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Parabola as Envelope

Parabola as Envelope
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2024-01-27 19:06
Geometric ConstructionLocus
In collections Coordinate Geometry
This example demonstrates the geometric construction of a parabola as an envelope. Key mathematical concepts include the equation of a parabola, the definition of an envelope, geometric construction techniques, and optimization in linear programming. Construction involves using GeoGebra commands such as Polygon, PointIn, Line, OrthogonalLine, Intersect, Parabola, and Angle, with objects like peep, DB, DA, dir, F, INT, displayDir, A, f, g, envel, B, go, btnAni, btnClear, speed, slo, con, par, p, I, α, btnPause, semi1, semi2. Suitable for high school students or college freshmen learning coordinate geometry and geometric construction, also useful for mathematical modeling exercises.

GeoGebra Commands & Parameters

  1. Polygon({(-12, -4.5), (13, -4.5), (13, 8.5), (-12, 8.5)}) → peep
  2. PointIn(peep) → DB
  3. PointIn(peep) → DA
  4. Line(DA, DB) → dir
  5. PointIn(peep) → F
  6. Segment(INT(1), INT(2)) → displayDir
  7. Point(displayDir) → A
  8. OrthogonalLine(A, dir) → f
  9. Segment(F, A) → g
  10. LineBisector(g) → envel
  11. Intersect(f, envel) → B
  12. Parabola(F, dir) → p
  13. Intersect(envel, g) → I
  14. Angle(A, I, B) → α
  15. Segment(F, I) → semi1
  16. Segment(I, A) → semi2

AI Drawing Prompt Reference

Construct a parabola as an envelope. First, draw a rectangle, then select two random points inside it and draw a line through them. Next, select a third point inside the rectangle and draw a line perpendicular to the previous line through this point. Find the intersection of these two perpendicular lines, then draw a parabola through this intersection and the initial line. Find the intersection of the parabola and the perpendicular line, and finally draw line segments from the initial point to the perpendicular line intersection and from the perpendicular line intersection to the parabola intersection.
The prompt above was compiled from the original drawing commands and is for reference only.