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Elliptic Curve and Rotational Geometry in a Barcelona Restaurant

Elliptic Curve and Rotational Geometry in a Barcelona Restaurant
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2016-11-09 18:07
Coordinate GeometryTransformationParametric EquationSlider AnimationLocus
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This example demonstrates a geometric construction inspired by a real Barcelona restaurant, featuring an elliptic curve, rotations, reflections, and intersections. The core mathematical concepts include parametric elliptic curves, geometric transformations (rotation, reflection, translation), and dynamic parameter adjustment via sliders. The construction uses GeoGebra commands such as Point, Mirror, Distance, Circle, Cylinder, CurveCartesian, Rotate, Intersect, IntersectPath, Segment, Angle, Sequence, Axes, Focus, and Translate, creating multiple objects (points, lines, circles, elliptic curves, cylinders) with a slider controlling key parameters. It is suitable for high school or university-level geometry and analytic geometry teaching, and for interactive exploration of parametric curves and transformations.

GeoGebra Commands & Parameters

  1. Point(zAxis) → A
  2. Mirror(A, xOyPlane) → A'
  3. Distance(A, A') → h
  4. PointIn(a) → B
  5. Circle(zAxis, B) → c
  6. Cylinder(c, h) → b, d, e
  7. Radius(c) → r
  8. CurveCartesian((r * cos(t)), (r * sin(t)), ((-r) * sin(t) / tan(β)), t, 0, (2 * π)) → g
  9. Rotate(g, π / 12, zAxis) → g'
  10. Intersect(yAxis, zAxis) → E
  11. IntersectPath(i, b) → k
  12. Corner(k) → F, G, H, I
  13. Segment(E, H) → n
  14. Angle(H, E, E_0) → γ
  15. CurveCartesian((r * cos(t)), (r * sin(t)), ((-r) * sin(t) / tan(β)), t, ang, (2 * π) - ang) → p
  16. Intersect(f, d) → C, D
  17. Angle(A', E, D) → α
  18. Sequence(Rotate[p, t, zAxis], t, 0, (2 * π), π / 12) → llista1
  19. Mirror(llista1, j) → llista1'
  20. Intersect(yAxis, k) → N,
  21. Axes(k) → l, m
  22. Intersect(n, k, 1) → L
  23. Focus(k) → P, Q
  24. Translate(d, Vector[(0, 0, ((-dc)) / 8)]) → d'
  25. If(z(A') - (quo * dc) < z(A), quo - 1, quo) → top
  26. Sequence(Translate[d', Vector[(k * (0, 0, (-dc)))]], k, 1, top) → llista2
  27. Segment(T, U) → t
  28. Angle(t, yAxis) → δ

AI Drawing Prompt Reference

Draw a three-dimensional geometric construction featuring an elliptic curve and rotational symmetry. Start with a vertical axis and a horizontal circle centered on it, which forms the base of a cylinder. On the cylinder's surface, draw a tilted elliptic curve that winds around the cylinder. Rotate this curve by one-twelfth of a full turn around the vertical axis to create a second curve. Inside the cylinder, intersect a plane with the cylinder to obtain a closed curve, and mark its four corner points. Connect one corner to a point on the vertical axis with a segment, and measure the angle between this segment and the horizontal plane. Additionally, generate a family of curves by rotating the elliptic curve in steps around the axis, and reflect this family across a plane. Finally, mark the foci, axes, and intersection points on the cylinder, and draw a translated copy of the cylinder's base curve. The whole scene includes points, lines, circles, cylinders, elliptic curves, intersection paths, and angle labels, with a slider to adjust key parameters.
The prompt above was compiled from the original drawing commands and is for reference only.