Witch of Agnesi -Stephanie Storer, Danny Livarchik

The Witch of Agnesi is a famous curve studied by an Italian mathematician, Maria Gaetana Agnesi (1718-1799). The name of this curve came about when translator John Colson misread Agnesi's description of the curve "la versiera" (which means "the turn") as "l'aversiera," which he thought meant to say "l'avversiera." This corrected translation means "witch" in Italian, so the curve adopted the name the Witch of Agnesi and has kept the name since then. \342\200\234The curve is obtained by drawing a line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSNPQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRidGK0Y6 from the origin through the circle of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= and center LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiKEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EldHJ1ZUYnLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZELUkjbW5HRiQ2JFEiMEYnRi8tRiw2LVEiLEYnRi8vRjNGNy9GNkY0L0Y5RjdGOkY8Rj5GQC9GQ1EmMC4wZW1GJy9GRlEsMC4zMzMzMzMzZW1GJy1JI21pR0YkNiVRImFGJy8lJ2l0YWxpY0dGNC9GMFEnaXRhbGljRidGLw==), then picking the point with the y coordinate of the intersection with the circle and the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo=coordinate of the intersection of the extension of line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSNPQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRidGK0Y6 with the line y =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvRjhRJ2l0YWxpY0YnRjdGK0Y3RitGNw==.\342\200\235

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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 I. Generation of the Witch of Agnesi Curve The Witch of Agnesi Curve is first plotted by taking the following two vector-valued functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0Y2RjlGQEZARkBGK0ZARitGQA== = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= and setting some arbitrary value for a which will help define the radius of our circle. In our presentation we set a = 1, so the curve is set on a more understandable scale. If we were to choose larger values for a, the circle would be larger forcing the curve to have a higher peak and it would still lie tangent to the line y = 2a and asymptotically to the x-axis. PkkiYUc2IiIiIg== QyY+SSNSYUc2Ii1JJXBsb3RHRiU2JTclLCQqJkkiYUdGJSIiIi1JJHRhbkdGJTYjSSZ0aGV0YUdGJSEiIiIiIywkRixGMy9GMTssJEkjUGlHJSpwcm90ZWN0ZWRHI0YtIiInLUkiKkdGOTYkLCRGOCIiJkY6L0koc2NhbGluZ0dGJUksQ09OU1RSQUlORURHRiUvSSZ0aXRsZUdGJVEjUmFGJUYySSIlR0YlRi0= QyY+SSNSYkc2Ii1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JTclKiZJImFHRiUiIiItSSRzaW5HRiU2IywkSSZ0aGV0YUdGJSIiI0YvLUYuNiMsJkYvRi8tSSRjb3NHRihGMiEiIi9GNDssJEkjUGlHRikjRi8iIictSSIqR0YpNiQsJEY/IiImRkAvSShzY2FsaW5nR0YlSSxDT05TVFJBSU5FREdGKC9JJnRpdGxlR0YlUSNSYkYlRjtJIiVHRiVGLw== QyU+SSJRRzYiLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkNyUsJComLUkkc2luR0YoNiNJJnRoZXRhR0YlIiIiLUkkY29zR0YoRjFGMyIiIy1JIipHRik2JCwkRi9GNkYvL0YyOyIiISwkSSNQaUdGKUY2L0kmdGl0bGVHRiVRIlFGJSEiIkYk II. Our Blueprints of the Curve The Witch of Agnesi curve is traced out by this equation: 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 QyU+SSZmaW5hbEc2Ii1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JTclLCQqJkkiYUdGJSIiIi1JJHRhbkdGKDYjSSZ0aGV0YUdGJSEiIiIiIyomRi9GMCwmRjBGMC1JJGNvc0dGKDYjLCRGNEY2RjVGMC9GNDssJEkjUGlHRikjRjAiIictSSIqR0YpNiQsJEZAIiImRkEvSShzY2FsaW5nR0YlSSxDT05TVFJBSU5FREdGKC9JJnRpdGxlR0YlUSZmaW5hbEYlRjVGJA== *Transition From Parametric to Rectangular. We can eliminate the parameter to better see the equation in terms of x and