open-source-silicon.dev

Where is this transform matrix and the coordinate calculation from
<http://opencircuitdesign.com/magic/manpages/mag_manpage.html> :
```The transform line gives the geometric transform from coordinates of the child filename into coordinates of the cell being read. The six integers a, b, c, d, e, and f are part of the following transformation matrix, which is used to postmultiply all coordinates in the child filename whenever their coordinates in the parent are required:

a

d

0

b

e

0

c

f

1```
defined in a way that the coordinate computation becomes understandable?

<@U01732WNM60> very good question. The display on the web page is misleading.
It’s been awhile but I think this explanation is close to being accurate.
The transformation matrix should be
```a b c
d e f
0 0 1```
This is the product of the translation matrix (Tx is the x offset and Ty is the y offset)
```1 0 Tx
0 1 Ty
0 0 1 ```
the rotation matrix (θ is 0, 90, 180, or 270)
``` cosθ sinθ 0
-sinθ cosθ 0
  0    0   1```
and the reflection matrix when mirrored about the x-axis
```1  0 0
0 -1 0
0  0 1```
or the y-axis
```-1 0 0
 0 1 0
 0 0 1```
So if we let Mx be 1 normally and -1 when mirrored about the x-axis,
let My be 1 normally and -1 when mirrored about the y-axis, we can use this reflection matrix
```My 0 0
0 Mx 0
0  0 1```
and assume the transformations are applied in the translation-rotation-reflection order (the order is relevant)
and ignore any scaling (which would be another matrix)
The combined matrix variables would be
```a = cosθ・My
b = sinθ・Mx
c = Tx
d = -sinθ・My
e = cosθ・Mx
f = Ty```
This matrix is cross multiplied by a vertical matrix of the coordinates
```x
y
1```
With the transformed coordinate being
```x' = cosθ・My・x + sinθ・Mx・y + Tx
y' = -sinθ・My・x + cosθ・Mx・y + Ty```
and the third value (always 1) is ignored.

xform.png

xformnumbers.png

<@U017X0NM2E7>, thanks!
After dumping this into my <https://www.wolfram.com/mathematica/|commercial Brain Extension>, this actually starts to make sense.

Follow-up remark:
Your explanation is already _way_ better than <http://opencircuitdesign.com/magic/manpages/mag_manpage.html|the published original>,
but compared to just writing down the equation in a readable format,
and maybe one example with numbers, it's _still_ a bit chatty.

That said, good
*Job x:y*