Line-line intersection

Line-line intersection

In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases, and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection.

The number and locations of possible intersections between two lines and the number of possible lines with no intersections (parallel) with a given line are the distinguishing features of Non-Euclidean geometry. The entry titled "Parallel postulate" provides additional background on this topic.

Mathematics

The intersection of two lines L_1, and L_2, in 2 dimensional space. With line L_1, being defined by two points (x_1,y_1), and (x_2,y_2),, and line L_2, being defined by two points (x_3,y_3), and (x_4,y_4),. ]

The intersection P, of line L_1, and L_2, can be defined using determinants.

:Px = frac{egin{vmatrix} egin{vmatrix} x_1 & y_1\x_2 & y_2end{vmatrix} & egin{vmatrix} x_1 & 1\x_2 & 1end{vmatrix} \\ egin{vmatrix} x_3 & y_3\x_4 & y_4end{vmatrix} & egin{vmatrix} x_3 & 1\x_4 & 1end{vmatrix} end{vmatrix} }{egin{vmatrix} egin{vmatrix} x_1 & 1\x_2 & 1end{vmatrix} & egin{vmatrix} y_1 & 1\y_2 & 1end{vmatrix} \\ egin{vmatrix} x_3 & 1\x_4 & 1end{vmatrix} & egin{vmatrix} y_3 & 1\y_4 & 1end{vmatrix} end{vmatrix,!Py = frac{egin{vmatrix} egin{vmatrix} x_1 & y_1\x_2 & y_2end{vmatrix} & egin{vmatrix} y_1 & 1\y_2 & 1end{vmatrix} \\ egin{vmatrix} x_3 & y_3\x_4 & y_4end{vmatrix} & egin{vmatrix} y_3 & 1\y_4 & 1end{vmatrix} end{vmatrix} }{egin{vmatrix} egin{vmatrix} x_1 & 1\x_2 & 1end{vmatrix} & egin{vmatrix} y_1 & 1\y_2 & 1end{vmatrix} \\ egin{vmatrix} x_3 & 1\x_4 & 1end{vmatrix} & egin{vmatrix} y_3 & 1\y_4 & 1end{vmatrix} end{vmatrix,!

The determinants can be written out as:

egin{align}P(x,y)= igg(&frac{(x_1 y_2-y_1 x_2)(x_3-x_4)-(x_1-x_2)(x_3 y_4-y_3 x_4)}{(x_1-x_2)(y_3-y_4)-(y_1-y_2)(x_3-x_4)}, \ &frac{(x_1 y_2-y_1 x_2)(y_3-y_4)-(y_1-y_2)(x_3 y_4-y_3 x_4)}{(x_1-x_2)(y_3-y_4)-(y_1-y_2)(x_3-x_4)}igg)end{align}

Note that the intersection point is for the infinitely long lines defined by the points, rather than the line segments between the points, and can produce an intersection point beyond the lengths of the line segments.

ee also

*Line segment intersection

References


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