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A series of hypocycloids is given. The first is a three pointed star marked a/b = 3. The second is a four pointed star marked a/b = 4. The third is a five pointed star marked a/b = 5. None of these first three figures has lines that cross each other. The fourth figure is a five pointed star but this one has lines which cross each other and looks like the star that children first learn to draw; it is marked a/b = 5/3. A similar sort of star with seven points is next and is marked a/b = 7/3. Then a similar star with eight points is next and is marked a/b = 8/3. The next figure is a complicated series of curves that ultimately creates a small rosette in the middle; this is marked a/b = π. Lastly, there is an even more complicated series of curves that creates a large rosette with sharper florets marked a/b = the square root of 2.
Graph of various hypocycloids corresponding to different values of a / b .

The witch of agnesi

Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi . Why a witch?

Maria Gaetana Agnesi (1718–1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versoria,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a “female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since.

The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points ( 0 , 0 ) and ( 0 , 2 a ) are points on the circle ( [link] ). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA . Let B denote the point at which the line OA intersects the horizontal line through ( 0 , 2 a ) . The vertical line through B intersects the horizontal line through A at the point P . As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle.

Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves.

A circle with bottom at point O (the origin) and top at point (0, 2a) is drawn. The x axis is drawn from point O, and the y axis is drawn up from point O through (0, 2a). Parallel to the x axis is a line drawn from (0, 2a); it has point B marked to the right. A line from point B to point O passes through the circle at point A. A line is drawn parallel to the x axis from point A, and it forms a right angle with a line drawn down from point B; these lines intersect at point P. There is a curve that is symmetric about the y axis that passes through the point P. This curve has its maximum at (0, 2a) and gently decreases through the point P.
As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle.
  1. On the figure, label the following points, lengths, and angle:
    1. C is the point on the x -axis with the same x -coordinate as A .
    2. x is the x -coordinate of P , and y is the y -coordinate of P .
    3. E is the point ( 0 , a ) .
    4. F is the point on the line segment OA such that the line segment EF is perpendicular to the line segment OA .
    5. b is the distance from O to F .
    6. c is the distance from F to A .
    7. d is the distance from O to B .
    8. θ is the measure of angle C O A .

    The goal of this project is to parameterize the witch using θ as a parameter. To do this, write equations for x and y in terms of only θ .
  2. Show that d = 2 a sin θ .
  3. Note that x = d cos θ . Show that x = 2 a cot θ . When you do this, you will have parameterized the x -coordinate of the curve with respect to θ . If you can get a similar equation for y , you will have parameterized the curve.
  4. In terms of θ , what is the angle E O A ?
  5. Show that b + c = 2 a cos ( π 2 θ ) .
  6. Show that y = 2 a cos ( π 2 θ ) sin θ .
  7. Show that y = 2 a sin 2 θ . You have now parameterized the y -coordinate of the curve with respect to θ .
  8. Conclude that a parameterization of the given witch curve is
    x = 2 a cot θ , y = 2 a sin 2 θ , < θ < .
  9. Use your parameterization to show that the given witch curve is the graph of the function f ( x ) = 8 a 3 x 2 + 4 a 2 .

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Brian Reply
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Rafiq
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Damian
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LITNING Reply
what is a peer
LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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Daniel
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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