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Index › Programming Questions & Help › OpenGL and 3D Libraries › Calculating  local coords for a 3d vector
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Calculating  local coords for a 3d vector (Read 5568 times)
sgsrules
Calculating  local coords for a 3d vector
Mar 31st, 2008, 3:10am  
Hi everyone, i need help.  is there any way to obtain vectors that represent the main vectors local coordinate space? Basically If i have a 3d vector, how can i obtain a vector that is perpendicular to it that points to it's positive Y direction and another vector that points to it's Z or X direction.  Basically the main vector is used to plot a path similar to a ribbon or light trail.  I need the vectors perpendicular to it in order to give the final shape some width and height by offsetting the points a certain distance.  The end result is supposed to be a cube extruded along a spline.

Thanks Again
Stephen.
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Stephen Shepard (sgsrules)
 
Classiclll
Re: Calculating  local coords for a 3d vector
Reply #1 - Mar 31st, 2008, 3:50am  
hi sgsrules.
although not finished, is this any help for you?  
http://www.geocities.jp/classiclll_newweb/Deku10/applet/index.html
see the class "Karakuri" and the class "Loc".
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JohnG
Re: Calculating  local coords for a 3d vector
Reply #2 - Mar 31st, 2008, 11:17am  
I think some code I wrote a few years ago should still be of use. The most recent copy on the boards I can find is here:

http://processing.org/discourse/yabb_beta/YaBB.cgi?board=OpenGL;action=display;num=1159148328;start=0#0
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sgsrules
Re: Calculating  local coords for a 3d vector
Reply #3 - Mar 31st, 2008, 6:05pm  
Thanks for the sample code guys, this will be useful.  I ended up changing my approach.  Instead of drawing the shape based around the vector i just drew the shape and then made a function that repositions and rotates it around the given vector.
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Stephen Shepard (sgsrules)
 
CongoZombie
Re: Calculating  local coords for a 3d vector
Reply #4 - Mar 31st, 2008, 9:03pm  
Sounds like exactly what I've just been doing.
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www.andybest.net
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sgsrules
Re: Calculating  local coords for a 3d vector
Reply #5 - Mar 31st, 2008, 11:44pm  
For anyone that's interested here's the basic idea:

given vector with components (x,y,z);

rotateZ(atan2(y,x));
rotateY(atan2(z,x));

This will rotate everything to coincide with direction of the vector.
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Stephen Shepard (sgsrules)
 
julapy
Re: Calculating  local coords for a 3d vector
Reply #6 - Apr 1st, 2008, 2:53pm  
hi, ive been looking for the same effect, as in creating a trail in 3D space much like a ribbon.

would you have any example code to get the brain cogs turning?
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davbol
Re: Calculating  local coords for a 3d vector
Reply #7 - Apr 1st, 2008, 10:46pm  
@sgsrules:  you might want to check that atan2/atan2 really gives you the rotation you want.  I think what you want is a cartesian to spherical coordinate conversion to get those rotations.

It's a common misconception that to point at for example <1,1,1> you'd rotate z by PI/4 (atan2(y/x)), then rotate y by PI/4 (atan2(z/x)).  45 degrees around, then tilt 45 degrees up -- simple, right?  Yep, but wrong.  ;)

Here are two ways to draw a line from 0,0,0 to x,y,z:

Quote:

 // with no rotations, draw a cartesian coordinate line
 line(0,0,0, x,y,z);

 // with rotations, draw a spherical coordinate line
 pushMatrix();
 float r = sqrt(x*x+y*y+z*z); // radial distance
 float theta = atan2(y,x); // zenith angle, range -pi..pi
 float phi = acos(z/r); // azimuth angle, range -pi/2..pi/2
 rotateZ(theta); // "heading" or "around"
 rotateY(phi);  // "tilt" or "elevation"
 line(0,0,0, 0,0,r); // the z-axis now points "at" x,y,z
 popMatrix();



hope that helps
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taifunbrowser
Re: Calculating  local coords for a 3d vector
Reply #8 - Apr 2nd, 2008, 2:18am  
thank you thank you thank you thank you thank you

x e^(654,321x10^23)

very very useful math! =)

edit: so, now I can finally do my "thickLine" pipe-draw function
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sgsrules
Re: Calculating  local coords for a 3d vector
Reply #9 - Apr 5th, 2008, 1:35am  
Ok here's a quick function i made to rotate a point so that it lines up with a vector.

