Sure. I see that my comment was about as clear as mud... :-/
In computer graphics programming we often represent a vertex as a 4D vector
{X,Y,Z,1}. The "1" is nothing more than a convenience variable used in the
matrix multiplication. The standard operations of translation, scaling, and
rotation are as follows (note: I'm using fixed width fonts):
| x' | |1 0 0 tx| |x|
translation | y' | = |0 1 0 ty| |y|
| z' | |0 0 1 tz| |z|
| 1' | |0 0 0 1| |1|
| x' | |sx 0 0 0| |x|
scaling | y' | = |0 sy 0 0| |y|
| z' | |0 0 sz 0| |z|
| 1' | |0 0 0 1| |1|
and rotation about x (where T=theta) =
| x' | |1 0 0 0| |x|
| y' | = |0 cosT -sinT 0| |y|
| z' | |0 sinT cosT 0| |z|
| 1' | |0 0 0 1| |1|
It turns out that you can combine any number of these together and do a single
4x4 matrix multiply to convert the original point into whatever projected view
you are currently using.
Hmmm... I still feel that is not very clear at all. Hmmm...
What I am trying to get at is that these transformation can be used to convert
polar notation into the conventional Cartesian g-code notation very
efficiently. Here are a couple of online presentation which might be helpful:
http://www.mcs.anl.gov/~disz/cs-341/colorvis/sld015.htm
http://www.cs.trinity.edu/~jhowland/cs2322/2d/2d/
and the end of http://en.wikipedia.org/wiki/Transformation_matrix on Affine
transformations and Perspective projections.
Hope that helps, but I kind of doubt it -- my fault for not being clear...
EBo --
Chris Radek <[email protected]> said:
> On Sun, Dec 27, 2009 at 03:48:00PM -0700, EBo wrote:
>
> > If I could add my 2c, it would be that I
> > would like to see the mathematical transformations between the proposed
> > instructions and standard g-codes.
>
> Could you elaborate on this? I don't understand what you mean.
>
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