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Rolling Equilateral Triangle

Rolling Equilateral Triangle
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2015-08-06 20:53
TriangleTransformationGeometric Construction
In collections Rolling Circles and Polygons Rolling Circles and Polygons Rolling Circles and Polygons
This example demonstrates the dynamic process of an equilateral triangle rolling on a plane. Key mathematical concepts include the properties of an equilateral triangle, rotation, and translation transformations. Using GeoGebra commands such as Polygon, Rotate, and CurveCartesian, the rolling trajectory of the triangle can be constructed. Suitable for students at the junior high school level and above to learn about geometric transformations and dynamic geometry.

GeoGebra Commands & Parameters

  1. Polygon((-1, 0), (1, 0), 3) → poly1
  2. Rotate(poly1, If[a < (2 * π / 3), (-a), -2.0944], (1, 0)) → poly1'
  3. Rotate(poly1', If[(2 * π / 3) < a < (4 * π / 3), (-(a - (2 * π / 3))), -2.0943951023931953], (3, 0)) → poly1''
  4. Rotate(poly1'', If[(4 * π / 3) < a < (6 * π / 3), (-(a - (4 * π / 3))), -2.0944], (5, 0)) → poly1'''
  5. Rotate(poly1''', If[(6 * π / 3) < a < (8 * π / 3), (-(a - (6 * π / 3))), -2.0944], (7, 0)) → poly2
  6. Rotate(poly2, If[(8 * π / 3) < a < (10 * π / 3), (-(a - (8 * π / 3))), -2.0944], (9, 0)) → poly2'
  7. Rotate(poly2', If[(10 * π / 3) < a < (12 * π / 3), (-(a - (10 * π / 3))), -2.0944], (11, 0)) → poly2''
  8. Rotate(A, If[a < (2 * π / 3), (-a), -2.0944], (1, 0)) → A'
  9. Rotate(A', If[(4 * π / 3) < a < (6 * π / 3), (-(a - (4 * π / 3))), -2.0944], (5, 0)) → A''
  10. Rotate(A'', If[(6 * π / 3) < a < (8 * π / 3), (-(a - (6 * π / 3))), -2.0944], (7, 0)) → A'''
  11. Rotate(A''', If[(10 * π / 3) < a < (12 * π / 3), (-(a - (10 * π / 3))), -2.0944], (11, 0)) → B
  12. CurveCartesian(1 + (2 * cos((2 * π / 3) - t)), (2 * sin((2 * π / 3) - t)), t, 0, If[a < (2 * π / 3), a, 2.0944]) → b
  13. CurveCartesian(5 + (2 * cos(π - t)), (2 * sin(π - t)), t, 0, If[a < (2 * π), a - (4 * π / 3), 2.0943951023931953]) → c
  14. CurveCartesian(7 + (2 * cos((2 * π / 3) - t)), (2 * sin((2 * π / 3) - t)), t, 0, If[a < (8 * π / 3), a - (2 * π), 2.0944]) → e
  15. CurveCartesian(11 + (2 * cos(π - t)), (2 * sin(π - t)), t, 0, If[a < (4 * π), a - (10 * π / 3), 2.0944]) → f

AI Drawing Prompt Reference

Draw the dynamic process of an equilateral triangle rolling on a plane. Start by drawing an equilateral triangle, then rotate it around different points with varying angles and rotation centers until the triangle completes one full roll. Finally, draw the curves representing the triangle's rolling path.
The prompt above was compiled from the original drawing commands and is for reference only.