Void orient (Vec3D vec, float x, float y, float z){
  // get rotation amount
 float rotX=(atan2(vec.y,vec.z));
 float rotY=(atan2(vec.z,vec.x));
 float rotZ=(atan2(vec.y,vec.x));

  // rotation around x axis
  float y1 = cos(rotX)*y - sin(rotX)*z;
  float z1 = sin(rotX)*y + cos(rotX)*z;

  // rotation around y axis
  float finalZ = cos(rotY)*z1 - sin(rotY)*x;
  float x1 = sin(rotY)*z1 + cos(rotY)*x;

  // rotation around z axis
  float finalX = cos(rotZ)*x1 - sin(rotZ)*y1;
  float finalY = sin(rotZ)*x1 + cos(rotZ)*y1;

  x=finalX;
  y=finalY;
  z=FinalZ;
}

I thought this would work but i haven't tested it yet since i'm at work (I'm sure there's a couple of mistakes).  My trig is pretty rusty so i'm guessing this isn't right.  According to davbol, and i trust his judgment more than mine it should be:


Void orient (Vec3D vec, float x, float y, float z){

 float r = sqrt(vec.x*vec,x+vec.y*vec,y+vec.z*vec.z);
 
   // get rotation amount
 float rotY=(atan2(vec.z,vec.x));
 float rotZ=(acos(vec.z/r));

    // rotation around y axis
  float finalZ = cos(rotY)*z - sin(rotY)*x;
  float x1 = sin(rotY)*z + cos(rotY)*x;

    // rotation around z axis
  float finalX = cos(rotZ)*x1 - sin(rotZ)*y;
  float finalY = sin(rotZ)*x1 + cos(rotZ)*y;

  x=finalX;
  y=finalY;
  z=finalZ;
}

I'm using trig functions to handle the rotations because i can't use rotate and translate while im creating a shape with beginshape.  Will this work or am i going about this all wrong?  Once again thanks for all the help guys.
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Stephen Shepard (sgsrules)
 
V
Re: Calculating  local coords for a 3d vector
Reply #10 - Apr 5th, 2008, 1:33pm  
a point is only a position in space it has no direction.
if you're trying to rotate a vector or an object's orientation vector to follow another vector's orientation here goes 2 ways:

let V = V2 - V1;
U = (0, 1, 0);

1. normalize V vector
2. take the cross-product of U and V, this will give you the axis of rotation
3. take the dot-product of U and V. the arccosine of the result is the angle of rotation.
4. with this information you can already build a matrix or a quaternion and rotate your vector

just pass the axis and the angle and build a quaternion. rotate your vector/object

to build a matrix is the same as the camera matrix is built.
you have a U, V, N vectors. make the matrix out of it and use it to transform your data.
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Victor Martins | www.pixelnerve.com/v
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sgsrules
Re: Calculating  local coords for a 3d vector
Reply #11 - Apr 5th, 2008, 4:46pm  
I'm rotating these points around the origin not themselves (that wouldn't accomplish much Tongue )  I kinda figured i'de have to use a matrix but was avoiding it,  time to do some more reading.  And do i really need to get into Quaternions?  Can't i just use vectors and matrices?
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« Last Edit: Apr 5th, 2008, 6:44pm by sgsrules »  
Stephen Shepard (sgsrules)
 
V
Re: Calculating  local coords for a 3d vector
Reply #12 - Apr 5th, 2008, 7:29pm  
sure you can use just vectors and matrices.
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Victor Martins | www.pixelnerve.com/v
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sgsrules
Re: Calculating  local coords for a 3d vector
Reply #13 - Apr 5th, 2008, 11:28pm  
I found this, which does exactly what i'm look for:

http://www.fundza.com/mel/axis_to_vector/align_axis_to_vector.html

This is pretty much what Davbol and some of the other guys have suggested.  This works but i think i'de rather use a matrix to do all my translations.  I have a vector that i'm using to specify the Z axis direction but i still need to obtain vectors for the X and Y axis which should be easy enough since they're at 90 degree angles from each other. This is what i've figured:

x axis = cross product of given Z vector and (0,1,0)
y axis = cross product of given Z vector and x axis

but since alot of my assumptions have been wrong so far i was going to check with the knowledgeable people on this board first. hopefully i've got it right this time so we can stop this thread Tongue
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« Last Edit: Apr 6th, 2008, 2:44am by sgsrules »  
Stephen Shepard (sgsrules)
 
V
Re: Calculating  local coords for a 3d vector
Reply #14 - Apr 6th, 2008, 12:53pm  
yes, here's something i use to compute local matrices for some project of mine. i hope this is what you really want to do.

void computeCSMatrix()
{
         // default up vector
         vec3 up = new vec3( 0, 1, 0 );

         // dir vector
   vec3 N = new vec3( z.x, z.y, z.z );
   N.normalize();

         // up vector
   vec3 U = up.cross( N );
 U.normalize();

         // right vector
 vec3 V = N.cross( U );
 V.normalize();

 mm[0] = U.x;
   mm[1] = U.y;
 mm[2] = U.z;
 mm[3] = 0;
 mm[4] = V.x;
 mm[5] = V.y;
 mm[6] = V.z;
 mm[7] = 0;
 mm[8] = N.x;
 mm[9] = N.y;
 mm[10] = N.z;
 mm[11] = 0;
 mm[12] = 0;
 mm[13] = 0;
 mm[14] = 0;
 mm[15] = 1;
}
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Victor Martins | www.pixelnerve.com/v
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