Witch of Agnesi

The versiera of Agnesi, or versiera of Maria Agnesi, is a special plane curve is an algebraic curve of 3rd order from the genus 0

It is named after the mathematician Maria Agnesi, which she published in 1748. The curve has been studied already in 1703 by Pierre de Fermat and Guido Grandi. Due to a translation error, the curve is called in English " witch of Agnesi ". The reason: In Italian, the curve la versiera di Agnesi is, which means " The Curve of Agnesi ". This was from the Cambridge professor John Colson as " l' awersiera di Agnesi " read where " awersiera " which means " woman who is against God " means, then as a "witch" ( "witch " ) has been interpreted, and the mistranslation prevailed.

Construction

Starting with a solid circle, a point O is chosen on the circle. For any other point A on the circle the secant OA is drawn. M is the point diametrically opposite to O. The line OA intersecting the tangent at the point M N. The line parallel to the OM by N, and the line perpendicular to the OM by A intersect at P. When the point A is changed, is the path of the P versiera of Agnesi.

, The curve is asymptotic to the tangent to the circle at the point O.

Equations of versiera of Agnesi

Suppose that the Cartesian coordinate system have the origin in O, and M lies on the positive y- axis; continue to be the diameter of the circle a Then, the following equations of versiera of Agnesi result:

• Cartesian coordinates: or
• Parameter equation:
• Parameter equation using the angle, when the angle between OA and OM (measured clockwise):
• Parameter equation using the angle, when the angle between OA and the x- axis, increasing in the counterclockwise direction:

Here, the parameters.

Properties

• Asymptote:
• Area between the curve and the asymptote:
• Volumes of revolution of the curve to its asymptote:
• Radius of curvature at the apex.
• Two turning points:
• If you compare the representation in Cartesian coordinates according to y, we obtain This is a primitive function of y ( x ), ie.

Credentials