Locus of Intersection

Q: Consider the circle and the parabola . First, find the intersection points of the circle and the parabola. Second, compute the chord of the circle through the two intersection points. Third, compute the line segment through perpendicular to the chord at point . Then, find the locus of the point . And, finally, animate the process as a function of . How can I do this?

A: Computing the intersection points of the circle and the parabola as a function of r is immediate. The result is rather complicated so we suppress the output.

Next, we compute the chord, which is the line segment joining the two real intersection points, using pattern matching.

To compute the line through the point perpendicular to the chord at point , we use the ExtendGraphics`Geometry` package by Tom Wickham-Jones, available from MathSource (library.wolfram.com/database/Books/3753). After putting the ExtendGraphics` folder into the AddOns`Applications` directory (or, alternatively, in the directory returned by evaluating $UserAddOnsDirectory), we can load it in the usual fashion.

After converting the chord to an ImplicitLine object (this implicit parametrization is the loci of points that satisfy the equation where is a point on the line and is the normal to the line), we can determine the point as a function of using ClosestPointOnLine.

We are now in a position to show the chord, perpendicular, and locus of the point as a function of .

Finding the exact locus of the point is more complicated. Because points on the parabola have coordinates , computing the locus of the point as a function of and —the two intersection points—is straightforward.

Returning to the original equations, after eliminating , the real values of at the two intersection points can be determined using CylindricalAlgebraicDecomposition.

Hence, we obtain an explicit expression for the locus of .

Although this expression can be simplified further (note that RootReduce fails here), it is sufficient for our purpose. After plotting the locus, we can display it along with the circle, parabola, chord, and perpendicular.