Medial Curve, Circle–Circle

mdcircir.html

Exploration

Show that the two quadratics whose equations are given by

where

,

,

,

,

and

and

,

,

and

s=±1

are equidistant from the two circles

and .

Approach

Create the two circles and form an equation by equating the distance to each circle from a generic point.

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,h1,k1,r1,h2,k2,r2];
P=Point2D[x,y];
C1=Circle2D[{h1,k1},r1];
C2=Circle2D[{h2,k2},r2];

Find the distances to the two circles, where and .

Clear[s1];
d1=s1*(Distance2D[P,Point2D[C1]]-r1)

Clear[s2];
d2=s2*(Distance2D[P,Point2D[C2]]-r2)

Equate the two distances and simplify by making substitutions.

Clear[E1];
eq1=d1-d2 /. Sqrt[E1_]:>Sqrt[Expand[E1]]

Clear[D1,D2];
eq2=eq1 /. {h1^2+k1^2->D1,h2^2+k2^2->D2}

Rearrange the equation and square both sides (twice).

{lhs=eq2[[1]]//Expand,rhs=eq2[[2]]//Expand}

Clear[s,R];
eq3=((lhs[[1]]+rhs[[1]])^2 - (lhs[[2]]+rhs[[2]])^2 //Expand) //.
    {s1^2->1, s2^2->1, s1*s2->s, r1^2-2*s*r1*r2+r2^2->R}

eq4=Drop[eq3,-1]^2-Last[eq3]^2

Form a quadratic and simplify.

Q1=Map[Factor,
       Quadratic2D[eq4 /. s^2->1,{x,y}]] //. {
      s^2->1,
      h1^2-2*h1*h2+h2^2->(h1-h2)^2,
      k1^2-2*k1*k2+k2^2->(k1-k2)^2,
      D1^2-2*D1*D2+D2^2->(D1-D2)^2}

By inspection, the resulting quadratic is the same as the desired one.

Q2=Quadratic2D[Q1[[1]],Q1[[2]],Q1[[3]],
      Collect[Q1[[4]],{h1,h2}],
      Collect[Q1[[5]],{k1,k2}],
      Q1[[6]] ]