旋转曲面 by Andrew Knauft

几何画板及绘图内容加载中，请稍等……

2016-04-02 05:58

①本范例演示将函数f(x)与g(x)围成的区域绕水平线旋转生成旋转曲面；②涉及的核心数学概念包括微积分中的旋转曲面体积计算、定积分、线性方程、坐标变换；③作图要点——从下方『构造摘要』提炼用到了Point（点）、Rotate（旋转）、Locus（轨迹）、Surface（曲面）、Textfield（文本框）等GeoGebra命令/对象；④适合高中及以上学生及数学爱好者学习旋转曲面生成原理及积分计算。

GeoGebra 命令及参数

Point(Segment[A, B]) → E
Rotate(C, α, rot) → C'
Rotate(D, α, rot) → D'
Locus(D', E) → loc3
Locus(C', E) → loc4
Surface(t, ((f(t) - R) * cos(u)) + R, ((f(t) - R) * sin(u)), t, a, b, u, (-α), 0) → e
Surface(t, ((g(t) - R) * cos(u)) + R, ((g(t) - R) * sin(u)), t, a, b, u, (-α), 0) → i
Surface(t, ((f(t) - R) * cos(u)) + R, ((f(t) - R) * sin(u)), t, x(C), x(C) + dx, u, (-α), 0) → l
Surface(t, ((g(t) - R) * cos(u)) + R, ((g(t) - R) * sin(u)), t, x(D), x(D) + dx, u, (-α), 0) → d
Surface(x(C), R - ((R - ((y(C) * (1 - t)) + (y(D) * t))) * cos(v)), ((R - ((y(C) * (1 - t)) + (y(D) * t))) * sin(v)), t, 0, 1, v, 0, α) → c
Surface(x(C) + dx, R - ((R - ((g(x(C) + dx) * (1 - t)) + (f(x(C) + dx) * t))) * cos(v)), ((R - ((g(x(C) + dx) * (1 - t)) + (f(x(C) + dx) * t))) * sin(v)), t, 0, 1, v, 0, α) → k
Surface(a, R - ((R - ((g(a) * (1 - t)) + (f(a) * t))) * cos(v)), ((R - ((g(a) * (1 - t)) + (f(a) * t))) * sin(v)), t, 0, 1, v, 0, α) → m
Surface(b, R - ((R - ((g(b) * (1 - t)) + (f(b) * t))) * cos(v)), ((R - ((g(b) * (1 - t)) + (f(b) * t))) * sin(v)), t, 0, 1, v, 0, α) → n
Surface(t, (g(t) * u) + (f(t) * (1 - u)), 0, t, a, b, u, 0, 1) → p
Surface(t, R + (((f(t) * u) + (g(t) * (1 - u)) - R) * cos(α)), ((((R - f(t)) * u) + ((R - g(t)) * (1 - u))) * sin(α)), t, a, b, u, 0, 1) → o
Surface(t, R + (((f(t) * u) + (g(t) * (1 - u)) - R) * cos(α)), ((((R - f(t)) * u) + ((R - g(t)) * (1 - u))) * sin(α)), t, x(C), x(C) + dx, u, 0, 1) → q
Surface(t, (g(t) * u) + (f(t) * (1 - u)), 0, t, x(C), x(C) + dx, u, 0, 1) → r
Textfield(f) → InputBox1
Textfield(g) → InputBox2
Textfield(R) → InputBox3
Textfield(a) → InputBox4
Textfield(b) → InputBox5