Degree 3: Witch of Agnesi
Algebraic Proof of the Witch of Agnesi
The Witch of Agnesi is a curve defined using a circle with diameter OM, where O is the origin (0,0) and M is the point (0,2a) on the positive y-axis. The construction leads to the following Cartesian equation of the
curve: y = 8a^3 / (x^2 +4a^2)
When a = 1/2, the equation simplifies as follows: y = 8(1/2)^3 /(x^2 + 4(1/2)^2) = 1 / (x^2 + 1)
Equivalently, by multiplying through by the denominator, we obtain: (x^2 + 1) y = 1
This simplified form shows that the Witch of Agnesi is the graph of the derivative of the arctangent function, since d/dx(arctan(x)) = 1/ (1 + x^2).
Parametric Representation
The Witch of Agnesi can also be represented parametrically,
where θ is the angle between OM and OA, measured clockwise:
x = 2a tan(θ)
y = 2a cos²(θ)
Thus, the Witch of Agnesi can be studied both algebraically
and geometrically, revealing deep connections between circle geometry, rational
functions, and calculus.
Drag Test:
Drag θ and watch point P trace out the Witch.
Adjust a to see the curve scale.