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Ellipse, Hyperbola and Circle as Envelopes

Ellipse, Hyperbola and Circle as Envelopes
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2024-01-27 19:42
CircleGeometric ConstructionLocus
In collections Coordinate Geometry
This example demonstrates the construction of envelopes for an ellipse, hyperbola, and circle. Key mathematical concepts include envelopes, geometric construction, and parametric equations. The construction involves GeoGebra commands such as Polygon, Circle, Hyperbola, Ellipse, LineBisector, and Tangent. Suitable for high school students or college undergraduates learning analytic geometry and geometric transformations.

GeoGebra Commands & Parameters

  1. Polygon({(-12, -4.5), (13, -4.5), (13, 8.5), (-12, 8.5)}) → peep
  2. Circle(O, 8) → c
  3. Point(c) → A
  4. Segment((-14, 0), (14, 0)) → domF
  5. Point(domF) → F
  6. Segment(F, A) → g
  7. LineBisector(g) → envel
  8. Intersect(envel, g) → I
  9. Segment(F, I) → f
  10. Segment(I, A) → h
  11. Rotate(F, 90°, I) → F'
  12. Angle(F, I, F') → α
  13. Tangent(A, c) → tgA,
  14. OrthogonalLine(A, tgA) → normA
  15. Intersect(envel, normA) → E
  16. Angle(tgA, normA) → β
  17. Hyperbola(F, O, Radius[c] / 2) → ip
  18. Ellipse(F, O, Radius[c] / 2) → d

AI Drawing Prompt Reference

Within a rectangular area, draw a circle with radius 8 centered at the origin. Pick a point A on this circle. Create a horizontal line segment spanning from left to right, and place a point F on this line. Connect F to A to form a line segment, then construct the perpendicular bisector of this segment. The bisector intersects the original segment at one point. Through point A on the circle, draw the tangent to the circle, then construct a line perpendicular to this tangent. The perpendicular bisector meets this perpendicular line at another point. Finally, using F and the circle's center O as foci, construct both an ellipse and a hyperbola.
The prompt above was compiled from the original drawing commands and is for reference only.