Intersection Point of Two Line Segments

lnsegpt.html

Exploration

Show that the intersection point of the lines underlying two line segments and in terms of the coordinates of the four points is given by

.

Approach

Construct the two lines underlying the line segments and intersect the lines.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Define the lines underlying the two line segments.

Clear[x1,y1,x2,y2,x3,y3,x4,y4];
L1=Line2D[p1={x1,y1},p2={x2,y2}];
L2=Line2D[p3={x3,y3},p4={x4,y4}];

Compute the intersection point.

pt=Point2D[L1,L2]

Discussion

Notice that the denominators of the abscissa and ordinate are equal, and that these denominators cannot be zero unless the line segments are parallel, in which case the underlying lines do not intersect.  The following is a plot of a numerical example.

Sketch2D[{Segment2D[p1,p2],Segment2D[p3,p4],
   pt,Map[Point2D,{p1,p2,p3,p4}]} /. {
      x1->2, y1->1,  x2->-2, y2->-2,
      x3->2, y3->-2, x4->-3, y4->1}]

Graphics saved as "lnsegp01.eps".