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 “versiera,” 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, 2a) are points on the circle (Figure 7.12). 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, 2a). 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. On the figure, label the following points, lengths, and angle: a. C is the point 011 the x-axis with the same x-Coordinate as A. b. x is the x-coordinate of P, and y is the y-coordinate of P. c. E is the point (0, a). d. F is the point on the line segment 0A such that the line segment EF is perpendicular to the line segment OA. e. b is the distance from O to F . f. c is the distance from F to A. g. d is the distance from O to B. h. 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.
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 “versiera,” 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, 2a) are points on the circle (Figure 7.12). 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, 2a). 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. On the figure, label the following points, lengths, and angle: a. C is the point 011 the x-axis with the same x-Coordinate as A. b. x is the x-coordinate of P, and y is the y-coordinate of P. c. E is the point (0, a). d. F is the point on the line segment 0A such that the line segment EF is perpendicular to the line segment OA. e. b is the distance from O to F . f. c is the distance from F to A. g. d is the distance from O to B. h. 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.
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 “versiera,” 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, 2a) are points on the circle (Figure 7.12). 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, 2a). 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.
On the figure, label the following points, lengths, and angle:
a. C is the point 011 the x-axis with the same x-Coordinate as A.
b. x is the x-coordinate of P, and y is the y-coordinate of P.
c. E is the point (0, a).
d. F is the point on the line segment 0A such that the line segment EF is perpendicular to the line segment OA.
e. b is the distance from O to F .
f. c is the distance from F to A.
g. d is the distance from O to B.
h. 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.
The problem comes from the textbook, Kiselev's Geometry. Book I. Planimetry by A. P. Kiselev:
Prove theorem: Through one of the two intersection points of two circles, a diameter in each of the circles is drawn. Prove that the line connecting the endpoints of these diameters passes through the other intersection point.
2. X
Law of Cosines Text.pd X
6 Witebox
appspot.com
ary Siddhart.
EChallenge Post-Tes..
E Untitled document -..
Air Traffic Control at a local airport keeps track of various aircraft on a radar screen that
has Control at the center. Currently on the screen is a Helicopter and an Airplane. If the
angle formed between the Helicopter and the Airplane with control at the vertex is 42
degrees, the helicopter is 44 miles from control, and the airplane is 40 miles from control,
then what is the distance between the helicopter and the airplane to the nearest
hundredth of a mile?
465 chars
The points A, B, C and D lie on the circle shown in the figure such that AB = BC and DC // AB. The tangent drawn to the circle at A is AL.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.