Distance Between Parallel Lines

lnsdst.html

Exploration

Demonstrate that the distance, d, between two parallel lines

and

is given by

Approach

Create two lines perpendicular to both given lines and passing through the origin. Find the points of intersection between the original lines and the perpendicular lines. Compute the distance between the intersection points, with is the distance between the parallel 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

Create the two given lines.

Clear[A,B,C1,C2];
l1=Line2D[A,B,C1];
l2=Line2D[A,B,C2];

Construct two lines perpendicular to the given lines.

L1=Line2D[Point2D[0,0],l1];
L2=Line2D[Point2D[0,0],l2];

Intersect the lines in pairs to find the intersection points.

p1=Point2D[l1,L1];
p2=Point2D[l2,L2];

Find the distance between the intersection points.

Distance2D[p1,p2] //Simplify

Discussion

Define a new function to compute the distance between two parallel lines.  A more general function could be developed that allows parallel lines whose linear coefficients are multiples of each other.

Distance2D[Line2D[A_,B_,C1_],
           Line2D[A_,B_,C2_]]:=
   Sqrt[(C1-C2)^2/(A^2+B^2)];

Find the distance between the two lines 2x+3y+4=0 and 2x+3y-3=0 using the new function.

Distance2D[Line2D[2,3,4],Line2D[2,3,-3]]