Sudipta Mondal

Computer Graphics 7:Transformation &

Viewing in 3-D

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3-D Coordinate Spaces

Remember what we mean by a 3-Dcoordinate space

x axis

y axis

z axis

P

y

zx

Right-HandReference System

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Translations In 3-D

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

x’ = x + dx y’ = y + dy z’ = z + dz

(x’, y’, z’)(x, y, z)

Translated Position

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Scaling In 3-D

To sale a point in three dimensions by sx, syand sz simply calculate the new points asfollows:

x’ = sx*x y’ = sy*y z’ = sz*z

(x, y, z)

Scaled Position

(x’, y’, z’)

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Rotations In 3-D

When we performed rotations in twodimensions we only had the choice ofrotating about the z axisIn the case of three dimensions we havemore options

– Rotate about x – pitch– Rotate about y – yaw– Rotate about z - roll

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Rotations In 3-D (cont…)

x’ = x·cosθ - y·sinθy’ = x·sinθ + y·cosθ

z’ = z

x’ = xy’ = y·cosθ - z·sinθz’ = y·sinθ + z·cosθ

x’ = z·sinθ + x·cosθy’ = y

z’ = z·cosθ - x·sinθ

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

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Homogeneous Coordinates In 3-D

Similar to the 2-D situation we can usehomogeneous coordinates for 3-Dtransformations - 4 coordinatecolumn vectorAll transformations canthen be representedas matrices

úúúú

û

ù

êêêê

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1zyx

x axis

y axis

z axis

P

y

zxP(x, y, z) =

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3D Transformation Matrices

úúúú

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1000100010001

dzdydx

úúúú

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1000000000000

z

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x

ss

s

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û

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-10000cos0sin00100sin0cos

qq

qq

Translation bydx, dy, dz

Scaling bysx, sy, sz

úúúú

û

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êêêê

ë

é-

10000cossin00sincos00001

qqqq

Rotate About X-Axis

úúúú

û

ù

êêêê

ë

é -

1000010000cossin00sincos

qqqq

Rotate About Y-Axis Rotate About Z-Axis

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What Are Projections?

Our 3-D scenes are all specified in 3-Dworld coordinatesTo display these we need to generate a 2-Dimage - project objects onto a picture plane

So how do we figure out these projections?

Picture Plane

Objects inWorld Space

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Converting From 3-D To 2-D

Projection is just one part of the process ofconverting from 3-D world coordinates to a2-D image

Clip againstview volume

Project ontoprojection

plane

Transform to2-D devicecoordinates

3-D worldcoordinate

outputprimitives

2-D devicecoordinates

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Types Of Projections

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

engineering drawings– Perspective: Realistic looking and used in

computer graphics

Perspective ProjectionParallel Projection

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Types Of Projections (cont…)

For anyone who did engineering or technicaldrawing

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Parallel Projections

Some examples of parallel projections

Orthographic Projection

Isometric Projection

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Isometric Projections

Isometric projections have been used incomputer games from the very early days ofthe industry up to today

Q*Bert Sim City Virtual Magic Kingdom

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Perspective Projections

Perspective projections are much morerealistic than parallel projections

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Perspective Projections

There are a number of different kinds ofperspective viewsThe most common are one-point and twopoint perspectives

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One Point PerspectiveProjection

Two-PointPerspectiveProjection

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Elements Of A Perspective Projection

VirtualCamera

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The Up And Look Vectors

The look vectorindicates the direction inwhich the camera ispointingThe up vectordetermines how the

camera is rotatedFor example, is the camera held vertically orhorizontally

Up vectorLook vector

Position

Projection ofup vector