"Bald Eagle" <cre### [at] netscapenet> wrote:
> "Kenneth" <kdw### [at] gmailcom> wrote:

> > The other mystery to me is what a 'basis vector' is (maybe specifically to
> > POV-Ray.) Am I correct in thinking that it's not a typical <...,...,...>-style
> > vector, but a matrix version of same?
>
> The basis vectors are just the "axis vectors" you're using.

https://en.wikipedia.org/wiki/Basis_(linear_algebra)

Imagine you had a piece of "graph paper" that had pivot points at every
intersection.  You could "tilt" it.
Rather than trying to calculate where points would be _based upon the original
position_ of the graph paper axes, you just "swivel" the y-axis so it's slanted
at a 30-degree angle, and now you can travel along that just as easy as
"translate y*10".  you do the trig to get the transform for the axis, ONCE, and
the whole rest of the space gets "calculated" for free.

Take a vector <0, 1, 0> and plot it, 20 times as long, as an arrow in POV-Ray.
Make a sphere and set it at the origin.
Then translate a copy of that sphere y*5.
Transform vector <0, 1, 0> by rotate -z*30.
draw that as an arrow 20 times as long.
Now translate the sphere along that vector by
translate (transform result)*5

You could also play with the camera vectors - right, up, direction


Sort of a variation on this idea is what I've playing with (for far too long)
here:
http://news.povray.org/povray.binaries.images/message/%3Cweb.5ba14dd7dc5d6572c437ac910%40news.povray.org%3E/#%3Cweb.5ba
14dd7dc5d6572c437ac910%40news.povray.org%3E

The bezier patch curves sort of act like a deformed rubber graph paper.
I only calculate a simple pattern to uv-map, and then the bezier patch splines -
the "local basis vectors" transform those points along its surface.  At least
that's the way I think about it in my head.  :D

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