Message Boards Message Boards

0
|
3124 Views
|
2 Replies
|
0 Total Likes
View groups...
Share
Share this post:

Calculating the closest point to a set of lines

Posted 9 years ago

Hi,

as titled, I am trying to calculate the closest point to a set of lines in 3D

I saw here that this guy pointed out some Mathematica code.

I put it into the Mathematica Kernel and I got a very long result.. so I decided to write an algorithm by myself.

Now I would like, instead, given four known lines (each of them defined by two points), to get the closest point..

The lines are the following:

Line[] lines = new Line[]{ new Line(new Vec3(77.007f, 32.46f, 177.79f), new Vec3(139.398f, -309.133f, -116.29f)), //
new Line(new Vec3(-79.504f, 31.969f, -179.158f), new Vec3(31.886f, -367.752f, 99.796f)), //
new Line(new Vec3(-77.401f, 32.855f, 174.843f), new Vec3(25.944f, -350.134f, -129.523f)), //
new Line(new Vec3(80.781f, 32.154f, -176.942f), new Vec3(-149.572f, -326.383f, 84.57f)) };

But I have no knowledge of Mathematica, could someone help me?

2 Replies

Hi Giuseppe,

... so I decided to write an algorithm by myself.

Excellent idea!

As a start you could have written at least the definition of the lines in WL ... Well, anyway, here my solution, it is pretty straightforward:

ClearAll["Global`*"]
p = {p1, p2, p3};
g = {g1, g2, g3}/Sqrt[g1^2 + g2^2 + g3^2];
(* function to calculate distance between line and point: *)    
lineDist[line : {g1_, g2_, g3_}, point : {p1_, p2_, p3_}] = 
  FullSimplify[Norm[p\[Cross]g], Thread[{p1, p2, p3, g1, g2, g3} \[Element] Reals]];

(* your four lines *)    
v = {{{77.007, 32.46, 177.79}, {139.398, -309.133, -116.29}},
   {{-79.504, 31.969, -179.158}, {31.886, -367.752, 99.796}},
   {{-77.401, 32.855, 174.843}, {25.944, -350.134, -129.523}},
   {{80.781, 32.154, -176.942}, {-149.572, -326.383, 84.57}}};

(* search for the point, where the max-distance is minimized *)
sol = Minimize[Max[lineDist[#1, p - #2] & @@@ ({#2 - #1, #1} & @@@ v)], p]

Graphics3D[{{Red, Line[v]}, Sphere[p /. sol[[2]], sol[[1]]]}, SphericalRegion -> True]

The result is:

enter image description here

Regards Henrik

POSTED BY: Henrik Schachner

Hi Henrik,

thanks for your code, one guy gave me this function:

void MathUtil::CalcBeamsClosestPoint(pfSeg* beams, int numBeams, pfVec3& p)
{
    float m00 = 0.0f;
    float m01 = 0.0f;
    float m02 = 0.0f;
    float m11 = 0.0f;
    float m12 = 0.0f;
    float m22 = 0.0f;
    float v0 = 0.0f;
    float v1 = 0.0f;
    float v2 = 0.0f;

    for (int i = 0; i < numBeams; i++) {
        pfSeg& beam = beams[i];
        pfVec3& dir = beam.dir;
        pfVec3& pos = beam.pos;
        const float ax = dir[0];
        const float ay = dir[1];
        const float az = dir[2];
        const float bx = pos[0];
        const float by = pos[1];
        const float bz = pos[2];

        m00 += ay * ay + az * az;
        m11 += az * az + ax * ax;
        m22 += ax * ax + ay * ay;
        m01 += -ax * ay;
        m02 += -az * ax;
        m12 += -ay * az;
        v0 += az * (az * bx - ax * bz) - ay * (ax * by - ay * bx);
        v1 += ax * (ax * by - ay * bx) - az * (ay * bz - az * by);
        v2 += ay * (ay * bz - az * by) - ax * (az * bx - ax * bz);
    }
    matrix33f m(vector3f(m00, m01, m02), vector3f(m01, m11, m12), vector3f(m02, m12, m22));
    float det = m.det();
    assert(det != 0.0f);
    vector3f pt = inverse(m) * vector3f(v0, v1, v2);
    p.set(pt.x, pt.y, pt.z);
}

it works with simple lines such as

       new Line(new Vec3(0, 0, 0), new Vec3(1, 1, 1)),

       new Line(new Vec3(2, 0, 0), new Vec3(1, 1, 1)),

but not with the ones I gave you previously (I obtain a different result).. I don't know why, I can't get anyway the theory behind his code

By the way, the formula is this one at the end http://math.stackexchange.com/a/61721/64943

I tried to wrote the algorithm, but I didn't make it

I should come to a form such as:

A x = b

but I don't have idea how to deal with sums and dot products..

Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract