自制圣诞树

自制圣诞树
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2022-12-12 00:28
①本范例演示如何利用参数方程和三维图形构造一个圣诞树形状;②涉及的核心数学概念包括参数曲线、立体几何、平移变换、三角形和向量运算;③作图要点——从下方『构造摘要』提炼用到了Point(点)、Circle(圆)、Vector(向量)、Cone(圆锥)、CurveCartesian(笛卡尔曲线)、Sequence(数列)、Sphere(球体)等GeoGebra命令/对象;④适合高中及以上学生学习和探索空间几何与参数方程的应用。

GeoGebra 命令及参数

  1. Point(zAxis) → V
  2. Point(Circle[O, r, Line[V, O]]) → C
  3. Circle(O, r, Line[V, O]) → d
  4. Vector(O, C) → w
  5. Cone(O, V, r) → tree, c, e
  6. CurveCartesian(x(V + (h * (O - V + (cos((30 * h)) * v) + (sin((30 * h)) * w)))), y(V + (h * (O - V + (cos((30 * h)) * v) + (sin((30 * h)) * w)))), z(V + (h * (O - V + (cos((30 * h)) * v) + (sin((30 * h)) * w)))), h, 0, draw) → a
  7. Sequence(Circle[V + (h * Vector[V, O]), (r * h), Line[V, O]], h, 0, orCin, 0.1) → cinOr
  8. Sequence(Segment[V, O + (cos(k) * v) + (sin(k) * w)], k, 0, verCin, pi / 12) → cinVert
  9. Point(cinOr) → P1
  10. Point(cinOr) → P2
  11. Point(cinOr) → P3
  12. Point(cinOr) → P4
  13. Point(cinOr) → P5
  14. Point(cinOr) → P6
  15. Point(cinOr) → P7
  16. Point(cinOr) → P8
  17. Point(cinOr) → P9
  18. Point(cinVert) → P10
  19. Point(cinVert) → P11
  20. Point(cinVert) → P12
  21. Point(cinVert) → P13
  22. Point(cinVert) → P14
  23. Point(cinVert) → P15
  24. Point(cinVert) → P16
  25. Point(cinVert) → P17
  26. Sphere(P8, light) → b
  27. Sphere(P1, light) → f
  28. Sphere(P2, light) → g
  29. Sphere(P3, light) → h
  30. Sphere(P4, light) → i
  31. Sphere(P5, light) → j
  32. Sphere(P6, light) → k
  33. Sphere(P7, light) → l
  34. Sphere(P8, light) → m
  35. Sphere(P9, light) → n
  36. Sphere(P10, light) → o
  37. Sphere(P11, light) → p
  38. Sphere(P12, light) → q
  39. Sphere(P13, light) → s
  40. Sphere(P14, light) → t
  41. Sphere(P15, light) → a_1
  42. Sphere(P16, light) → b_1
  43. Sphere(P17, light) → c_1

AI 绘图提示语参考

请绘制一个3D圣诞树形状:在三维空间建立一根竖直轴方向,并从该轴上选取基准圆与半径,形成以圆心为起点的向量与方向。用一个圆锥体叠加生成树干/树体,再沿高度参数画出具有周期侧向摆动的空间曲线(呈螺旋样式),曲线围绕轴线变化并连接到圆锥形轮廓。最后在曲线所对应的离散高度位置上放置多组小球作为“彩球/灯饰”,每个球相对固定在螺旋与竖直框架的交会点处,整体呈现从下到上逐渐延展的树形装饰。
以上提示语为根据原始绘图命令整理而来,仅供参考。