Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!cs.utexas.edu!usc!ucsd!ames!sun-barr!decwrl!sgi!bruceh@brushwud.sgi.com
From: bruceh@brushwud.sgi.com (Bruce R. Holloway)
Newsgroups: comp.graphics
Subject: Re: Question
Message-ID: <45596@sgi.sgi.com>
Date: 2 Dec 89 03:02:58 GMT
References: <871@ccssrv.UUCP>
Sender: bruceh@brushwud.sgi.com
Organization: Silicon Graphics, Inc., Mountain View, CA
Lines: 33

In article <871@ccssrv.UUCP>, brian@ccssrv.UUCP (Brian Lantz) writes:
> 
> The transformation matrix:
> 
>    | cos(a)  -sin(b)    0 |
>    | sin(a)   cos(b)    0 |             a != b
>    |     0        0     1 |
> 
> has me mystified.  In empirical tests with lines, it rotates through
> a different angle depending on the slope of the line.  I can see that
> that property should be expected after having it pushed in my face,
> but I still don't have an intuitive grasp of the full function of
> this transform.  Can it be constructed as the product of the basic
> translate, scale, and rotate transforms?  How can one summerize
> its characteristics? All answers will be appreciated.

For what it's worth, the matrix can be factored as

   | cos(a)  -sin(a)   0|   | 1   sin(a)cos(b) - sin(b)cos(a)   0 |
 = | sin(a)   cos(a)   0| * | 0   sin(a)sin(b) + cos(a)cos(b)   0 |
   |     0        0    1|   | 0                0                1 |

   | cos(a)  -sin(a)   0|   | 1   sin(a - b)   0 |
 = | sin(a)   cos(a)   0| * | 0   cos(a - b)   0 |
   |     0        0    1|   | 0       0        1 |

   | cos(a)  -sin(a)   0|   | 1   tan(a - b)   0 |   | 1       0        0 |
 = | sin(a)   cos(a)   0| * | 0       1        0 | * | 0   cos(a - b)   0 |,
   |     0        0    1|   | 0       0        1 |   | 0       0        1 |

so it is the product of a rotation by angle a, a shear which depends on
(a - b), and a minification of y which depends on (a - b).  Thus,
there is no translation or uniform scaling.


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