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Ellipse and Hyperbola Geometric Construction

Ellipse and Hyperbola Geometric Construction
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2023-03-17 04:16
FunctionTrigonometryCoordinate GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the geometric construction of ellipses and hyperbolas through the rotation of a circle and the locus of points. Key math concepts include conic sections, function graphs, rotational transformations, and analytical geometry. The construction uses GeoGebra commands such as Circle, Point, Angle, CurveCartesian, Ray, Intersect, Rotate, and Segment. Suitable for high school mathematics education to understand the generation principles of conic sections and the application of rotation in geometry.

GeoGebra Commands & Parameters

  1. Circle((0, 0), 6.4) → d
  2. Point(d) → A
  3. Angle((1, 0), (0, 0), A) → α
  4. CurveCartesian((((2 + r) * cos(t)) * cos(α)), ((2 + r) * sin(t)), t, 0, 6.3) → e
  5. Point(e) → B
  6. CurveCartesian(((2 * cos(t)) * cos(α)), (2 * sin(t)), t, 0, 6.28319) → f
  7. Angle((1, 0), O, B0) → β
  8. CurveCartesian(((2 * (((a + 1) * cos(t)) - (b * cos(((a + 1) * t))))) * cos(α) / a), (2 * (((a + 1) * sin(t)) - (b * sin(((a + 1) * t)))) / a), t, 0, 6.283185307179586) → c
  9. CurveCartesian(x(B) + ((r * cos(t)) * cos(α)), y(B) + (r * sin(t)), t, 0, 6.28319) → g
  10. Ray(O, B0) → i
  11. Circle(O, 2 - (r * (b - 1))) → k
  12. Intersect(k, i) → D,
  13. Rotate(D, (a * β), B0) → D'
  14. Segment(B, D'') → h
  15. CurveCartesian(((2 * (((a + 1) * cos(t)) - (b * cos(((a + 1) * t))))) * cos(α) / a), (2 * (((a + 1) * sin(t)) - (b * sin(((a + 1) * t)))) / a), t, 6.283185307179586, 62.83185307179586) → j
  16. CurveCartesian(x(B) + (((b * r) * cos(t)) * cos(α)), y(B) + ((b * r) * sin(t)), t, 0, 6.283185307179586) → l

AI Drawing Prompt Reference

Draw a geometric construction of an ellipse and a hyperbola, including a circle, multiple points, angles, curves, rays, and segments. The circle has a center at the origin and a radius of 6.4. Construct an ellipse and a hyperbola by rotating the circle and the points. The hyperbola's foci are located on either side of the circle, and the ellipse's foci are located inside the circle. Draw the asymptotes of the hyperbola and the ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.