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抛物线截面演示

抛物线截面演示
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2019-03-07 00:27
立体几何微积分积分几何作图
①本范例演示了通过抛物线底部的不同截面来展示其三维形态；②涉及的核心数学概念包括微积分、积分、抛物线几何性质、三维立体图形相似变换；③作图要点提炼自GeoGebra命令：IndexOf用于索引对象，Intersect用于求交点，Polygon用于构建多边形，CurveCartesian用于绘制笛卡尔曲线，Surface用于创建曲面，Segment用于画线段，Midpoint用于找中点，CircumcircleSector用于构建圆心角；④适合高中及以上学生及数学爱好者用于理解抛物线相关几何与微积分知识，可作为教学演示工具。

GeoGebra 命令及参数

  1. IndexOf(SelectedElement[shape], shape) → a
  2. Intersect(f, g, 1) → A
  3. Intersect(f, g, 2) → B
  4. Polygon((b, f(b), 0), (b, 4, 0), (b, 4, 4 - f(b)), (b, f(b), 4 - f(b))) → q1, h, i, j, k
  5. CurveCartesian(t, 4, 4 - f(t), t, x(A), x(B)) → c
  6. CurveCartesian(t, f(t), 4 - f(t), t, x(A), x(B)) → d
  7. Surface(t, f(t), (u * z(d(t))), t, x(A), x(B), u, 0, 1) → e
  8. Surface(t, 4, (u * z(c(t))), t, x(A), x(B), u, 0, 1) → l
  9. Surface(t, (u * f(t)) + ((1 - u) * 4), z(c(t)), t, x(A), x(B), u, 0, 1) → m
  10. Polygon((b, f(b), 0), ((-b), f((-b)), 0), ((-b), f((-b)), (2 * abs(b))), (b, f(b), (2 * abs(b)))) → q2, n, q, r, s
  11. CurveCartesian(t, f(t), (2 * t), t, 0, x(B)) → o
  12. CurveCartesian((-t), f((-t)), (-2 * (-t)), t, 0, x(B)) → tt
  13. Surface(t, f(t), (u * z(o(t))), t, 0, x(B), u, 0, 1) → uu
  14. Surface((-t), f(t), ((-u) * z(tt((-t)))), t, 0, x(B), u, 0, 1) → vv
  15. Surface((u * x(o(t))) + ((1 - u) * x(tt(t))), (u * y(o(t))) + ((1 - u) * y(tt(t))), (u * z(o(t))) + ((1 - u) * z(tt(t))), t, 0, x(B), u, 0, 1) → u
  16. CurveCartesian(t, t(t), ((t(t) - f(t)) * sqrt(3)), t, x(A), x(B)) → v
  17. Surface(t, (u * f(t)) + ((1 - u) * y(v(t))), ((1 - u) * z(v(t))), t, x(A), x(B), u, 0, 1) → w
  18. Surface(t, (u * g(t)) + ((1 - u) * y(v(t))), ((1 - u) * z(v(t))), t, x(A), x(B), u, 0, 1) → a_1
  19. Polygon((b, f(b), 0), (b, 4, 0), v(b)) → t1, f_1, g_1, h_1
  20. Surface(t, t(t) + ((t(t) - f(t)) * cos(u)), ((t(t) - f(t)) * sin(u)), t, x(A), x(B), u, 0, π) → b_1
  21. Segment((b, f(b), 0), (b, 4, 0)) → i_1
  22. Midpoint(i_1) → C
  23. CurveCartesian(x(C), (i_1 / 2 * cos(t)) + y(C), (i_1 / 2 * sin(t)), t, 0, π) → c_1
  24. CircumcircleSector((b, f(b), 0), c_1(π / 2), (b, 4, 0)) → d_1
  25. CurveCartesian(t, f(t), 4 - f(t), t, x(A), x(B)) → e_1
  26. Surface(t, f(t), (u * z(e_1(t))), t, x(A), x(B), u, 0, 1) → j_1
  27. Surface(t, (u * f(t)) + ((1 - u) * g(t)), (u * z(e_1(t))), t, x(A), x(B), u, 0, 1) → k_1
  28. Polygon((b, f(b), 0), (b, 4, 0), e_1(b)) → t2, l_1, m_1, n_1

AI 绘图提示语参考

绘制一个抛物线的三维形状，通过展示其基座上的各种截面来可视化。首先画出抛物线与两条线的交点，然后基于这些点构建底部的多边形。接着，用这些点生成两条曲线，并利用这些曲线创建两个曲面，分别代表抛物线的两侧。最后，添加一些辅助图形，如截面和圆弧，以增强三维效果。
以上提示语为根据原始绘图命令整理而来，仅供参考。