- If 0 <= t <= 1, then they intersect at exactly one point.
- If t < 0 or t > 1, they do not intersect.
- If t is NaN, then the line segment is right on top of line and they intersect at an infinite number of points.
Line and a Line segment intersection
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Does anybody know how to check whether a given line (y=mx+c) intersects with a line segment denoted by end points P1 (x1, y1) and P2(x2, y2)?
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Teja Muppirala
le 29 Sep 2011
Define
t = (c - y1 + m*x1)/(y2 - y1 + m*(x1 - x2))
How can you derive this result?
Well, first parameterize the line segment by a parameter T.
x = x1+(x2-x1)*T
y = y1+(y2-y1)*T
The line segment goes from T = 0 --> 1. Assume the line segment intersects the line y=mx+c as some time t.
Then [y1+(y2-y1)*t] = m*[x1+(x2-x1)*t] + c
Solving for t (using MATLAB to do the symbolic math of course)
t = solve('(y1+(y2-y1)*t) - m*(x1+(x2-x1)*t) + c','t')
you obtain the expression above. If t is outside the range [0, 1] then the lines do not intersect.
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