MATLAB Central Newsreader - Angle between two vectors

Angle between two vectors

y Mehta — Mon, 09 Jul 2007 17:16:29 -0400

How do I find the angle between two unit vectors a and b? I know I
can find cosine theta by the following formula:

theta = acos(dot(a,b));

However, how do I know whether the angle is actually theta, or -theta
or pi-theta or pi+theta??

Notice that the vectors are in three dimension (3d).

Thanks,
-YM

Re: Angle between two vectors

Greg Heath — Tue, 10 Jul 2007 00:30:26 -0400

On Jul 9, 5:16 pm, "y Mehta" <mehtayogesh@gmail.(DOT).com> wrote:
> How do I find the angle between two unit vectors a and b? I know I
> can find cosine theta by the following formula:
>
> theta = acos(dot(a,b));
>

Invalid since it is possible that abs(dot(a,b)) > 1.

costheta = dot(a,b)/(norm(a)*norm(b));
theta = acos(costheta);

will give you the anser in the interval [0,pi].

> However, how do I know whether the angle is
> actually theta, or -theta or pi-theta or pi+theta??

Angles between vectors only lie in the interval [0,pi].

> Notice that the vectors are in three dimension (3d).

Dimensionality of the original space is irrelevant. As long as
norm(a)*norm(b) > 0, the vectors uniquely define a 2-d space when
dot(a,b) ~= 0 and a unique 1-d space otherwise.

Hope this helps.

Greg

Re: Angle between two vectors

ellieandrogerxyzzy@mindspring.com.invalid (Roger Stafford) — Tue, 10 Jul 2007 02:57:36 -0400

In article <ef5ce9c.-1@webcrossing.raydaftYaTP>, "y Mehta"
<mehtayogesh@gmail.(DOT).com> wrote:

> How do I find the angle between two unit vectors a and b? I know I
> can find cosine theta by the following formula:
>
> theta = acos(dot(a,b));
>
> However, how do I know whether the angle is actually theta, or -theta
> or pi-theta or pi+theta??
>
> Notice that the vectors are in three dimension (3d).
>
> Thanks,
> -YM
---------------------
It is usually understood that the angle between two three-dimensional
vectors is measured by the shortest great circle path between them, which
means that it must lie between 0 and pi radians. To get such an answer,
the best method, in my opinion, is this:

angle = atan2(norm(cross(a,b)),dot(a,b));

Since the first argument must be non-negative, the angle will lie
somewhere in the two first quadrants, and thus be between 0 and pi. This
formula remains valid even if a and b are not unit vectors.

Your 'acos' formula gives the correct answer with unit vectors as it
stands, but it encounters an accuracy problem for angles that are near 0
or pi. This is the main reason for my preference for the 'atan2' method.

Roger Stafford

Re: Angle between two vectors

ellieandrogerxyzzy@mindspring.com.invalid (Roger Stafford) — Tue, 10 Jul 2007 08:27:08 -0400

In article <1184052626.324470.175540@n2g2000hse.googlegroups.com>, Greg
Heath <heath@alumni.brown.edu> wrote:

> On Jul 9, 5:16 pm, "y Mehta" <mehtayogesh@gmail.(DOT).com> wrote:
> > How do I find the angle between two unit vectors a and b? I know I
> > can find cosine theta by the following formula:
> >
> > theta = acos(dot(a,b));
>
> Invalid since it is possible that abs(dot(a,b)) > 1.
>
> costheta = dot(a,b)/(norm(a)*norm(b));
> theta = acos(costheta);
>
> will give you the anser in the interval [0,pi].
>
> > However, how do I know whether the angle is
> > actually theta, or -theta or pi-theta or pi+theta??
>
> Angles between vectors only lie in the interval [0,pi].
>
> > Notice that the vectors are in three dimension (3d).
>
> Dimensionality of the original space is irrelevant. As long as
> norm(a)*norm(b) > 0, the vectors uniquely define a 2-d space when
> dot(a,b) ~= 0 and a unique 1-d space otherwise.
>
> Hope this helps.
>
> Greg
---------------
Greg, Y Mehta did specifically state that a and b are unit vectors, so
his formula is in fact correct as it stands, though subject to increasing
errors as its dot product approaches +1 or -1. When a and b are nearly
parallel, the same kind of trouble occurs with the formula you have given
here. In such cases there is a real need to combine the scalar dot
product with the vector cross product in order to make use of both the
sine and cosine in calculating the angle accurately, which is what the
'tan2' formula does.

