Exploring Analyic Geometry with Mathematica®

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Parabola Through Four Points

pb4pts.html

Exploration

Describe a method for finding the two parabolas passing through four points. Show that the technique produces the correct results for the points (2,1), (-1,1), (-2,-1) and (4,-3) by plotting the parabola and the four points.

Approach

Form a quadratic, parameterized by the variable k, representing the pencil of quadratics passing through the four points. The first three coefficients of the quadratic, a, b, and c must satisfy the relationship "pb4pts_1.gif" because the quadratic is a parabola. Solve the equation for k.

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Solution

Define a function that implements the approach.

Quadratic2D[
   p1:Point2D[{x1_,y1_}],
   p2:Point2D[{x2_,y2_}],
   p3:Point2D[{x3_,y3_}],
   p4:Point2D[{x4_,y4_}],
   Parabola2D] :=
Module[{q1,k,a,b,c},
      q1=Quadratic2D[p1,p2,p3,p4,k,Pencil2D];
      {a,b,c}=List @@ Take[q1,3];
      ans=Solve[b^2==4*a*c,k];
      Map[(q1 /. #)&, ans] ];

Discussion

The following is a numerical example using the points specified.

pts={p1=Point2D[2,1],
     p2=Point2D[-1,1],
     p3=Point2D[-2,-1],
     p4=Point2D[4,-3]};
q1=Quadratic2D[p1,p2,p3,p4,Parabola2D] //N

"pb4pts_2.gif"

par1=Map[Loci2D,q1]

"pb4pts_3.gif"

Sketch2D[{pts,par1},CurveLength2D->20]

"pb4pts_4.gif"

Graphics saved as "pb4pts01.eps".


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