Witch of Agnesi

 

Here we will deal with another cubic curve, the Witch of Agnesi (or versiera).
Witch of Agnesi[4]. Consider a fixed circle centered at C(0, a) and tangent to the x-axis at the origin O. For each secant through O, let Q be the intersection of the secant and the circle; let R be the intersection of the secant and the line y = 2a; and let P be the intersection of a line through Q parallel to the x-axis and a line through R parallel to the y-axis. The locus of P, for all such secants, is the Witch of Agnesi (see the screenshot below).

 

Let A be the point (0, 2a) and let the line OR be given by y = mx. Since the circle is x2 + (y - a)2 = a2 , the ordinate of Q will be

  2am2
Qy= ------------
  m2 + 1

(You can carry out the computation by yourself, or find the 1st elimination ideal of <x2 + (y - a)2 - a2, y - mx>by computing the Groebner basis w.r.t. lex order and y>m>a>x). Now, AR and OR intersect at A(2a/m, 2a). Hence, P is given by

  ( 2a   2am2 ).
P = ----- , ------------
  m   m2 + 1

Now that we have a parametrization of the curve, we can find the Cartesian equation by applying the rational implicitization procedure. Since we have only one parameter m, we just need to compute the 1st elimination ideal of <mx - 2a, (m2 + 1)y - 2am2>. Using our Groebner applet and lex order with m>x>y>a, we get the Cartesian equation

x2y + 4a2y - 8a3 = 0.

 

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