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.
For all integers a and b if a is congruent to 0(mod n) and b is congruent to 0(mod n) then a+b is congruent 0(mod n)
DRAW A KNOW-SHOW TABLE:
Phase 1C: Question Writing and Approval
Based on either your own discussion post or ideas sparked from what others mentioned, select two questions you’d like to answer by analyzing data from Census at School.
You will need to select one question from the qualitative category, and one question from the quantitative category. Remember the intent of these questions is to make comparisons and analyze data to eventually make inferences about and possibly draw conclusions about the larger population. You should make notes as you gather your data on what things might be missing, what factors might be contributing to this data, and what questions you still have.
Qualitative Only Options
How are males and females similar or different in their favorite subjects in school?
Quantitative Options
Do the number of texts sent differ between freshmen and seniors in high school?
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.