CODEWITHSUNDEEP
mathgraphics3d
HT XU
HT XU

Reputation: 187

Rotation About an Arbitrary Axis in 3 Dimensions Using Matrix

I come accross a math problem about Interactive Computer Graphics. I summarize and abstract this problem as follows:

I'm going to rotation a 3d coordinate P(x1,y1,z1) around a point O(x0,y0,z0)

and there are 2 vectors u and v which we already know.

u is the direction to O before transformation.

v is the direction to O after transformation.

I want to know how to conduct the calculation and get the coordinate of Q

enter image description here

Thanks a lot.

Solution:

Rotation About an Arbitrary Axis in 3 Dimensions using the following matrix:

rotation axis vector (normalized): (u,v,w)

position coordinate of the rotation center: (a,b,c)

rotation angel: theta

enter image description here

Reference:

https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxnbGVubm11cnJheXxneDoyMTJiZTZlNzVlMjFiZTFi

Upvotes: 0

Views: 4130

Answers (4)

Spektre
Spektre

Reputation: 51923

for just single point no rotations is needed ... so knowns are:

u,v,O,P

so we now the distance is not changing:

|P-O| = |Q-O|

and directions are parallel to u,v so:

Q = O + v*(|P-O|/|v|)

But I suspect you want to construct rotation (transform matrix) such that more points (mesh perhaps) are transformed. If that is true then you need at least one known to get this right. Because there is infinite possible rotations transforming P -> Q but the rest of the mesh will be different for each ... so you need to know at least 2 non trivial points pair P0,P1 -> Q0,Q1 or axis of rotation or plane parallel to rotation or any other data known ...

Anyway in current state you can use as rotation axis vector perpendicular to u,v and angle obtained from dot product:

axis = cross (u,v)
ang = +/-acos(dot(u,v))

You just need to find out the sign of angle so try both and use the one for which the resultinq Q is where it should be so dot(Q-O,v) is max. To rotate around arbitrary axis and point use:

Also this might be helpfull:

Upvotes: 1

Dorota Kadłubowska
Dorota Kadłubowska

Reputation: 675

By computing dot product between v and u get the angle l between the vectors. Do a cross product of v and u (normalized) to produce axis of rotation vector a. Let w be a vector along vector u from O to P. To rotate point P into Q apply the following actions (in pseudo code) having axis a and angle l computed above:

float4 Rotate(float4 w, float l, float4 a) 
{
    float4x4 Mr = IDENTITY;
    quat_t quat = IDENTITY;
    float4 t = ZERO;

    float xx, yy, zz, xy, xz, yz, wx, wy, wz;

    quat[X] = a[X] * sin((-l / 2.0f));
    quat[Y] = a[Y] * sin((-l / 2.0f));
    quat[Z] = a[Z] * sin((-l / 2.0f));
    quat[W] = cos((-l / 2.0f));

    xx = quat[X] * quat[X];
    yy = quat[Y] * quat[Y];
    zz = quat[Z] * quat[Z];

    xy = quat[X] * quat[Y];
    xz = quat[X] * quat[Z];

    yz = quat[Y] * quat[Z];

    wx = quat[W] * quat[X];
    wy = quat[W] * quat[Y];
    wz = quat[W] * quat[Z];

    Mr[0][0] = 1.0f - 2.0f * (yy + zz);
    Mr[0][1] = 2.0f * (xy + wz);
    Mr[0][2] = 2.0f * (xz - wy);
    Mr[0][3] = 0.0f;
    Mr[1][0] = 2.0f * (xy - wz);
    Mr[1][1] = 1.0f - 2.0f * (xx + zz);
    Mr[1][2] = 2.0f * (yz + wx);
    Mr[1][3] = 0.0f;
    Mr[2][0] = 2.0f * (xz + wy);
    Mr[2][1] = 2.0f * (yz - wx);
    Mr[2][2] = 1.0f - 2.0f * (xx + yy);
    Mr[2][3] = 0.0f;
    Mr[3][0] = 0.0f;
    Mr[3][1] = 0.0f;
    Mr[3][2] = 0.0f;
    Mr[3][3] = 1.0f;

    w = Mr * w;
    return w;
}

Point Q is at the end of the rotated vector w. Algorithm used in the pseudo code is quaternion rotation.

Upvotes: 1

Jerome Demantke
Jerome Demantke

Reputation: 335

P and Q are at the same distance R to O
R = sqrt( (x1-x0)^2 + (y1-y0)^2 + (z1-z0)^2 )

and OQ is collinear to v, so OQ = v * R / ||v|| where ||v|| is the norm of v
||v|| = sqrt( xv^2 + yv^2 + zv^2 )

So the coordinates of Q(xq,yq,zq) are:

xq= xo + xv * R / ||v||
yq= yo + yv * R / ||v||
zq= zo + zv * R / ||v||

Upvotes: 0

AGN Gazer
AGN Gazer

Reputation: 8388

If you know u, v, P, and O then I would suggest that you compute |OP| which should be preserved under rotations. Then multiply this length by the unit vector -v (I assumed u, v are unit vectors: if not - normalize them) and translate the origin by this -|OP|v vector. The negative sign in front of v comes from the description given in your question:"v is the direction to O after transformation".

Upvotes: 0

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