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Viviani Curve

Viviani Curve
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2022-02-19 18:29
Parametric EquationGeometric ConstructionLocus
This example demonstrates the generation of the Viviani curve. The Viviani curve is formed by the locus of the midpoint of the line segment joining a fixed point on a circle to the circle's center. Key mathematical concepts include parametric equations and loci. It utilizes commands such as Surface, CurveCartesian, UnitVector, Vector, If, and IntegralBetween in GeoGebra. Suitable for high school mathematics education and learning about parametric equations and loci.

GeoGebra Commands & Parameters

  1. Surface(((2 * sin(θ)) * cos(φ)), ((2 * sin(θ)) * sin(φ)), (2 * cos(θ)), θ, 0, (2 * π), φ, 0, (2 * π)) → b
  2. CurveCartesian(1 + cos(t), sin(t), (2 * sin(t / 2)), t, 0, (4 * π)) → c
  3. Surface(cos(u) + 1, sin(u), v, u, 0, (2 * π), v, -2, 2) → a
  4. Surface(v, t - 1, (-sqrt(t)), t, 0, 4, v, -4, 4) → e
  5. CurveCartesian(sin(t), (-cos(t)), (sqrt(2) * sin(t / 2)), t, 0, (4 * π)) → f
  6. CurveCartesian(1 + cos(t), sin(t), (2 * sin(t / 2)), t, 0, t1) → h
  7. CurveCartesian(sin(t), (-cos(t)), (sqrt(2) * sin(t / 2)), t, 0, t2) → k
  8. UnitVector(r1p) → eu1
  9. UnitVector(r1pp) → u
  10. Vector(P1, P11) → v
  11. Vector(P1, P12) → w
  12. Vector(P1, P13) → n
  13. UnitVector(r2p) → eu2
  14. UnitVector(r2pp) → r
  15. Vector(P2, P21) → s
  16. Vector(P2, P22) → t
  17. Vector(P2, P23) → u_1
  18. If(o ≟ 1 ∧ i ≟ 1 ∧ j ≟ 1, true, false) → KADr1
  19. If(p ≟ 1 ∧ l ≟ 1 ∧ m ≟ 1, true, false) → KADr2
  20. Surface(v, t - 1, sqrt(t), t, 0, 4, v, -4, 4) → d
  21. Surface(cos(u), sin(u), v, u, 0, (2 * π), v, (-sqrt(2)), sqrt(2)) → g
  22. IntegralBetween(sqrt(1 + (cos(x / 2))^(2)), 0, 0, t1) → s1Int
  23. IntegralBetween(sqrt(1 + (1 / 2 * (cos(x / 2))^(2))), 0, 0, t2) → s2Int

AI Drawing Prompt Reference

Create the Viviani curve and its related geometric setup. First draw a parametric space curve with coordinates (1+cos t, sin t, 2·sin(t/2)), using t from 0 to 4π. Then construct supportive 3D reference surfaces based on a circle parametrization such as (cos u, sin u, v) with u from 0 to 2π and v spanning a small interval, so the curve is visualized in the correct locus/trajectory context. Include auxiliary vector/UnitVector-based directions to indicate the midpoint/locus construction environment, and incorporate two length-related integral quantities computed over the corresponding t1 and t2 contexts as additional measurement aids.
The prompt above was compiled from the original drawing commands and is for reference only.