2D Intersecting Lines
Author
Hi,
How do I find the point of intersection of two lines in the form Ax+By=C. I know how the basic method works (both subtraction and substitution) but in either case there are numerous possibilities for divide by 0 errors (horizontal and vertical lines). Is there any way to do this without having so many checks for 0 divisors?
THanks,
tj963
I calculated this method a week ago. I could have made mistakes, but it seems to work. It assumes that you have for two lines: xinitial, yinitial, xdelta, and the ydelta. It returns the "time" of intersection:
Hopefully the comes out correctly. If the denominator is zero the lines are parallel.
x2i(x1v - x2v) + x1i(x2v - x1v) - y2i(y1v - y2v) - y1i(y1v + y2v)t = - ----------------------------------------------------------------- (x1v - x2v)^2 + (y1v - y2v)^2Hopefully the comes out correctly. If the denominator is zero the lines are parallel.
Hi !
Just make a special case to check if the two lines are parallel. I highly recommend you to treat lines with linear algebra, it becomes much easier when it comes to planes, R^n subspaces, and so on.
If the two director vectors (a director vector for a line is simply a vector from a point to another point on that same line, that is a vector that tells the direction) are colinear (that is, u = kv, where u and v are the vectors and k a scalar), then your lines are not parallel.
This works just fine with lines in R^2 since any non-parallel lines in R^2 always intersect. If you want to intersect two lines in R^3 (that is, in 3D) then you''ll also have to check the special case in wich the two lines simply don''t intersect and are not parallel. But since you wrote ax + by = c, I won''t explain it.
Hope this helps.
Just make a special case to check if the two lines are parallel. I highly recommend you to treat lines with linear algebra, it becomes much easier when it comes to planes, R^n subspaces, and so on.
If the two director vectors (a director vector for a line is simply a vector from a point to another point on that same line, that is a vector that tells the direction) are colinear (that is, u = kv, where u and v are the vectors and k a scalar), then your lines are not parallel.
This works just fine with lines in R^2 since any non-parallel lines in R^2 always intersect. If you want to intersect two lines in R^3 (that is, in 3D) then you''ll also have to check the special case in wich the two lines simply don''t intersect and are not parallel. But since you wrote ax + by = c, I won''t explain it.
Hope this helps.
Well, if you make that a 3 x 2 matrix then you can make three 2 x 2 matrices by removing one column. If you call those m# where the # is replaced by the number of the column removed then x=-|m0|/||m2| and y=|m1|/|m2| where |a| is the determinant of a. It works similarly for 3D with x=|m0|/|m3|, y=-|m1|/|m3| and z=|m2|/|m3|. You only have to check for one zero either way.
quote:
If the two director vectors (a director vector for a line is simply a vector from a point to another point on that same line, that is a vector that tells the direction) are colinear (that is, u = kv, where u and v are the vectors and k a scalar), then your lines are not parallel.
Besides this there''s quite simple way of determining if the lines are parallel:
If you have 2 lines, which are given by the following equations:
y=k*x+b and y=K*x+B (which can be derived from initial ones "Ax+By=C"
, then these lines are parralel if k=K, else if k*K=-1 they are perpendicular, and so on...
Author
Hi,
The problem is that to figure out what "k" or the delta is, it requires another divide for each line.
tj963
The problem is that to figure out what "k" or the delta is, it requires another divide for each line.
tj963
...so first of all you check if in first equation (ax+by=c) b==0, and also in the second (Ax+By=C) B==0 - that means lines are parralel! if only one of B or b is equaled zero, then lines are not parallel, and if neither b nor B is zero, you consider k and K! That is it!
If a/b = A/B then the lines are parallel. If you multiply both sides by b*B you get a*B = A*b. Then if you subtract A*b from both sides you get a*B-A*b=0 and a*B-A*b is the determinant. So if the determinant is zero the lines are parallel. Ignore what I said about 3D since that would be planes, not lines. You can find the intersection of two lines in 3D in a similar manner, but you have to create three planes from the two lines.
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