This word "witch" has nothing to do with witchcraft or sorcery. From the Merriam-Webster Dictionary at:

https://www.merriam-webster.com/dictionary/witch%20of%20Agnesi

Maria Gaetana *Agnesi* †1799 Italian mathematician; *witch*, translation of Italian *versiera* cubic curve (influenced by Italian *versiera* female demon).

There is a whole discourse about this geometric entity at: https://en.wikipedia.org/wiki/Witch_of_Agnesi

To understand the derivation of the Witch of Agnesi curve, we have the following:

We begin by drawing a circle centered symmetrically about the y-axis and tangent to the x-axis.

To find an individual point on the curve, we draw a chord from the origin where x=0 and y=0 and extend that chord until it crosses the horizontal line that is tangent to the circle at the very top where y equals twice the circle's radius.

From where the chord crosses the circle, we draw a horizontal line going away from the y-axis and where the chord intersects the upper horizontal line, we draw a vertical line downward toward the x-axis. Where these two lines meet is a point on our curve. In the above sketch, we show that several times with a little red dot placed where each point is found to be.

You can see a bell curve taking shape from this.

If we draw lots and lots of chords, we get lots and lots of points from which (no pun intended) we can trace out the curve itself as follows:

This curve is tangent to the top of the starting circle and asymptotically approaches tangency as the "x" value goes to infinity in each direction.

The highlighted equation for "x" as a function of "y" can be algebraically reversed to find "y" as a function of "x" for which we get: y = 2 * rad / ( 1 + ( 0.5 * x / rad )^2 )

If we integrate the curve to get the total area under the curve for "x" going from minus infinity to plus infinity, that area is four times the area of the starting circle. Therefore, if the radius of the starting circle is rad = sqrt ( 0.25 / pi ), the area under the curve is unity and this curve describes a non-Gaussian probability distribution function.

Please see the Wikipedia URL for further information.

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