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Dupin Cyclide

Dupin Cyclide
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2015-02-26 03:28
CircleCoordinate GeometryGeometric Construction
In collections surface 3D surface 3D
This example demonstrates the construction of Dupin cyclide, a special type of quadratic surface consisting of four intersecting circles. By adjusting parameters a, b, c, and d, the positions and sizes of these circles can be changed. The example uses Surface, CurveCartesian, and Circle commands for construction, suitable for high school geometry and coordinate geometry courses.

GeoGebra Commands & Parameters

  1. Surface(((d * (c - ((a * cos(u)) * cos(v)))) + (b^(2) * cos(u))) / (a - ((c * cos(u)) * cos(v))), ((b * sin(u)) * (a - (d * cos(v))) / (a - ((c * cos(u)) * cos(v)))), ((b * sin(v)) * ((c * cos(u)) - d) / (a - ((c * cos(u)) * cos(v)))), u, (-U), U, v, (-V), V) → e
  2. CurveCartesian(t, 0, sqrt(((2 * p) * t)), t, -2, 2) → g
  3. Surface((p / 2 * ((2 * v^(2)) + (k * (1 - u^(2) - v^(2)))) / (1 + u^(2) + v^(2))), ((p * u) * (v^(2) + k) / (1 + u^(2) + v^(2))), ((p * v) * (1 + u^(2) - k) / (1 + u^(2) + v^(2))), u, (-Um), Um, v, (-Vm), Vm) → s
  4. CurveCartesian(t, sqrt(p^(2) - ((2 * p) * t)), 0, t, ((-p)) / 2, p / 2) → f
  5. Circle(((p / 2 * (2 + (k * (1 - UU^(2) - 1))) / (1 + UU^(2) + 1)), ((p * UU) * (1 + k) / (1 + UU^(2) + 1)), ((-p) * (1 + UU^(2) - k) / (1 + UU^(2) + 1))), ((p / 2 * ((k * (1 - UU^(2)))) / (1 + UU^(2))), ((p * UU) * k / (1 + UU^(2))), 0), ((p / 2 * (2 + (k * (1 - UU^(2) - 1))) / (1 + UU^(2) + 1)), ((p * UU) * (1 + k) / (1 + UU^(2) + 1)), (p * (1 + UU^(2) - k) / (1 + UU^(2) + 1)))) → r
  6. Circle(((p / 2 * ((2 * VV^(2)) + (k * (-VV^(2)))) / (2 + VV^(2))), ((-p) * (VV^(2) + k) / (2 + VV^(2))), ((p * VV) * (2 - k) / (2 + VV^(2)))), ((p / 2 * ((2 * VV^(2)) + (k * (1 - VV^(2)))) / (1 + VV^(2))), 0, ((p * VV) * (1 - k) / (1 + VV^(2)))), ((p / 2 * ((2 * VV^(2)) + (k * (-VV^(2)))) / (2 + VV^(2))), (p * (VV^(2) + k) / (2 + VV^(2))), ((p * VV) * (2 - k) / (2 + VV^(2))))) → q
  7. CurveCartesian(((d * (c - ((a * cos(Uu)) * cos(v)))) + (b^(2) * cos(Uu))) / (a - ((c * cos(Uu)) * cos(v))), ((b * sin(Uu)) * (a - (d * cos(v))) / (a - ((c * cos(Uu)) * cos(v)))), ((b * sin(v)) * ((c * cos(Uu)) - d) / (a - ((c * cos(Uu)) * cos(v)))), v, (-V), V) → j
  8. CurveCartesian(((d * (c - ((a * cos(u)) * cos(Vv)))) + (b^(2) * cos(u))) / (a - ((c * cos(u)) * cos(Vv))), ((b * sin(u)) * (a - (d * cos(Vv))) / (a - ((c * cos(u)) * cos(Vv)))), ((b * sin(Vv)) * ((c * cos(u)) - d) / (a - ((c * cos(u)) * cos(Vv)))), u, (-U), U) → l

AI Drawing Prompt Reference

Draw a special quadratic surface, the Dupin Cyclide, consisting of four intersecting circles. Adjust parameters to change their positions and sizes. Start by plotting the surface, then add two curves, and finally draw four circles that intersect both the surface and curves, highlighting the geometric properties of the Dupin Cyclide.
The prompt above was compiled from the original drawing commands and is for reference only.