Roger Stafford

Re: Angle between two vectors

Greg Heath — Tue, 10 Jul 2007 14:55:26 -0400

On Jul 10, 4:27 am, ellieandrogerxy...@mindspring.com.invalid (Roger
Stafford) wrote:
> In article <1184052626.324470.175...@n2g2000hse.googlegroups.com>, Greg
>
>
>
>
>
> Heath <h...@alumni.brown.edu> wrote:
> > On Jul 9, 5:16 pm, "y Mehta" <mehtayogesh@gmail.(DOT).com> wrote:
> > > How do I find the angle between two unit vectors a and b? I know I
> > > can find cosine theta by the following formula:
>
> > > theta = acos(dot(a,b));
>
> > Invalid since it is possible that abs(dot(a,b)) > 1.
>
> > costheta = dot(a,b)/(norm(a)*norm(b));
> > theta = acos(costheta);
>
> > will give you the anser in the interval [0,pi].
>
> > > However, how do I know whether the angle is
> > > actually theta, or -theta or pi-theta or pi+theta??
>
> > Angles between vectors only lie in the interval [0,pi].
>
> > > Notice that the vectors are in three dimension (3d).
>
> > Dimensionality of the original space is irrelevant. As long as
> > norm(a)*norm(b) > 0, the vectors uniquely define a 2-d space when
> > dot(a,b) ~= 0 and a unique 1-d space otherwise.
>
> > Hope this helps.
>
> > Greg
>
> ---------------
> Greg, Y Mehta did specifically state that a and b are unit vectors, so
> his formula is in fact correct as it stands, though subject to increasing
> errors as its dot product approaches +1 or -1. When a and b are nearly
> parallel, the same kind of trouble occurs with the formula you have given
> here. In such cases there is a real need to combine the scalar dot
> product with the vector cross product in order to make use of both the
> sine and cosine in calculating the angle accurately, which is what the
> 'tan2' formula does.

Thanks.

For some problem in the past (probably single precision?) I got better
accuracy using

sign(sintheta)*acos(costheta)

instead of atan2.

Hope this helps.

Greg

Hope this helps.

Re: Angle between two vectors

ellieandrogerxyzzy@mindspring.com.invalid (Roger Stafford) — Tue, 10 Jul 2007 23:22:36 -0400

In article <1184104526.774530.54800@n60g2000hse.googlegroups.com>, Greg
Heath <heath@alumni.brown.edu> wrote:

> For some problem in the past (probably single precision?) I got better
> accuracy using
>
> sign(sintheta)*acos(costheta)
>
> instead of atan2.
---------------
  I don't know why that would be better, Greg. The acos(costheta) is
subject to the same problem as before. When costheta is very near 1, the
accuracy is very much inferior to that of 'atan2'. I would think this
would be equally true in single precision.

  Here's a concrete example of what I am referring to:

format long
a = pi*(1-1/10^8); % Choose an angle a little below pi
a2 = atan2(sin(a),cos(a)); % Use atan2
a3 = acos(cos(a)); % Use acos
[a;a2;a3;a-a2;a-a3] % Compare results

a = pi/2*1.123; % Now select an angle near pi/2
a2 = atan2(sin(a),cos(a)); % atan2 again
a3 = acos(cos(a)); % Then acos
[a;a2;a3;a-a2;a-a3] % Make the same comparisons

ans =

   3.14159262217387
   3.14159262217387
   3.14159262378747
                  0
  -0.00000000161360

ans =

   1.76400427499067
   1.76400427499067
   1.76400427499067
   0.00000000000000
   0.00000000000000

As you see, there is a very distinct loss of accuracy in 'acos' for angles
near pi. Some seven entire decimal places have been lost - that is,
errors are several million times as large as normal. On the other hand,
the angle near pi/2 yields the customary 1 in 2^52 accuracy.

