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Reflection about z=0

Reflection about z=0
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2020-09-29 22:36
Coordinate GeometryTransformationGeometric Construction
In collections Reflecting about Planes
This example demonstrates reflection about the z=0 plane. Key mathematical concepts include reflection transformation and coordinate geometry. Using GeoGebra commands such as Plane, Mirror, Line, Intersect, etc., the construction of symmetric figures about the z-axis is illustrated. Suitable for students at the junior high school level and above to learn about reflection transformations and coordinate geometry.

GeoGebra Commands & Parameters

  1. Plane(A, B, C) → p
  2. Mirror(O, p) → O'
  3. Line(O, O') → f
  4. Intersect(p, f) → D
  5. OrthogonalLine(D, Vector[D, O], Vector[D, B]) → j
  6. Plane(f, j) → q
  7. Dilate(O, If[0 < SlideMe ≤ 1, 1 - SlideMe, 0], D) → O'_{1}
  8. Segment(O, O'_{1}) → g
  9. Dilate(D, If[1 < SlideMe ≤ 2, 2 - SlideMe, 0], O') → D'
  10. Segment(D, D') → h
  11. OrthogonalLine(D, Vector[O, O'], xOyPlane) → i
  12. Point(i) → E
  13. Rotate(O, If[2 < SlideMe ≤ 3, ((SlideMe - 2) * π), π], D, q) → O'_{2}
  14. Rotate(O'_{1}, If[2 < SlideMe ≤ 3, ((SlideMe - 2) * π), π], D, q) → F
  15. Segment(O'_{2}, F) → g'
  16. Point(j) → G
  17. Angle(G, D, O) → α

AI Drawing Prompt Reference

Construct a 3D reflection setup about the plane z = 0: first define the reflecting plane using three points, then use the origin as the reference for its mirrored point across that plane and connect the origin and its mirror with a line. Intersect the reflecting plane with the line to obtain an intersection point D. From D, draw an orthogonal line determined by the given direction vectors to define an auxiliary plane. Use a slide parameter to dilate the origin-related point and the D-related mirror point in different ranges, producing two moving symmetric points, and draw corresponding segments between them. On the auxiliary direction, mark an additional point and finally display the angle α defined by three points.
The prompt above was compiled from the original drawing commands and is for reference only.