3D Transformations: Homogeneous Coordinates & Matrix Projections in Graphics, Slides of Computer Graphics

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Viewing in 3-D

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Contents

• In today’s lecture we are going to have a look at: – Transformations in 3-D

• • How do transformations in 3-D work?3-D homogeneous coordinates and matrix based
• Projectionstransformations
• • HistoryGeometrical Constructions
• • Types of ProjectionProjection in Computer Graphics Docsity.com

Translations In 3-D

• To translate a point in three dimensions by dx ,

dy follows: and dz simply calculate the new points as

• x’ = x + dx y’ = y + dy z’ = z + dz

( x, y, z ) Translated Position ( x’, y’, z’ ) Images taken from Hearn & Baker, “Computer Graphics with OpenGL” (2004) Docsity.com

Scaling In 3-D

• To sale a point in three dimensions by sx , sy

andfollows: sz simply calculate the new points as

• x’ = sxx y’ = syy z’ = sz*z

( x, y, z ) Scaled Position

( x’, y’, z’ ) Images taken from Hearn & Baker, “Computer Graphics with OpenGL” (2004) Docsity.com

Images taken from Hearn & Baker, “Computer Graphics with OpenGL” (2004)

Rotations In 3-D (cont…)

• The equations for the three kinds of rotationsin 3-D are as follows:

y’ = x·^ x’ = x· sincosθθ +^ - y·y· cossinθθ z’ = z y’ = y· cos^ x’ = x θ - z· sinθ z’ = y· sinθ + z· cosθ

x’ = z· sin y’ = y θ + x· cosθ z’ = z· cosθ - x· Docsity.comsinθ

Homogeneous Coordinates In 3-D



 

 

 

 1^ z

yx

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

transformations - 4 coordinatecolumn vector

• All transformations canthen be represented

as matrices

x axis

y axis

z axis

P

y

P( x, y, z ) =^ z x Docsity.com

Remember The Big Idea

Images taken from Hearn & Baker, “Computer Graphics with OpenGL” (2004) Docsity.com

What Are Projections?

• Our 3-D scenes are all specified in 3-D worldcoordinates

• To 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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Types Of Projections

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

• engineering drawingsPerspective: Realistic looking and used in computer graphics

Parallel Projection Perspective Projection Docsity.com

Types Of Projections (cont…)

• For anyone who did engineering or technicaldrawing

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Isometric Projections

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

the industry up to today

QBert Sim City Virtual Magic Kingdom* Docsity.com

Perspective Projections

• Perspective projections are much morerealistic than parallel projections

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Elements Of A Perspective Projection

Camera^ Virtual Docsity.com

The Up And Look Vectors

• indicates the direction inThe look vector

pointingwhich the camera is

• The determines how the up vector

camera is rotated

• For example, is the camera held vertically orhorizontally

Up vector (^) Look vector Position

Projection ofup vector

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