Roger Stafford

Re: Angle between two vectors

Yogesh Mehta — Mon, 16 Jul 2007 01:46:40 -0400

Thanks Greg and Roger. Your inputs were very useful for me.

Re: Angle between two vectors

salih tuna — Mon, 10 Dec 2007 11:59:47 -0500

hello,
how can i calculate the angles so that they are in the range
0-360 degrees?
thanks
salih

"y Mehta" <mehtayogesh@gmail.(DOT).com> wrote in message
<ef5ce9c.-1@webcrossing.raydaftYaTP>...
> How do I find the angle between two unit vectors a and b?
I know I
> can find cosine theta by the following formula:
>
> theta = acos(dot(a,b));
>
> However, how do I know whether the angle is actually
theta, or -theta
> or pi-theta or pi+theta??
>
> Notice that the vectors are in three dimension (3d).
>
> Thanks,
> -YM

Re: Angle between two vectors

salih tuna — Mon, 10 Dec 2007 12:00:24 -0500

Re: Angle between two vectors

Roger Stafford — Mon, 10 Dec 2007 18:35:28 -0500

"salih tuna" <salihtuna@gmail.com> wrote in message <fjj9nj$fia
$1@fred.mathworks.com>...
> hello,
> how can i calculate the angles so that they are in the range 0-360 degrees?
> thanks
> salih
>
> "y Mehta" <mehtayogesh@gmail.(DOT).com> wrote in message
> <ef5ce9c.-1@webcrossing.raydaftYaTP>...
> > How do I find the angle between two unit vectors a and b? I know I
> > can find cosine theta by the following formula:
> >
> > theta = acos(dot(a,b));
> >
> > However, how do I know whether the angle is actually theta, or -theta
> > or pi-theta or pi+theta??
> >
> > Notice that the vectors are in three dimension (3d).
> >
> > Thanks,
> > -YM
--------
Y Mehta's question involved angles between vectors in three-dimensional
space. I can think of no reasonable definition for a canonical angle between
such vectors which ranges from 0 to 360 degrees (0 to 2*pi radians.)

However, if you are in two-dimensional space, then you can speak of the
non-negative angle measured counterclockwise from vector a to vector b,
and this would give the range you have requested. If a = [x1,y1] and b =
[x2,y2], then such an angle is given in matlab by:

angle = mod(atan2(y2-y1,x2-x1),2*pi); % Range: 0 to 2*pi radians

(Multiply this answer by 180/pi to get degrees.)

Roger Stafford

Re: Angle between two vectors

salih tuna — Tue, 11 Dec 2007 11:20:21 -0500

Hi,
thanks a lot for your reply. yes they are in 2d, sorry i
forgot to mention.
i tried to apply the formula but i am getting wrong result.
for example i want to calculate the angle between a = [1 1]
and b = [0 -1] which is 225 degrees. with this formulae i
got 243.4. i couldn't see where i am doing the mistake.
thanks a lot in advance
salih

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
wrote in message <fjk0tg$jli$1@fred.mathworks.com>...
> "salih tuna" <salihtuna@gmail.com> wrote in message
<fjj9nj$fia
> $1@fred.mathworks.com>...
> > hello,
> > how can i calculate the angles so that they are in the
range 0-360 degrees?
> > thanks
> > salih
> >
> > "y Mehta" <mehtayogesh@gmail.(DOT).com> wrote in message
> > <ef5ce9c.-1@webcrossing.raydaftYaTP>...
> > > How do I find the angle between two unit vectors a and
b? I know I
> > > can find cosine theta by the following formula:
> > >
> > > theta = acos(dot(a,b));
> > >
> > > However, how do I know whether the angle is actually
theta, or -theta
> > > or pi-theta or pi+theta??
> > >
> > > Notice that the vectors are in three dimension (3d).
> > >
> > > Thanks,
> > > -YM
> --------
> Y Mehta's question involved angles between vectors in
three-dimensional
> space. I can think of no reasonable definition for a
canonical angle between
> such vectors which ranges from 0 to 360 degrees (0 to 2*pi
radians.)
>
> However, if you are in two-dimensional space, then you
can speak of the
> non-negative angle measured counterclockwise from vector a
to vector b,
> and this would give the range you have requested. If a =
[x1,y1] and b =
> [x2,y2], then such an angle is given in matlab by:
>
> angle = mod(atan2(y2-y1,x2-x1),2*pi); % Range: 0 to 2*pi
radians
>
> (Multiply this answer by 180/pi to get degrees.)
>
> Roger Stafford

