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Degrees of Freedom, Local Rigidity, and Global Rigidity

Degrees of Freedom, Local Rigidity, and Global Rigidity
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2022-02-24 23:49
TrigonometryCoordinate GeometryCalculusGeometric Construction
In collections Linkages
This example demonstrates the geometric relationships between degrees of freedom, local rigidity, and global rigidity in linkages. It involves core mathematical concepts such as geometric constructions, trigonometric functions, coordinate geometry, and calculus. The construction utilizes commands like Point, Segment, Circle, and Intersect, along with objects like P3, Q3, r3, P1, Q1, r1, O, c1, U, P2, Q2, r2, c2, E, op1, op2, op3, op4, op5a, op6a, b1, P4, Q4, r4, c3a, C2a, C2b, c3b, D1, b3c, c4, F, F', op6b, op5b, c3c, D1b, D2, b2, f. It is suitable for high school and above students to understand complex geometric and physical concepts.

GeoGebra Commands & Parameters

  1. Point(Ray[P3, P3 + (1, 0)]) → Q3
  2. Segment(P3, Q3) → r3
  3. Point(Ray[P1, P1 + (1, 0)]) → Q1
  4. Segment(P1, Q1) → r1
  5. Circle(O, r1) → c1
  6. Point(c1) → U
  7. Point(Ray[P2, P2 + (1, 0)]) → Q2
  8. Segment(P2, Q2) → r2
  9. Circle(O, r2) → c2
  10. Point(c2) → E
  11. Segment(O, U) → b1
  12. Point(Ray[P4, P4 + (1, 0)]) → Q4
  13. Segment(P4, Q4) → r4
  14. Circle(U, r3) → c3a
  15. Intersect(c2, c3a) → C2a, C2b, ,
  16. Circle(E, r3) → c3b
  17. Point(c3b) → D1
  18. Segment(E, D1) → b3c
  19. Circle(U, r4) → c4
  20. Intersect(c3b, c4) → F, F', ,
  21. Circle(O, r3) → c3c
  22. Point(c3c) → D1b
  23. Point(c4) → D2
  24. Segment(O, E) → b2
  25. Segment(O, D1b) → f
  26. Segment(E, F) → b3d
  27. Segment(E, F') → b3d'
  28. Segment(U, F) → b4a
  29. Segment(U, F') → b4b
  30. Segment(U, C2a) → b2a
  31. Segment(U, C2b) → b2b
  32. Segment(O, C2a) → b3a
  33. Segment(O, C2b) → b3b
  34. Segment(D2, U) → g

AI Drawing Prompt Reference

Draw a geometric figure with multiple circles and segments, including points P3, P1, and P2, generating parallel segments Q3, Q1, and Q2, and creating circles c1, c2, and c3a with these segments as radii. Use the intersection of circles to generate points U, E, C2a, C2b, D1, F, F', D1b, and D2. Connect these points and the centers with segments b1, b2, b3c, b3d, b3d', b4a, b4b, b2a, b2b, b3a, b3b, and g to illustrate the geometric relationships of local and global rigidity.
The prompt above was compiled from the original drawing commands and is for reference only.