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Viviani's Curve

Viviani's Curve
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2022-02-18 20:17
Coordinate GeometrySolid GeometryParametric Equation
In collections Curves and Surfaces Curves and Spirals on Spheres
This example demonstrates the construction of Viviani's Curve, a type of parametric curve involving trigonometry and solid geometry. By using the parametric equations of a cylinder and a sphere, we can construct Viviani's Curve. The example uses commands such as CurveCartesian, Cylinder, and Sphere, suitable for high school mathematics education or students learning about parametric equations.

GeoGebra Commands & Parameters

  1. CurveCartesian(((r * cos(u)) * cos(u)), ((r * cos(u)) * sin(u)), (r * sin(u)), u, (-π), π) → b
  2. CurveCartesian(((r * cos(u + π / 4)) * cos(u)), ((r * cos(u + π / 4)) * sin(u)), (r * sin(u + π / 4)), u, (-π), π) → c
  3. Cylinder(A, B, r / 2) → d, d_{2}, d_{3}, d_{4}
  4. CurveCartesian(((r * cos(u + π / 2)) * cos(u)), ((r * cos(u + π / 2)) * sin(u)), (r * sin(u + π / 2)), u, (-π), π) → f
  5. CurveCartesian(((r * cos(u + ((3 * π)) / 4)) * cos(u)), ((r * cos(u + ((3 * π)) / 4)) * sin(u)), (r * sin(u + ((3 * π)) / 4)), u, (-π), π) → g
  6. CurveCartesian(((r * cos(u + π)) * cos(u)), ((r * cos(u + π)) * sin(u)), (r * sin(u + π)), u, (-π), π) → h
  7. CurveCartesian(((r * cos(u + ((5 * π)) / 4)) * cos(u)), ((r * cos(u + ((5 * π)) / 4)) * sin(u)), (r * sin(u + ((5 * π)) / 4)), u, (-π), π) → e
  8. CurveCartesian(((r * cos(u + ((6 * π)) / 4)) * cos(u)), ((r * cos(u + ((6 * π)) / 4)) * sin(u)), (r * sin(u + ((6 * π)) / 4)), u, (-π), π) → i
  9. CurveCartesian(((r * cos(u + ((7 * π)) / 4)) * cos(u)), ((r * cos(u + ((7 * π)) / 4)) * sin(u)), (r * sin(u + ((7 * π)) / 4)), u, (-π), π) → j
  10. Sphere((0, 0, 0), r) → a

AI Drawing Prompt Reference

Draw Viviani's curve, which is constructed using the parametric equations of a cylinder and a sphere. First, draw a sphere with radius r centered at the origin. Then, draw four cylinders, each with a radius of r/2 and axes tangent to the sphere. Use the parametric equations to draw Viviani's curve, adjusting the range and values of parameter u to get different parts of the curve.
The prompt above was compiled from the original drawing commands and is for reference only.