Download Day 3

CS 497: Computer
Graphics
James Money
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Points and
Homogeneous Form
Given a Point in the form:
x
y
z
we can represent them as a
homogeneous point in 4D space:
x x
y y

z
z
w 1
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Homogenous Form and
Matrices
We represent a 3D coordinate in
homogenous form to allow easy
manipulation with matrices. We use 4x4
matrices to act on these points. I for
these matrices is:
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
From XYZ to
Homogeneous and Back
• To make a point into homogeneous
x
form:
y
( x, y , z ) 
z
1.0
• To make a homogeneous vector into
XYZ form again: a
b
 a / w, b / w, c / w; w  0
c
w
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Translation
To move a point from its current position
to (x+a,y+b,z+c) use the matrix:
1
0
0
0
0
1
0
0
0
0
1
0
a x xa
b y yb
 
c z
zc
1 w
w
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Scaling
To scale a point by (a,b,c):
a
0
0
0
0
b
0
0
0
0
c
0
0 x a*x
0 y b* y
 
0 z c* z
1 w
w
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Rotating in X
To rotate a point around the x axis:
1
0
0
0 cos()  sin( )
0 sin( ) cos()
0
0
0
0
0
0
1
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Rotating in Y
To rotate a point around the Y axis:
cos()
0
 sin( )
0
0 sin( )
1
0
0 cos()
0
0
0
0
0
1
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Rotating in Z
To rotate a point around the Z axis:
cos()  sin( )
sin( ) cos()
0
0
0
0
0
0
1
0
0
0
0
1
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Combining Matrices
• You can combine matrices to simplify the
processing of the points. However you must
remember, order is important!(ABBA)
• If you translate then scale, that is not
necessarily the same as scaling then
translating.
• Also, rotations must be done around (0,0,0) in
most cases. In this case, you must translate
to the origin, rotate, and then translate back.
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.
Homework
Continuation of Last Time!
•
•
•
•
•
Read Chapter 1
Read section 3.6,3.14
Read Chapter 4 if needed.
Read Chapter 5.
Read Chapter 6.
Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the author.