Witch of Agnesi
The bell-shaped cubic y = 8a³/(x² + 4a²), misnamed “witch” by a translation slip, and the Cauchy distribution’s profile.
Adjust parameters
Export
Formulas and key values
- Cartesian equation
- Asymptote
- Area · between the curve and the x-axisIn the graph ≈ 8.042
- Inflection points
a is the radius of the generating circle; the peak is at height 2a.
Explainer video
Make a video of the Witch of Agnesi
Opens a video template with the parameters on screen. Every number in the video is computed before it renders; choose landscape or vertical.
Start the video01 · Definition
What is the Witch of Agnesi?
Start with a circle of radius a sitting on the x-axis. For each ray from the origin, combine the height where it meets the circle’s top tangent with the x-position where it crosses the circle: the resulting point traces the bell-shaped curve y = 8a³/(x² + 4a²). Fermat studied it in 1630; Maria Gaetana Agnesi presented it clearly in her landmark 1748 calculus textbook.
The English name is a famous mistranslation: Agnesi called the curve la versiera (the turning curve), which translator John Colson misread as l’avversiera — “the witch”. Under its statistician’s alias the same shape is the Cauchy–Lorentz distribution, the heavy-tailed density whose mean does not exist, and physicists meet it as the Lorentzian line shape of resonance.
02 · Properties
Key properties
- Peak height 2a at x = 0; the x-axis is the asymptote.
- The area between curve and asymptote is 4πa² — four times the generating circle’s area.
- Inflection points occur at x = ±2a/√3.
- Normalized, it is the Cauchy distribution: undefined mean, infinite variance.
- The volume of revolution about the asymptote is 4π²a³.
03 · Questions
Witch of Agnesi: common questions
- What is the equation of the Witch of Agnesi?
- Cartesian equation: y = (8a^3)/(x^2 + 4a^2). a is the radius of the generating circle; the peak is at height 2a.
- What is the area of the Witch of Agnesi?
- Area (between the curve and the x-axis): A = 4π a^2. a is the radius of the generating circle; the peak is at height 2a.
04 · Keep exploring