Subject:
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Re: Rotating a part?
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Newsgroups:
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lugnet.cad
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Date:
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Thu, 21 Jan 1999 18:17:23 GMT
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Viewed:
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841 times
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On Thu, 21 Jan 1999 15:00:57 GMT, "John VanZwieten"
<john_vanzwieten@msn.com> wrote:
> L-cad is down, and I really would like help with this:
>
> I have point A (x,y,z) and point B (x',y',z'). I want to rotate a part
> (with its insertion point at A) so that it is perpendicular to the line AB.
> How do I calculate the angles of rotation about each axis? Do I need to
> rotate on three axes, or is two sufficient?
Here's a better answer than the one I sent to L-CAD (which I assume is
still in listserv limbo). This is a quote from the
comp.graphics.algorithms FAQ. But first, info on where to get the
complete FAQ:
==============================================================================
Subject 0.03: How can I get this FAQ?
Here are a few locations that have the FAQ:
ftp://rtfm.mit.edu/pub/usenet-by-group/news.answers/graphics/algorithms-faq
ftp://ftp.seas.gwu.edu/pub/rtfm/comp/graphics/algorithms/comp.graphics.algorithms_Frequently_Asked_Questions
http://www.cis.ohio-state.edu/hypertext/faq/usenet/graphics/algorithms-faq/faq.html
The (busy) rtfm.mit.edu site lists many alternative "mirror" sites.
The ohio-state site is often out of date.
Also can reach the FAQ from
http://www.geom.umn.edu/software/cglist/,
which is worth visiting in its own right.
==============================================================================
And here's your requested information:
==============================================================================
Subject 5.01: How do I rotate a 3D point?
Assuming you want to rotate vectors around the origin of your
coordinate system. (If you want to rotate around some other point,
subtract its coordinates from the point you are rotating, do the
rotation, and then add back what you subtracted.) In 3-D, you need
not only an angle, but also an axis. (In higher dimensions it gets
much worse, very quickly.) Actually, you need 3 independent
numbers, and these come in a variety of flavors. The flavor I
recommend is unit quaternions: 4 numbers that square and add up to
+1. You can write these as [(x,y,z),w], with 4 real numbers, or
[v,w], with v, a 3-D vector pointing along the axis. The concept
of an axis is unique to 3-D. It is a line through the origin
containing all the points which do not move during the rotation.
So we know if we are turning forwards or back, we use a vector
pointing out along the line. Suppose you want to use unit vector u
as your axis, and rotate by 2t degrees. (Yes, that's twice t.)
Make a quaternion [u sin t, cos t]. You can use the quaternion --
call it q -- directly on a vector v with quaternion
multiplication, q v q^-1, or just convert the quaternion to a 3x3
matrix M. If the components of q are {(x,y,z),w], then you want
the matrix M = {{1-2(yy+zz), 2(xy-wz), 2(xz+wy)},
{ 2(xy+wz),1-2(xx+zz), 2(yz-wx)},
{ 2(xz-wy), 2(yz+wx),1-2(xx+yy)}}.
Rotations, translations, and much more are explained in all basic
computer graphics texts. Quaternions are covered briefly in
[Foley], and more extensively in several Graphics Gems, and the
SIGGRAPH 85 proceedings.
/* The following routine converts an angle and a unit axis vector
* to a matrix, returning the corresponding unit quaternion at no
* extra cost. It is written in such a way as to allow both fixed
* point and floating point versions to be created by appropriate
* definitions of FPOINT, ANGLE, VECTOR, QUAT, MATRIX, MUL, HALF,
* TWICE, COS, SIN, ONE, and ZERO.
* The following is an example of floating point definitions.
