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Nephroid Curve: Geometric Construction and Locus Demonstration

Nephroid Curve: Geometric Construction and Locus Demonstration
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2023-03-17 02:32
CircleTransformationParametric EquationGeometric ConstructionLocus
In collections geometriadinamica.es
This GeoGebra example demonstrates the generation of the nephroid curve, a special epicycloid-like shape, by tracing intersections of rotated points on circles. It visually reveals the heart-like contour through dynamic rotation and locus tracing. Core mathematical concepts include circle geometry, rotation symmetry, locus of points, and parametric curve families. The construction uses GeoGebra commands such as Intersect, Circle, Point, Ray, Angle, Rotate, Segment, CircleArc, and Locus, with sliders n and m controlling the number of rotations and angular steps. Multiple rotated point pairs (e.g., A', C', etc.) are generated and connected, and the final nephroid emerges via the Locus tool. Suitable for high school and undergraduate students studying analytic geometry, parametric curves, or engaging in exploratory mathematics, as well as for teachers demonstrating dynamic curve generation in the classroom.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 1 + m / n) → d
  3. Point(d) → A
  4. Ray(O, A) → b
  5. Angle(B, O, A) → α
  6. Circle(O, B) → c
  7. Intersect(c, b) → D,
  8. Circle(A, m / n) → e
  9. Rotate(D, (n * α / m), A) → C
  10. Segment(A, C) → a
  11. CircleArc(A, D, C) → f
  12. CircleArc(O, B, D) → g
  13. Segment(O, A) → h
  14. Rotate(A, 29.999999999999996°, A) → A'
  15. Rotate(C, 29.999999999999996°, A) → C'
  16. Segment(A', C') → i
  17. Rotate(A', 29.999999999999996°, A) → A''
  18. Rotate(C', 29.999999999999996°, A) → C''
  19. Segment(A'', C'') → j
  20. Rotate(A'', 29.999999999999996°, A) → A'''
  21. Rotate(C'', 29.999999999999996°, A) → C'''
  22. Segment(A''', C''') → k
  23. Rotate(A''', 29.999999999999996°, A) → A''''
  24. Rotate(C''', 29.999999999999996°, A) → C''''
  25. Segment(A'''', C'''') → l
  26. Rotate(A'''', 29.999999999999996°, A) → A'''''
  27. Rotate(C'''', 29.999999999999996°, A) → C'''''
  28. Segment(A''''', C''''') → m_1
  29. Rotate(A''''', 29.999999999999996°, A) → A''''''
  30. Rotate(C''''', 29.999999999999996°, A) → C''''''
  31. Segment(A'''''', C'''''') → n_1
  32. Rotate(A'''''', 29.999999999999996°, A) → A'''''''
  33. Rotate(C'''''', 29.999999999999996°, A) → C'''''''
  34. Segment(A''''''', C''''''') → p
  35. Rotate(A''''''', 29.999999999999996°, A) → A''''''''
  36. Rotate(C''''''', 29.999999999999996°, A) → C''''''''
  37. Segment(A'''''''', C'''''''') → q
  38. Rotate(A'''''''', 29.999999999999996°, A) → A'''''''''
  39. Rotate(C'''''''', 29.999999999999996°, A) → C'''''''''
  40. Segment(A''''''''', C''''''''') → s
  41. Rotate(A''''''''', 29.999999999999996°, A) → A''''''''''
  42. Rotate(C''''''''', 29.999999999999996°, A) → C''''''''''
  43. Segment(A'''''''''', C'''''''''') → t
  44. Rotate(A'''''''''', 29.999999999999996°, A) → A'''''''''''
  45. Rotate(C'''''''''', 29.999999999999996°, A) → C'''''''''''
  46. Segment(A''''''''''', C''''''''''') → a_1
  47. Locus(C, A) → loc1
  48. Rotate(D, (n * α / m), A) → E

AI Drawing Prompt Reference

On a coordinate plane, set the origin as the common center. Construct a circle centered at the origin with a radius determined by two parameters. Create a second circle whose radius is based on the same parameter ratio, and take a ray from the origin along the chosen initial direction. Define the angle between the ray and the reference radius, then build the circle associated with that angle and find an intersection point of the two circles. Using this intersection point, repeatedly rotate it about the origin by a step angle determined from the parameters and the defined angle, generating successive rotated points. Connect each pair of corresponding rotated points to form a chain, and trace the locus of one rotated point relative to the other to reveal the nephroid (heart-like) curve shape.
The prompt above was compiled from the original drawing commands and is for reference only.