Re: Angle between two vectors

Roger Stafford — Tue, 11 Dec 2007 13:56:20 -0500

"salih tuna" <salihtuna@gmail.com> wrote in message <fjlrpl$gii
$1@fred.mathworks.com>...
> Hi,
> thanks a lot for your reply. yes they are in 2d, sorry i
> forgot to mention.
> i tried to apply the formula but i am getting wrong result.
> for example i want to calculate the angle between a = [1 1]
> and b = [0 -1] which is 225 degrees. with this formulae i
> got 243.4. i couldn't see where i am doing the mistake.
> thanks a lot in advance
> salih
>
> "Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
> wrote in message <fjk0tg$jli$1@fred.mathworks.com>...
> > angle = mod(atan2(y2-y1,x2-x1),2*pi); % Range: 0 to 2*pi
--------
  I certainly owe you an apology, Salih. That formula I gave you is very, very
wrong. I can't imagine what I was thinking about when I wrote it. Chalk it up
to momentary insanity! :-) The correct computation should be as follows.

  Assuming a = [x1,y1] and b = [x2,y2] are two vectors with their bases at the
origin, the non-negative angle between them measured counterclockwise
from a to b is given by

angle = mod(atan2(x1*y2-x2*y1,x1*x2+y1*y2),2*pi);

  As you can see, this bears a close relationship to the three-dimensional
formula I wrote last July 10. The quantities, x1*y2-x2*y1 and x1*x2+y1*y2
are, respectively, the sine and cosine of the counterclockwise angle from
vector a to vector b, multiplied by the product of their norms - that is, their
cross product and the dot product restricted to two dimensions. The 'atan2'
function then gives the angle between them ranging from -pi to +pi, and the
'mod' operation changes this so as to range from 0 to 2*pi, as you requested.

Roger Stafford

Re: Angle between two vectors

salih tuna — Tue, 11 Dec 2007 15:29:00 -0500

Roger hi,
thanks a lot for your help but i am afraid something is
still missing. i tried the formulae on the same example of
a = [1 1]and b = [0 -1] (both passing through origin).
the answer i got is 315 instead of 225.
sorry i am taking a lot of your time :)
thanks
salih

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
wrote in message <fjm4u4$i8o$1@fred.mathworks.com>...
> "salih tuna" <salihtuna@gmail.com> wrote in message
<fjlrpl$gii
> $1@fred.mathworks.com>...
> > Hi,
> > thanks a lot for your reply. yes they are in 2d, sorry i
> > forgot to mention.
> > i tried to apply the formula but i am getting wrong result.
> > for example i want to calculate the angle between a = [1 1]
> > and b = [0 -1] which is 225 degrees. with this formulae i
> > got 243.4. i couldn't see where i am doing the mistake.
> > thanks a lot in advance
> > salih
> >
> > "Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
> > wrote in message <fjk0tg$jli$1@fred.mathworks.com>...
> > > angle = mod(atan2(y2-y1,x2-x1),2*pi); % Range: 0 to 2*pi
> --------
> I certainly owe you an apology, Salih. That formula I
gave you is very, very
> wrong. I can't imagine what I was thinking about when I
wrote it. Chalk it up
> to momentary insanity! :-) The correct computation
should be as follows.
>
> Assuming a = [x1,y1] and b = [x2,y2] are two vectors
with their bases at the
> origin, the non-negative angle between them measured
counterclockwise
> from a to b is given by
>
> angle = mod(atan2(x1*y2-x2*y1,x1*x2+y1*y2),2*pi);
>
> As you can see, this bears a close relationship to the
three-dimensional
> formula I wrote last July 10. The quantities, x1*y2-x2*y1
and x1*x2+y1*y2
> are, respectively, the sine and cosine of the
counterclockwise angle from
> vector a to vector b, multiplied by the product of their
norms - that is, their
> cross product and the dot product restricted to two
dimensions. The 'atan2'
> function then gives the angle between them ranging from
-pi to +pi, and the
> 'mod' operation changes this so as to range from 0 to
2*pi, as you requested.
>
> Roger Stafford
>
>

