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三维贝塞尔曲线

三维贝塞尔曲线
几何画板及绘图内容加载中，请稍等……
2015-12-29 05:58
向量函数立体几何参数方程几何作图
①本范例演示三维空间中的贝塞尔曲线生成与展示；②涉及核心数学概念包括参数方程、空间几何、向量运算；③作图要点：使用Segment（线段）、CurveCartesian（笛卡尔曲线）、Surface（曲面）等命令，构建点A、B、C、D、E及参数a、b、c、d等对象；④适合高中及以上学生及数学爱好者学习空间曲线与参数化曲面。

GeoGebra 命令及参数

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, D) → c
  4. Segment(D, E) → d
  5. CurveCartesian(((1 - t)^(2) * A) + (((2 * (1 - t)) * t) * B) + (t^(2) * C), t, 0, 1) → c11
  6. CurveCartesian(((1 - t)^(2) * B) + (((2 * (1 - t)) * t) * C) + (t^(2) * D), t, 0, 1) → c12
  7. CurveCartesian(((1 - t)^(2) * C) + (((2 * (1 - t)) * t) * D) + (t^(2) * E), t, 0, 1) → c13
  8. CurveCartesian(((1 - t)^(3) * A) + (((3 * (1 - t)^(2)) * t) * B) + (((3 * (1 - t)) * t^(2)) * C) + (t^(3) * D), t, 0, 1) → c21
  9. CurveCartesian(((1 - t)^(3) * B) + (((3 * (1 - t)^(2)) * t) * C) + (((3 * (1 - t)) * t^(2)) * D) + (t^(3) * E), t, 0, 1) → c22
  10. CurveCartesian(((1 - t)^(4) * A) + (((4 * (1 - t)^(3)) * t) * B) + (((6 * (1 - t)^(2)) * t^(2)) * C) + (((4 * (1 - t)) * t^(3)) * D) + (t^(4) * E), t, 0, 1) → c3
  11. Surface(x((((A * (1 - p)) + (p * B)) * (1 - t)) + (t * ((B * (1 - p)) + (p * C)))), y((((A * (1 - p)) + (p * B)) * (1 - t)) + (t * ((B * (1 - p)) + (p * C)))), z((((A * (1 - p)) + (p * B)) * (1 - t)) + (t * ((B * (1 - p)) + (p * C)))), t, 0, U, p, 0, 1) → s11
  12. Surface(x((((B * (1 - p)) + (p * C)) * (1 - t)) + (t * ((C * (1 - p)) + (p * D)))), y((((B * (1 - p)) + (p * C)) * (1 - t)) + (t * ((C * (1 - p)) + (p * D)))), z((((B * (1 - p)) + (p * C)) * (1 - t)) + (t * ((C * (1 - p)) + (p * D)))), t, 0, U, p, 0, 1) → s12
  13. Surface(x((((C * (1 - p)) + (p * D)) * (1 - t)) + (t * ((D * (1 - p)) + (p * E)))), y((((C * (1 - p)) + (p * D)) * (1 - t)) + (t * ((D * (1 - p)) + (p * E)))), z((((C * (1 - p)) + (p * D)) * (1 - t)) + (t * ((D * (1 - p)) + (p * E)))), t, 0, U, p, 0, 1) → s13
  14. Surface(x((c11(u) * (1 - v)) + (v * c12(u))), y((c11(u) * (1 - v)) + (v * c12(u))), z((c11(u) * (1 - v)) + (v * c12(u))), u, 0, U, v, 0, 1) → s112
  15. Surface(x((c12(u) * (1 - v)) + (v * c13(u))), y((c12(u) * (1 - v)) + (v * c13(u))), z((c12(u) * (1 - v)) + (v * c13(u))), u, 0, U, v, 0, 1) → s123
  16. Surface(x((c21(u) * (1 - v)) + (v * c3(u))), y((c21(u) * (1 - v)) + (v * c3(u))), z((c21(u) * (1 - v)) + (v * c3(u))), u, 0, U, v, 0, 1) → s213
  17. Surface(x((c22(u) * (1 - v)) + (v * c3(u))), y((c22(u) * (1 - v)) + (v * c3(u))), z((c22(u) * (1 - v)) + (v * c3(u))), u, 0, U, v, 0, 1) → s223