Re: [Algorithms] How to derive transformation matrices

[Martin G. <bz...@wi...>]

Rotation transforms are easily derived when one expands the multiplication of
a column vector A(x, y, z) by a rotation (say in the XZ plane) matrix rot( cos
a,      0,       -sin a     0 )
                                                                  (     0   ,
1,              0,    0 )
                                                                  (  sin a,
0,       cos a,    0 ).

The matrix multiplication of A' = rot3x3 * A is:

| x' = x * rot(0,0) + y * rot(0,1) + z * rot(0,2)
| y' = x * rot(1,0) + y * rot(1,1) + z * rot(1,2)
| z' = x * rot(2,0) + y * rot(2,1) + z * rot(1,2)

which is simplfied to:

| x' = x * cos a  - z * sin a
| y' = y
| z' = x * sin a  + z * cos a

which in turn can be derived by:

| x' = dist * cos ( phi + a )
| y' = y
| z' = dist * sin ( phi + a )

where dist = sqrt( x*x + z*z ) and phi is the initial rotation angle of A.

As far as I remember (did it too long ago) shearing is derived in the same way
but the equations are different.

P.S. complex rotation transformations are obtained by expressing them as
sequence of simple rotations in XY, XZ or YZ planes.

-----Original Message-----
From: Lorrimar <lor...@tr...>
To: gda...@li...
<gda...@li...>
Date: 23 Þëè 2000 ã. 13:18
Subject: [Algorithms] How to derive transformation matrices


>I'm curious on how to derive transformation matrices in 2d and 3d for things
>like shear and rotate. I've had an attempt at the maths but it didn't really
>work out. I'm wondering if anyone knows any internet resources that derive
>these matrices, and not just provide them.
>
>Cheers
>/Rodney
>
>
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