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Intersection Parameters of Two Line Segments

lnsegint.html

Exploration

Show that the parameter values, "lnsegint_1.gif"and "lnsegint_2.gif" of the intersection point of two line segments in terms of the end point coordinates is given by

"lnsegint_3.gif"

"lnsegint_4.gif"

where

"lnsegint_5.gif".

What is the significance of the values of "lnsegint_6.gif"and "lnsegint_7.gif" 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 "lnsegint_8.gif" and "lnsegint_9.gif"

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]]}

"lnsegint_10.gif"

Equate the abscissas and ordinates and solve for the parameters.

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

"lnsegint_11.gif"

Discussion

The significance of the values of  "lnsegint_12.gif" and "lnsegint_13.gif" lies in the range of values which determine if the two line segments actually intersect. If "lnsegint_14.gif" at the intersection point, then the intersection point is on the first line segment; if "lnsegint_15.gif" at the intersection point, then the intersection point is on the second line segment.


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