y. L0kieEc2IiwkKiZJImFHRiQiIiJJInRHRiRGKCIiIw== L0kieUc2IiwkKiZJImFHRiQiIiIsJkYoRigqJEkidEdGJCIiI0YoISIiRiw= L0kidEc2IiomSSJ4R0YkIiIiLUkiL0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCRJImFHRiQiIiNGJw== L0kieUc2IiwkKiZJImFHRiQiIiIsJkYoRigtSSJeRyUqcHJvdGVjdGVkRzYkKiZJInhHRiRGKC1JIi9HNiRGLEkoX3N5c2xpYkdGJDYjLCRGJyIiI0YoRjZGKCEiIkY2 L0kieUc2IiwkKiZJImFHRiQiIiQsJiokSSJ4R0YkIiIjIiIiKiRGJ0YsIiIlISIiIiIp QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJComSSJhR0YoIiIkLCYqJEkieEdGKCIiIyIiIiokRixGMSIiJSEiIiIiKS9GMDshIiRGLUYy QyQtSShkaXNwbGF5RzYiNiM3JkkjUmFHRiVJIlFHRiVJI1JiR0YlSSZmaW5hbEdGJSEiIg== PkkiYUc2IiIiIg== QyQ+SSt3aXRjaFRyYWNlRzYiLUktYW5pbWF0ZWN1cnZlR0YlNiQ3JSwkKiZJImFHRiUiIiItSSR0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiUhIiIiIiMqJkYsRi0sJkYtRi0tSSRjb3NHRjA2IywkRjRGNkY1Ri0vRjQ7LCRJI1BpR0YxI0YtIiInLUkiKkdGMTYkLCRGQCIiJkZBL0koc2NhbGluZ0dGJUksQ09OU1RSQUlORURHRjBGNQ== III. Bringing the components together!!! In this animation, we will show the path which the chord traces out. The chord is only going to be traced out for the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjwtRiM2JC1JI21uR0YkNiRRIjZGJ0Y8RjwvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkovJSliZXZlbGxlZEdGO0Y8RitGPA== to 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 because at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNUYrRjU= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3 the value of the vector-valued function is undefined 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 QyU+SStjaG9yZFRyYWNlRzYiLUkoYW5pbWF0ZUdGJTYpNyUqKEkiYUdGJSIiIkkic0dGJUYsLUkkdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJInRHRiUhIiJGLS9GLTsiIiEiIiMvRjQ7LCRJI1BpR0YxI0YsIiInLUkiKkdGMTYkLCRGPSIiJkY+L0koc2NhbGluZ0dGJUksY29uc3RyYWluZWRHRiUvSSZjb2xvckdGJUkmYmxhY2tHRiUvSSdmcmFtZXNHRiUiJCsiL0kldmlld0dGJTckOyEiJSIiJTtGOCIiJC9JKnRoaWNrbmVzc0dGJUY5RjVGJA== Pkkmd2l0Y2hHNiI3JCwkKiZJImFHRiQiIiItSSR0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kmdGhldGFHRiQhIiIiIiMqJkYoRiksJkYpRiktSSRjb3NHRiw2IywkRjBGMkYxRik= QyU+SSt3aXRjaFRyYWNlRzYiLUktYW5pbWF0ZWN1cnZlR0YlNiY3JSZJJndpdGNoR0YlNiMiIiImRis2IyIiIy9JJnRoZXRhR0YlOywkSSNQaUclKnByb3RlY3RlZEcjRi0iIictSSIqR0Y2NiQsJEY1IiImRjcvSSV2aWV3R0YlNyQ7ISIlIiIlOyIiISIiJC9JJ2ZyYW1lc0dGJSIkKyIvSSp0aGlja25lc3NHRiVGMCEiIkYk **We will now animate the vertical line drawn through the various values of the intersections from the origin to the line 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. QyU+SS52ZXJ0TGluZVRyYWNlRzYiLUkoYW5pbWF0ZUdGJTYpNyUsJComSSJhR0YlIiIiLUkkdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJInRHRiUhIiIiIiNJInNHRiUvRjc7IiIhRjYvRjQ7LCRJI1BpR0YxI0YtIiInLUkiKkdGMTYkLCRGPiIiJkY/L0koc2NhbGluZ0dGJUksY29uc3RyYWluZWRHRiUvSSZjb2xvckdGJUklYmx1ZUdGJS9JJ2ZyYW1lc0dGJSIkKyIvSSV2aWV3R0YlNyQ7ISIlIiIlO0Y6IiIkL0kqdGhpY2tuZXNzR0YlRjZGNUYk QyQ+SS10b3BIb3JpekxpbmVHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQ3JUkic0dGJSIiIy9GLTshIiUiIiUvSSZjb2xvckdGJUkmZ3JlZW5HRiUhIiI= QyQ+SSx3aXRjaENpcmNsZUc2Ii1JKnBvbGFycGxvdEdGJTYkLCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiUiIiMvRjA7IiIhSSNQaUdGLSEiIg== QyU+SStob3JpelRyYWNlRzYiLUkoYW5pbWF0ZUdGJTYqNyUsJiooSSJhR0YlIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJInRHRiVGLS1JJGNvc0dGMEYzRi0iIiMqJkkic0dGJUYtLCYqJkYsRi0tSSR0YW5HRjBGMyEiIkY3RishIiNGLUYtLCQqJEYuRjdGNy9GOTsiIiFGLS9GNDssJEkjUGlHRjEjRi0iIictSSIqR0YxNiQsJEZIIiImRkkvSShzY2FsaW5nR0YlSSxDT05TVFJBSU5FREdGMC9JJmNvbG9yR0YlSSZibGFja0dGJS9JKmxpbmVzdHlsZUdGJUkoZGFzaGRvdEdGJS9JJ2ZyYW1lc0dGJSIkKyIvSSV2aWV3R0YlNyQ7ISIlIiIlO0ZEIiIkL0kqdGhpY2tuZXNzR0YlRjdGLUYk The final result!!!!!! LUkoZGlzcGxheUc2IjYjPChJK2Nob3JkVHJhY2VHRiRJK2hvcml6VHJhY2VHRiRJLHdpdGNoQ2lyY2xlR0YkSSt3aXRjaFRyYWNlR0YkSS10b3BIb3JpekxpbmVHRiRJLnZlcnRMaW5lVHJhY2VHRiQ=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=