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Ellipse in Three Dimensions

Ellipse in Three Dimensions
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2016-12-09 06:18
Coordinate GeometrySolid GeometryTransformationGeometric Construction
In collections Ellipse Analysis
This example demonstrates how to construct an ellipse in three-dimensional space and explores its geometric relationships with cones, spheres, lines, and planes. The core mathematical concepts include the definition of a 3D ellipse as the intersection of a plane with a cone, spatial geometry involving intersections, perpendicularity, distance measurement, midpoints, and reflections. The construction uses GeoGebra commands such as Point, Intersect, IntersectPath, Cone, Corner, OrthogonalLine, Distance, Midpoint, Ellipse, Segment, Circle, Line, and Mirror, creating numerous auxiliary objects (A, B, c, d, g, c_1, c_2, c_3, e, C, D, E, F, G, H, f, I, J, K, L, M, N, O, P, Q) and displaying key lengths via text objects (distanceJI, TextLK). Suitable for high school and above for studying solid geometry, analytic geometry, and conic sections, as well as for dynamic demonstrations in university geometry courses.

GeoGebra Commands & Parameters

  1. Point(zAxis) → A
  2. Intersect(yAxis, xAxis) → B
  3. IntersectPath(d, xOyPlane) → g
  4. Cone(A, B, 8) → c_1, c_2, c_3
  5. IntersectPath(d, c_1) → e
  6. Corner(e) → C, D, E, F
  7. Intersect(zAxis, e) → G,
  8. Intersect(zAxis, d) → H
  9. OrthogonalLine(H, g, space) → f
  10. Intersect(f, e) → I, J
  11. Distance(J, I) → distanceJI
  12. OrthogonalLine(I, xOyPlane) → h
  13. OrthogonalLine(J, xOyPlane) → i
  14. Intersect(h, xOyPlane) → K
  15. Intersect(i, xOyPlane) → L
  16. Midpoint(K, L) → M
  17. Distance(L, K) → distanceLK
  18. Ellipse(F_2, F_1, N) → k
  19. Point(k) → O
  20. Segment(F_2, O) → j
  21. Segment(F_1, O) → l
  22. Circle(F_1, r, xOyPlane) → p
  23. Line(F_1, O) → m
  24. Intersect(m, p) → P, Q
  25. Segment(F_1, P) → n
  26. Mirror(g, M) → g'
  27. Point(p) → R
  28. Segment(R, F_1) → q
  29. Intersect(q, k) → S, T
  30. Segment(F_1, T) → s
  31. Segment(F_2, T) → t

AI Drawing Prompt Reference

Construct a three-dimensional cone with its apex on the z-axis and its base centered at the origin with radius 8. Intersect this cone with a plane to obtain an ellipse as the intersection curve. Identify the four corner points of this ellipse and find where the plane meets the z-axis. From that point, draw a line perpendicular to the base plane, intersecting the ellipse at two points; measure the distance between them. Project these two points vertically onto the base plane, take the midpoint of the projected segment, and measure its length. Using the ellipse's two foci and a point on it, construct the ellipse, then pick a point on it and connect it to both foci. On the base plane, draw a circle centered at one focus, and through that focus and the ellipse point draw a line meeting the circle; connect the focus to the intersection. Finally, mirror the base intersection line about the midpoint, and on the circle pick a point, connect it to the focus, intersect with the ellipse, and connect the focus to that intersection, illustrating the spatial relationships among the cone, plane, ellipse, circle, lines, and points.
The prompt above was compiled from the original drawing commands and is for reference only.