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Hart's A-frame linkage produces straight-line motion

Hart's A-frame linkage produces straight-line motion
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2019-02-13 20:57
Coordinate GeometryTransformationGeometric ConstructionSlider AnimationLocus
① This example demonstrates how Hart's A-frame linkage, a classic mechanical linkage, generates an exact straight-line motion through hinged bars. ② The core mathematical concepts involve geometric loci, straight-line motion, degrees of freedom in linkages, and analytic geometry of lines, verified by tracing the path of a moving point. ③ Construction highlights: using GeoGebra commands such as Circle, Point, Midpoint, Segment, Intersect, Mirror, Line, and LocusEquation, the model builds multiple points (e.g., A, B, C, D, E) and segments, applies mirroring and intersection operations to form the linkage, and finally uses a locus equation to confirm the point moves along a straight line. ④ This is suitable for high school or college-level geometry and mechanism principles, useful for math modeling or engineering demonstrations, helping students intuitively understand the kinematics of linkage mechanisms.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Midpoint(A, C) → D_1
  4. Midpoint(D_1, C) → R
  5. Segment(A, D_1) → a
  6. Segment(A, R) → b
  7. Circle(B, b) → d
  8. Circle(R, a) → e
  9. Intersect(d, e, 2) → S
  10. Segment(B, S) → f
  11. Mirror(B, E) → B'
  12. Mirror(E, B') → E'
  13. Segment(S, E') → g
  14. Line(B', g) → h
  15. Intersect(h, f) → F
  16. Mirror(F, S) → D
  17. Circle(C, a) → k
  18. Circle(D, a) → p
  19. Intersect(k, p) → G, H, ,
  20. Segment(A, C) → i
  21. Segment(C, G) → j
  22. Segment(G, D) → l
  23. Segment(D, B) → m
  24. Segment(R, S) → n
  25. LocusEquation(G, C) → o
  26. Point(xAxis) → I
  27. Point(xAxis) → J
  28. Segment(I, J) → q

AI Drawing Prompt Reference

Draw a Hart's A-frame linkage made of hinged bars to demonstrate how it generates straight-line motion through geometric paths. Start by creating two circles and finding their intersections, then determine multiple midpoints and segments. Use mirroring and intersection operations to construct the linkage structure, and trace the path of a moving point to verify its linear motion. Finally, emphasize the straight-line path with a line and a locus equation, and add a horizontal segment as a reference.
The prompt above was compiled from the original drawing commands and is for reference only.