Medial Curve, Point–Line

mdptln.html

Exploration

Show that the quadratic equation

where

,

,

,

,

and

is equidistant from the point and the line , assuming that L is normalized ().

Approach

Create the point and the line. Compute 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 point and the line.

Clear[x,y,x1,y1,A2,B2,C2];
P=Point2D[x,y];
p1=Point2D[x1,y1];
l2=Line2D[A2,B2,C2];

Form an equation by setting the distances (squared) equal to each other.

eq1=Distance2D[P,p1]^2==
    Distance2D[P,l2]^2 //Simplify

Form the quadratic and simplify.

Q1=Quadratic2D[eq1,{x,y}] //. {
      A2^2+B2^2->1,
      1-A2^2->B2^2,
      1-B2^2->A2^2};
Map[Factor,Q1]