Test if point coordinates are colinear

How would I test if a series of coordinates are colinear i.e. on the same line, I do not have line segments to work with, because there is no class for it..
steenpatAsked:
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jkrCommented:
Calculate the linear equation y = mx + b (http://en.wikipedia.org/wiki/Linear_equation) for two pairs of points and check whether m and b are equivalent. If so, they are colinear. I.e.

m = (y2 - y1) / (x2 - x1);
b = y1 - m * x1;
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NovaDenizenCommented:
Assuming you're working with floating point numbers, you have to keep in mind that you can't calculate this exactly.  You have to decide on an error range, or at least work with a measure of how collinear the points are.
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WaterStreetCommented:
Pick one point and make sure that the slope (m) of all lines between that point and all the points is the same (using the coordinates for each point).  
Then if the slopes are the same for all, the points are colinear
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siddhant3sCommented:
Oh man, what I can guess is this,
There is a simple formula that calculates the area of a triangle(or any polygon) with coordinate of the points given set of points given.
The given points will be collinear if the area comes out to be zero.

Now I am giving the formula for the area of triangle, which will be zero when all the three points are collinear:
Area=(1/2)*(x1*y2 - x2*y1+ x2*y3 - x3*y2 + x3*y1 - x1*y3)
where x1,y1 are the coordinate of first point; x2,y2 are the coordinate of second point and so on.
So, if the area is zero we have
x1* y2 + x2* y3 + x3* y1 = x2* y1+ x3* y2 +x1* y3
 
So what you should do is take the first three points and check for colinearity, then the next three and so on.
Enjoy,


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NovaDenizenCommented:
@WaterStreet:
Using slopes instead of 2-d linear algebra is almost always a bad idea.  You have to set up special cases for when the lines are vertical.  It's worth the effort to do it the proper way.
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jkrCommented:
Um a maybe simpler approach from vector algebra: Calculate the normalized spatial vectors between two points and compare these (within reasonable limits) - i.e.


xn = (x2 - x1) / sqrt(pow(x2,2.0) + pow(x1,2.0));
yn = (y2 - y1) / sqrt(pow(y2,2.0) + pow(y1,2.0));

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jkrCommented:
Huh?
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WaterStreetCommented:
Nova,
I see what you mean.  Good to mention that.  Thanks.
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ozoCommented:
the area method should be the simplest.
If you set a threshold to also accept points that are almost colinear,
you may want to scale the threshhold by the distance between the points, since it gets harder to have a small area with widely separated points.
The other methods also become more difficult to meet a threshold when the points are far apart (and in the ones that check slope, when the slope becomes large) so they would also need be scaled if you wanted to be equally fair to almost collinear points at any size/angle
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siddhant3sCommented:
ozo, I think they have selected their choice, and they wont listen too.
Tell them what about these three points : (2,3)   (2,2)  ,  (2,10)
You cant tell if these points are collinear using the slope method.
Slope of the lines joining these points are not defined
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steenpatAuthor Commented:
siddhant3s you are right,
but i set up a special case for that like nova said.
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siddhant3sCommented:
go ahead for that speacial case, i cant stop you.
but there is one more glitch.
your method requires division of float point numbers which are less efficient.
Also, by the slope method you will never be accurate if the points are far apart.

By the way enjoy.
PS: Good programing lies in efficient algorithms.

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