Medial Curve, Line–Line

mdlnln.html

Exploration

Show that the pair of lines whose equations are

is equidistant from the two lines and .

Approach

Create both lines. Compute the distances to an arbitrary point. Form an equation by setting the distances equal to each other.

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 lines.

Clear[x,y,A1,B1,C1,A2,B2,C2];
P=Point2D[x,y];
l1=Line2D[A1,B1,C1];
l2=Line2D[A2,B2,C2];

Compute the distance from the first line. Use to eliminate the radical.

Clear[E1,E2,s1];
d1=Distance2D[P,l1] /.
      Sqrt[E1_^2/E2_]:>s1*E1/Sqrt[E2]

Compute the distance from the second line. Use to eliminate the radical.

Clear[s2];
d2=Distance2D[P,l2] /.
      Sqrt[E1_^2/E2_]:>s2*E1/Sqrt[E2]

Form the equation.

eq1=d1==d2

Combine and into a single sign constant s=±1.

Clear[s];
eq1 /. {s1->1,s2->s}