I wish OpenSCAD had a rotation function which operates on a vector, mimicking
the existing rotate operator which operates on 3-D model material. Like
this:

rotatev(deg_a, [x, y, z], v)

It would transform v into another vector which is rotated around the given
axis by the given number of degrees, as we are accustomed to. Even better if
it could take a list of vectors and rotate them all in the same manner.

This can be written as user code. But before I go and figure out the math
for myself, has anyone out there written it already?

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Should be quite simple.  To do individual rotations on each axis would be
like this:
http://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions
http://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions
function rotate(rv) =
[[1,0,0],[0,cos(rv[0]),-sin(rv[0])],[0,sin(rv[0]),cos(rv[0])]] *
[[cos(rv[1]),0,sin(rv[1])],[0,1,0],[-sin(rv[1]),0,cos(rv[1])]] *
[[cos(rv[2]),-sin(rv[2]),0],[sin(rv[2]),cos(rv[2]),0],[0,0,1]];
So to do arbitrary, you'd do this:
http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
identity = [[1,0,0],[0,1,0],[0,0,1]];function length(v) =
sqrt(pow(v[0],2)+pow(v[1],2)+pow(v[2],2));function unit_vector(v) =
[v[0]/length(v), v[1]/length(v), v[2]/length(v)];function uv(v) =
unit_vector(v);function rotate(a, v) = cos(a)*identity +
sin(a)*unit_vector(v) + cross((1 - cos(a))*unit_vector(v), unit_vector(v));
To join these together, you'd need a way to determine if a is an array.
There are a few ways to do this, but there's no built in function that does
this definitively that I've found.  So I wrote my own that should work in
all cases:
// FUNCTION: is_String(x)//  Returns true if x is a string, false
otherwise.function is_string(x) = x == undef || len(x) == undef ? false //
if undef, a boolean or a number : len(str(x,x)) == len(x)*2; // if an
array, this is false// FUNCTION: is_array(x)//  Returns true if x is an
array, false otherwise.function is_array(x) = is_string(x) ? false : len(x)
!= undef;
With that you can join the two rotation functions like this:
function rotate(a, v) = is_vector(a) ?
[[1,0,0],[0,cos(a[0]),-sin(a[0])],[0,sin(a[0]),cos(a[0])]] *
[[cos(a[1]),0,sin(a[1])],[0,1,0],[-sin(a[1]),0,cos(a[1])]] *
[[cos(a[2]),-sin(a[2]),0],[sin(a[2]),cos(a[2]),0],[0,0,1]] : cos(a)*identity

• sin(a)*unit_vector(v) + cross((1 - cos(a))*unit_vector(v),
  unit_vector(v));
  To do this on a list of vectors, you'd use a  for loop
  http://en.wikibooks.org/wiki/OpenSCAD_User_Manual/The_OpenSCAD_Language#For_Loop
  .
  Hope that helps.A

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Sorry, I meant:
function rotate(a, v) = is_array(a) ?
[[1,0,0],[0,cos(a[0]),-sin(a[0])],[0,sin(a[0]),cos(a[0])]] *
[[cos(a[1]),0,sin(a[1])],[0,1,0],[-sin(a[1]),0,cos(a[1])]] *
[[cos(a[2]),-sin(a[2]),0],[sin(a[2]),cos(a[2]),0],[0,0,1]] : cos(a)*identity

• sin(a)*unit_vector(v) + cross((1 - cos(a))*unit_vector(v),
  unit_vector(v));
  A

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This sort of response is why the OpenSCAD community is so
[great/fantastic/cooperative/friendly/intelligent/supportive/etc].

Unless specifically shown otherwise above, my contribution is in the Public Domain; To the extent possible under law, I have waived all copyright and related or neighbouring rights to this work. This work is published globally via the internet. :) Inclusion of works of previous authors is not included in the above.

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Oh, looks like I've not done vector algebra is a looooooooong time as I
didn't understand the notation.  For getting the arbitrary rotation, it
would look like this:
function identity(d)        = d == 0 ? 1 : [for(y=[1:d]) [for(x=[1:d]) x ==
y ? 1 : 0] ];function length(v)          = let(x=v[0], y=v[1], z=v[2])
sqrt(xx + yy + zz);function unit_vector(v)    = let(x=v[0], y=v[1],
z=v[2]) [x/length(v), y/length(v), z/length(v)];function skew_symmetric(v)
= let(x=v[0], y=v[1], z=v[2]) [[0, -z, y], [z, 0, -x], [-y, x, 0]];function
tensor_product1(u) = let(x=u[0], y=u[1], z=u[2]) [[xx, xy, xz], [xy,
yy, yz], [xz, yz, zz]];function rotate(a, v) = is_array(a) ?
let(rx=a[0], ry=a[1], rz=a[2])   [[1, 0, 0],              [0, cos(rx),
-sin(rx)], [0, sin(rx), cos(rx)]] * [[cos(ry), 0, sin(ry)],  [0, 1, 0],
[-sin(ry), 0, cos(ry)]] * [[cos(rz), -sin(rz), 0], [sin(rz), cos(rz), 0],
[0, 0, 1]] : let(uv=unit_vector(v))   cos(a)*identity(3) +
sin(a)*skew_symmetric(uv) + (1 - cos(a))*tensor_product1(uv);
If you'll note, I've updated my notation to the new one as it is more
flexible and I find clearer.  Make sure you download 2015.01.23 from the
development snapshots.  I didn't and didn't understand why some of the newer
features weren't working.
Also, you wanted:
rotatev(deg_a, [x, y, z], v)
but I gave you
rotate(a, v)
To use the rotate function I stated, you would multiply the vector that
comes out of the rotate function.  I.e.
echo(rotate(90, [1,0,0]) * [1, 0, 0]);echo(rotate(90, [1,0,0]) * [0, 1,
0]);echo(rotate(90, [1,0,0]) * [0, 0, 1]);
would give you:
ECHO: [1, 0, 0]ECHO: [0, 0, 1]ECHO: [0, -1, 0]
If you want to do against a list of vectors, you'd do this:
echo([for(i=[[1,0,0],[0,1,0],[0,0,1]]) rotate(90, [1,0,0]) * i ]);
which happens to be the same as:
echo([for(i=identity(3)) rotate(90, [1,0,0]) * i ]);
which would give you a similar result as before:
ECHO: [[1, 0, 0], [0, 0, 1], [0, -1, 0]]
This language is new to me and I've learned quite a bit from answering this
question.  Hopefully I've not totally confused you in the process. o.O :)
 
A

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I like the way you spaced the text to clearly show the rotation matrix.

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