Intersection Parameters of Two Line Segments

lnsegint.html

Exploration

Show that the parameter values, and of the intersection point of two line segments in terms of the end point coordinates is given by

where

.

What is the significance of the values of and with respect to the standard parameter range for a line segment?

Approach

Create the two line segments and express points on each parametrically. Set the x- and y-coordinates equal to each other and solve for and

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

Create the two line segments.

Clear[x1,y1,x2,y2,x3,y3,x4,y4];
L1=Segment2D[{x1,y1},{x2,y2}];
L2=Segment2D[{x3,y3},{x4,y4}];

Find the point coordinates in terms of parameters.

Clear[t1,t2];
{pt1=Point2D[L1[t1]],pt2=Point2D[L2[t2]]}

Equate the abscissas and ordinates and solve for the parameters.

ans=Solve[{XCoordinate2D[pt1]==XCoordinate2D[pt2],
           YCoordinate2D[pt1]==YCoordinate2D[pt2]},
          {t1,t2}] //FullSimplify

Discussion

The significance of the values of   and lies in the range of values which determine if the two line segments actually intersect. If at the intersection point, then the intersection point is on the first line segment; if at the intersection point, then the intersection point is on the second line segment.