onto a 2-D image

plane

Lots of ways to do this

Parallel projections Perspective projections

The virtual camera


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23Whos Choosing Graphics?

A couple of quick questions for you: Who is choosing graphics as an option?

Are there any problems with option time-

tabling? What do you think of the course so far?

Is it too fast/slow?

Is it too easy/hard?

Is there anything in particular you want to cover?

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