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Ellipse Envelope

Ellipse Envelope
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2024-01-27 19:29
Coordinate Geometry
In collections Coordinate Geometry
This example demonstrates the envelope of an ellipse. Key mathematical concepts include the ellipse, envelope, and geometric construction. Using GeoGebra commands such as Polygon, Circle, PointIn, Segment, LineBisector, Intersect, Rotate, Angle, Tangent, OrthogonalLine, etc., the envelope of the ellipse can be constructed. Suitable for high school mathematics education and learning coordinate geometry.

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. PointIn(c) → F
  5. Segment(F, A) → g
  6. LineBisector(g) → envel
  7. Intersect(envel, g) → I
  8. Segment(F, I) → f
  9. Segment(I, A) → h
  10. Rotate(F, 90°, I) → F'
  11. Angle(F, I, F') → α
  12. Tangent(A, c) → tgA,
  13. OrthogonalLine(A, tgA) → normA
  14. Intersect(envel, normA) → E
  15. Ellipse(F, O, E) → d
  16. Angle(tgA, normA) → β

AI Drawing Prompt Reference

Draw an ellipse and its envelope. Start by creating a rectangle as a background, then draw a circle centered within the rectangle. Select a point on the circle, connect it to the center, and find the perpendicular bisector of this line. Take another point on the line and rotate it 90 degrees around the intersection point to get a new point. Connect these two points. Next, draw the tangent line to the circle at the center and its perpendicular line. The intersection of these two lines with the perpendicular bisector gives a point. Finally, draw an ellipse using the center, this point, and the intersection point to demonstrate the envelope.
The prompt above was compiled from the original drawing commands and is for reference only.