#define FPOINT double #define ANGLE FPOINT
#define VECTOR QUAT typedef struct {FPOINT x,y,z,w;}
QUAT;
enum Indices {X,Y,Z,W}; typedef FPOINT MATRIX[4][4];
#define MUL(a,b) ((a)*(b)) #define HALF(a) ((a)*0.5)
#define TWICE(a) ((a)*2.0) #define COS cos #define
SIN sin
#define ONE 1.0 #define ZERO 0.0 */
QUAT MatrixFromAxisAngle(VECTOR axis, ANGLE theta, MATRIX m) {
QUAT q; ANGLE halfTheta = HALF(theta);
FPOINT cosHalfTheta = COS(halfTheta);
FPOINT sinHalfTheta = SIN(halfTheta);
FPOINT xs, ys, zs, wx, wy, wz, xx, xy, xz, yy, yz, zz;
q.x = MUL(axis.x,sinHalfTheta); q.y =
MUL(axis.y,sinHalfTheta);
q.z = MUL(axis.z,sinHalfTheta); q.w = cosHalfTheta;
xs = TWICE(q.x); ys = TWICE(q.y); zs = TWICE(q.z);
wx = MUL(q.w,xs); wy = MUL(q.w,ys); wz = MUL(q.w,zs);
xx = MUL(q.x,xs); xy = MUL(q.x,ys); xz = MUL(q.x,zs);
yy = MUL(q.y,ys); yz = MUL(q.y,zs); zz = MUL(q.z,zs);
m[X][X] = ONE - (yy + zz); m[X][Y] = xy - wz; m[X][Z] = xz + wy;
m[Y][X] = xy + wz; m[Y][Y] = ONE - (xx + zz); m[Y][Z] = yz - wx;
m[Z][X] = xz - wy; m[Z][Y] = yz + wx; m[Z][Z] = ONE - (xx + yy);
/* Fill in remainder of 4x4 homogeneous transform matrix. */
m[W][X] = m[W][Y] = m[W][Z] = m[X][W] = m[Y][W] = m[Z][W] =
ZERO;
m[W][W] = ONE; return (q); }
/* The routine just given, MatrixFromAxisAngle, performs rotation
about
* an axis passing through the origin, so only a unit vector was
needed
* in addition to the angle. To rotate about an axis not
containing the
* origin, a point on the axis is also needed, as in the
following. For
* mathematical purity, the type POINT is used, but may be
defined as:
#define POINT VECTOR */
QUAT MatrixFromAnyAxisAngle(POINT o, VECTOR axis, ANGLE theta,
MATRIX m)
{ QUAT q; q = MatrixFromAxisAngle(axis,theta,m);
m[X][W] =
o.x-(MUL(m[X][X],o.x)+MUL(m[X][Y],o.y)+MUL(m[X][Z],o.z));
m[Y][W] =
o.y-(MUL(m[Y][X],o.x)+MUL(m[Y][Y],o.y)+MUL(m[Y][Z],o.z));
m[Z][W] =
o.x-(MUL(m[Z][X],o.x)+MUL(m[Z][Y],o.y)+MUL(m[Z][Z],o.z));
return (q); }
/* An axis can also be specified by a pair of points, with the
direction
* along the line assumed from the ordering of the points.
Although both
* the previous routines assumed the axis vector was unit length
without
* checking, this routine must cope with a more delicate
possibility. If
* the two points are identical, or even nearly so, the axis is
unknown.
* For now the auxiliary routine which makes a unit vector
ignores that.
* It needs definitions like the following for floating point.
#define SQRT sqrt #define RECIPROCAL(a) (1.0/(a)) */
VECTOR Normalize(VECTOR v) { VECTOR u;
FPOINT norm = MUL(v.x,v.x)+MUL(v.y,v.y)+MUL(v.z,v.z);
/* Better to test for (near-)zero norm before taking reciprocal.
*/
FPOINT scl = RECIPROCAL(SQRT(norm));
u.x = MUL(v.x,scl); u.y = MUL(v.y,scl); u.z = MUL(v.z,scl);
return (u); }
QUAT MatrixFromPointsAngle(POINT o, POINT p, ANGLE theta, MATRIX m)
{
QUAT q; VECTOR diff, axis;
diff.x = p.x-o.x; diff.y = p.y-o.y; diff.z = p.z-o.z;
axis = Normalize(diff);
return (MatrixFromAnyAxisAngle(o,axis,theta,m)); }
==============================================================================
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Message has 1 Reply:
 | | Re: Rotating a part?
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| This seems designed to rotate points given an angle and a rotation axis. I'm trying to determine what angle(s) to use given a line between two points. Also, I'm not a programmer, so I'm basically stuck with formulas I can use in a spreadsheet. (...) (26 years ago, 21-Jan-99, to lugnet.cad)
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Message is in Reply To:
| | Rotating a part?
|
| L-cad is down, and I really would like help with this: I have point A (x,y,z) and point B (x',y',z'). I want to rotate a part (with its insertion point at A) so that it is perpendicular to the line AB. How do I calculate the angles of rotation about (...) (26 years ago, 21-Jan-99, to lugnet.cad)
|
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