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Subject: Re: Angle between two vectors
Date: Sat, 17 Mar 2012 23:10:29 +0000 (UTC)
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"Amit " <av_pachore@yahoo.com> wrote in message <jk1qvp$492$1@newscl01ah.mathworks.com>...
> Hello Roger,
> I want to find the angle between line1 as vertex-p2 and line2 as vertex-points stored in array either in clockwise or anticlockwise.But did not getting angles with following code:
> 
> vertex=[186,130];
> p2=[278,155];
> x=[87    91   103   175];%x co-ordinates of points
> y=[134   104    76    31];%y co-ordinates of points
> 
> [r,c]=size(x);
> ang=zeros(size(x));
> 
> for i=1:c
>     pt=[x(i),y(i)];
>     ang(i)=mod(atan2(abs(det([p2-vertex;pt-vertex])),dot(p2-vertex,pt-vertex)),2*pi);
> end;
> 
> ang=180/pi*ang;
> display(ang);
> 
> This gives the result as
> ang =
> 
>   162.4839  179.8963  162.1543  111.5426
> 
> if these angles are only anticlockwise then pt[175,31] would have angle more than 180 degrees
> or if it is clockwise [87,134] would have angle more than 180 degrees.
> Please help me with this.
> Thanks in advance.
- - - - - - - - - -
  In two dimensional space there is a difference between, on the one hand finding the angles, say, within a triangle which always lie between 0 and pi radians (0 and 180 degrees), and on the other hand finding the angle between two vectors with a common base starting from one of them and rotating counterclockwise (or sometimes clockwise) until first encountering the other one.  The latter gives an angle that is anywhere between 0 and 2*pi radians (0 and 360 degrees), or according to a different convention, between -pi and +pi radians (-180 and +180 degrees.)  In using matlab's 'atan2' function properly it is immportant to distinguish between these different meanings.

  In your problem if you want to rotate counterclockwise from the vector p2-v (using 'v' instead of 'vertex') until first encountering the vector pt-v, expecting an angle between 0 and 360 degrees, the appropriate formula would be:

 ang = mod(atan2(det([p2-v;pt-v]),dot(p2-v,pt-v)),2*pi)*180/pi;

  Note that the 'abs' operation is missing here.  That only applies if you are seeking results that are restricted to lie between 0 and pi as in the angles of a triangle.  The 'mod' operation is necessary here since 'atan2' yields answer between -pi and +pi, so the negative answers need to have 2*pi added to them.  If the rotation is counterclockwise from pt-v to p2-v or if it is clockwise from p2-v to pt-v, the two arguments pt-v and p2-v would be reversed in the above formula.  If your convention is that going beyond pi radians is regarded as negative rotation, then you would leave out the 'mod' operation.

 In the example you gave with

 pt = [ 87,134;
        91,104;
       103, 76;
       175, 31];
 p2 = [278,155];
 v = [186,130];

the formula above for rotating counterclockwise from p2-v to pt-v 
would give the four answers:

 162.4839    180.1037    197.8457    248.4574

Reversing the arguments (that is, going clockwise or from pt-v to p2-v) 
would give:

 197.5161    179.8963    162.1543    111.5426
 
Without the 'mod' operation the counterclockwise from p2-v to pt-v answers would be:

 162.4839   -179.8963   -162.1543   -111.5426

Roger Stafford