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Witch of Agnesi - A Classical Construction

Witch of Agnesi - A Classical Construction
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2024-06-29 17:03
CircleFunctionParametric EquationGeometric ConstructionLocus
In collections History of Maths
① This example demonstrates the classical geometric construction of the famous 'Witch of Agnesi' curve, generated by intersections of a circle, lines, and perpendiculars. ② Core mathematical concepts include parametric curves, locus in analytic geometry, and the definition of an algebraic curve through geometric construction; the curve is also relevant in calculus for its asymptote and area properties. ③ Construction highlights: from the summary, GeoGebra commands such as Circle, Point, Intersect, Line, and OrthogonalLine are used, with objects like r (radius), c (circle), A, O, B (points), lineAB, parA, C, D, V (intersection/foot), plus sliders and buttons (speed, btnStart, btnStop) to animate the trace. ④ Suitable for high school to university levels in analytic geometry, calculus, or history of mathematics, and for exploring parametric equations and geometric constructions.

GeoGebra Commands & Parameters

  1. Circle((0, 0), r) → c
  2. Point(c) → A
  3. Intersect(c, Ray[O, (0, -10)]) → B,
  4. Line(B, A) → lineAB
  5. Intersect(c, Ray[O, (0, 10)]) → C,
  6. Intersect(lineAB, tgC) → D
  7. OrthogonalLine(D, tgC) → perpD
  8. Intersect(parA, perpD) → V

AI Drawing Prompt Reference

Draw the geometric construction of the Witch of Agnesi curve. Start with a circle centered at the origin. Take a point A on the circle and a point B on the negative y-axis, then draw a line through A and B. Take another point C on the positive y-axis and draw a horizontal line through C. Find the intersection of this horizontal line with line AB, then draw a vertical line through that intersection. Finally, draw a line parallel to AB that intersects the vertical line; as point A moves around the circle, the locus of these intersection points traces the Witch of Agnesi curve. Include animation to show the curve being generated.
The prompt above was compiled from the original drawing commands and is for reference only.