Re: Angle between two vectors

Roger Stafford — Tue, 11 Dec 2007 19:32:07 -0500

"salih tuna" <salihtuna@gmail.com> wrote in message <fjmabs$9fh
$1@fred.mathworks.com>...
> Roger hi,
> thanks a lot for your help but i am afraid something is
> still missing. i tried the formulae on the same example of
> a = [1 1]and b = [0 -1] (both passing through origin).
> the answer i got is 315 instead of 225.
> sorry i am taking a lot of your time :)
> thanks
> salih
-----
Hi Salih. I get 225 degrees for that example with a = [1,1] and b = [0,-1],
which is correct. Are you sure you didn't have x2 and y1 interchanged by
mistake? That would get you an erroneous answer of 315. You should have x1
= 1, y1 = 1, x2 = 0, and y2 = -1.

Roger Stafford

Re: Angle between two vectors

baris kazar — Fri, 28 Dec 2007 16:11:38 -0500

"Roger Stafford"
<ellieandrogerxyzzy@mindspring.com.invalid> wrote in
message <fjk0tg$jli$1@fred.mathworks.com>...
> "salih tuna" <salihtuna@gmail.com> wrote in message
<fjj9nj$fia
> $1@fred.mathworks.com>...
> > hello,
> > how can i calculate the angles so that they are in the
range 0-360 degrees?
> > thanks
> > salih
> >
> > "y Mehta" <mehtayogesh@gmail.(DOT).com> wrote in
message
> > <ef5ce9c.-1@webcrossing.raydaftYaTP>...
> > > How do I find the angle between two unit vectors a
and b? I know I
> > > can find cosine theta by the following formula:
> > >
> > > theta = acos(dot(a,b));
> > >
> > > However, how do I know whether the angle is actually
theta, or -theta
> > > or pi-theta or pi+theta??
> > >
> > > Notice that the vectors are in three dimension (3d).
> > >
> > > Thanks,
> > > -YM
> --------
> Y Mehta's question involved angles between vectors in
three-dimensional
> space. I can think of no reasonable definition for a
canonical angle between
> such vectors which ranges from 0 to 360 degrees (0 to
2*pi radians.)
>
> However, if you are in two-dimensional space, then you
can speak of the
> non-negative angle measured counterclockwise from vector
a to vector b,
> and this would give the range you have requested. If a
= [x1,y1] and b =
> [x2,y2], then such an angle is given in matlab by:
>
> angle = mod(atan2(y2-y1,x2-x1),2*pi); % Range: 0 to
2*pi radians
>
> (Multiply this answer by 180/pi to get degrees.)
>
> Roger Stafford

Hi,-
how can we generalize this to 3-D vectors? Think of a
plane on 3-D space and you have vectors on this plane. I
wanna know the angle between 2 vectors in the range 0-2pi.
or at least -pi to pi. I have 3 vectors. One (First)
vector is the same all the time. I wanna know the relative
positions of the other two vector wrt the firsy one. Thus,
i need angles in the range 0 to 2pi or -pi to pi.
Thanks in advance

Re: Angle between two vectors

Bruno Luong — Fri, 28 Dec 2007 18:22:57 -0500

"baris kazar" <mbkazar.nospam@gmail.com> wrote in message
<fl377q$4ip$1@fred.mathworks.com>...

>
> Hi,-
> how can we generalize this to 3-D vectors? Think of a
> plane on 3-D space and you have vectors on this plane. I
> wanna know the angle between 2 vectors in the range 0-2pi.

The problem is depends on where you look (from above or from
below the plane), you will see the angle between vectors on
this plane reverse the sign (draw them on a transparent
glass and try to look from both sides). In 3D there is
nothing that could tell us whereas looking from one way or
from another is more "correct".

Think like a fish or a bird, not like a man.

Bruno