Viewing Transformations: Homogeneous Coordinates and 3D Viewing, Slides of Computer Graphics

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Viewing Transformations

Lecture Set 11

Homogeneous Coordinates

An infinite number of points correspond to (x,y,1). They constitute the whole line (tx,ty,t).

x

w

y

w = 1

(tx,ty,t) (x,y,1) Docsity.com

Simple Viewing TransformationExample

Points A B C D E F G H X -1 1 1 -1 -1 1 1 - Y 1 1 -1 -1 1 1 -1 - Z -1 -1 -1 -1 1 1 1 1

Simple Cube Viewedfrom (6,8,7.5)

x

z

A=(-1,1,-1) y

C=(1,-1,-1) B=(1,1,-1)

D=(-1,-1,-1)^ G=(1,-1,1)

H=(-1,-1,1) F=(1,1,1) E=(-1,1,1)

Topology of Cube

A: B: (^) A B (^) D C E F C: D: (^) AB DC GH E: F: AB EF HG G: H: (^) DC (^) EF HG C^ B

H E D

G F A

Simple Example

• • Give a Cube with cornersView from Eye Position (6,8,7.5)
• • Look at Origin (0,0,0)“Up” is in z-direction

Simple Viewing Transformation Example

00 00 01 17.^5

T 1 01 10 00 86

x

z

y

z ˆ e y ˆ e x ˆ e

Build LH Coord with T 2

(6,8,0)

x ˆ

z ˆ y ˆ

x

z

y

ze ye xe

Rotate about y with T 3

 (6,8,0) 6 8

10

z ˆ y ˆ x ˆ

Simple Viewing Transformation Example



 

 

 

  

. 06 00. 08 10 T 3.^08100.^600

sin( )^610

cos( )^810





where

Look at the (3-4-5)Right Triangle



10

sin( )^35

cos( )^45





(4) (3)

Simple Viewing Transformation Examle



 

 

 

  000 . 08. 6 .. 086 100

1 0 0 0 T 4

sin( )^35

cos( )^45





where

Map to canonical frustum

  45 20

20

Scale x,y by 2 for normalization



 

 

 

  00 00 01 10

N^02020000

Will view a 20”x20” screen from 20” away. Scale to standard viewing frustum.