Deriving the Witch of Agnesi Curve

In summary, the Witch of Agnesi Curve is a symmetrical mathematical curve first studied by Italian mathematician Maria Gaetana Agnesi in the 18th century. It can be derived using the reciprocal of a circle's equation and has applications in physics, engineering, and medicine. It can also be generalized to higher dimensions and has been studied by other famous mathematicians such as Pierre de Fermat and Isaac Newton.
  • #1
IvanCow
2
0
Hello,
I have encountered an exercise in my differential geometry book: to derive the parametrisation and the cartesian equation of the curve witch of Agnesi.

To see how it is constructed please see:

http://en.wikipedia.org/wiki/Witch_of_Agnesi"

How would one derive the results? I am lost. Please help. Thanks.
 
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  • #2
From the geometric construction and Phytagora's theorem.
 

1. What is the Witch of Agnesi Curve?

The Witch of Agnesi Curve, also known as the Witch of Maria Agnesi, is a mathematical curve that was first studied by Italian mathematician Maria Gaetana Agnesi in the 18th century. It is a symmetrical curve that resembles a bell shape, and is defined by the equation y = (8a^3) / (x^2 + 4a^2).

2. How is the Witch of Agnesi Curve derived?

The Witch of Agnesi Curve can be derived using various methods, but one common approach is by taking the reciprocal of the equation of a circle. This results in a hyperbola, which can then be shifted and scaled to obtain the final equation of the Witch of Agnesi Curve.

3. What are the real-life applications of the Witch of Agnesi Curve?

The Witch of Agnesi Curve has several real-life applications, including being used in physics to model the trajectory of a bullet, in engineering to describe the shape of a suspension bridge, and in medicine to study the shape of blood flow in arteries.

4. Can the Witch of Agnesi Curve be generalized to higher dimensions?

Yes, the Witch of Agnesi Curve can be generalized to higher dimensions. In 3D, it is known as the Witch of Agnesi Surface and in n-dimensional space, it is referred to as the Witch of Agnesi Hypersurface.

5. Are there any other famous mathematicians who have studied the Witch of Agnesi Curve?

Besides Maria Agnesi, other famous mathematicians who have studied the Witch of Agnesi Curve include Pierre de Fermat, who derived a similar curve in the 17th century, and Isaac Newton, who used it to approximate the shape of the Earth's orbit around the Sun.

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