Medial Curve, Line–Circle

mdlncir.html

Exploration

Show that the two quadratics whose equations are given by

where

,

,

,

,

,

and

s=±1.

are equidistant from the line

and the circle

assuming .

Approach

Create the line and the circle. Form an equation of the distances from a generic point to the line and circle.

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 geometry.

Clear[x,y,A1,B1,C1,h2,k2,r2];
P=Point2D[x,y];
l1=Line2D[A1,B1,C1];
c2=Circle2D[{h2,k2},r2];

Find the distance from the point to the line, where .

Clear[s1,E1];
d1=s1*Distance2D[P,l1] //.
    {A1^2+B1^2->1,Sqrt[E1_^2]->E1}

Find the distance from the point to the circle, where .

Clear[s2];
d2=s2*(Distance2D[P,Point2D[c2]]-r2) //Expand

Rearrange the equation and square both sides.

eq1=(d1-d2[[1]])^2==d2[[2]]^2 /. {s1^2->1, s2^2->1}

Form a quadratic and simplify.

Q1=Quadratic2D[eq1,{x,y}] //.
         {s1^2->1,
          s2^2->1,
          A1^2-1->-B1^2,
          B1^2-1->-A1^2}

Put the quadratic into the desired form, and use

Clear[s,a,b,c];
Q2=(Map[Factor[-1*#]&,Q1] /. s1*s2->s) /. a_*b_+a_*c_->a(b+c)