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余弦极坐标曲线 r=cos(a/b·t) 的参数化展示

余弦极坐标曲线 r=cos(a/b·t) 的参数化展示
几何画板及绘图内容加载中，请稍等……
2024-04-08 17:41
函数三角函数参数方程函数图像
所属专题 intersection: surface-surface, curve-curve
该范例演示在极坐标系中绘制极坐标曲线 r=cos(a/b·t) 时，曲线如何随参数 a/b 的取值而呈现周期性形态与幅值变化。核心数学概念包括极坐标方程、三角函数（余弦）、参数 t 的变化所引出的参数化曲线，以及由函数连续性带来的平滑绘制效果。作图要点是：利用 GeoGebra 的 Sequence 生成参数序列（t 或等价自变量采样），再用 CurveCartesian 将由参数映射得到的点连成曲线；同时使用多组可调参数 a、b（以及辅助常量如 c）来控制角频率与曲线形状，并配合 Text 与 Slider/显示对象（如 Text1、Tw、Ts、Text2）呈现公式与关键参数。适合高中到大学基础数学课程中用于三角函数与参数曲线可视化、建模与作图训练，也可用于课堂演示或探索型学习。

GeoGebra 命令及参数

  1. Sequence(Sequence[CurveCartesian[(2 * j) + (cos(t) * cos((j / k * t))), (2 * k) + (sin(t) * cos((j / k * t))), t, 0, (20 * pi)], j, 1, 10], k, 1, 10) → K, {{((2 * 1) + (cos(t) * cos((1 / 1 * t))), (2 * 1) + (sin(t) * cos((1 / 1 * t)))), ((2 * 2) + (cos(t) * cos((2 / 1 * t))), (2 * 1) + (sin(t) * cos((2 / 1 * t)))), ((2 * 3) + (cos(t) * cos((3 / 1 * t))), (2 * 1) + (sin(t) * cos((3 / 1 * t)))), ((2 * 4) + (cos(t) * cos((4 / 1 * t))), (2 * 1) + (sin(t) * cos((4 / 1 * t)))), ((2 * 5) + (cos(t) * cos((5 / 1 * t))), (2 * 1) + (sin(t) * cos((5 / 1 * t)))), ((2 * 6) + (cos(t) * cos((6 / 1 * t))), (2 * 1) + (sin(t) * cos((6 / 1 * t)))), ((2 * 7) + (cos(t) * cos((7 / 1 * t))), (2 * 1) + (sin(t) * cos((7 / 1 * t)))), ((2 * 8) + (cos(t) * cos((8 / 1 * t))), (2 * 1) + (sin(t) * cos((8 / 1 * t)))), ((2 * 9) + (cos(t) * cos((9 / 1 * t))), (2 * 1) + (sin(t) * cos((9 / 1 * t)))), ((2 * 10) + (cos(t) * cos((10 / 1 * t))), (2 * 1) + (sin(t) * cos((10 / 1 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 2 * t))), (2 * 2) + (sin(t) * cos((1 / 2 * t)))), ((2 * 2) + (cos(t) * cos((2 / 2 * t))), (2 * 2) + (sin(t) * cos((2 / 2 * t)))), ((2 * 3) + (cos(t) * cos((3 / 2 * t))), (2 * 2) + (sin(t) * cos((3 / 2 * t)))), ((2 * 4) + (cos(t) * cos((4 / 2 * t))), (2 * 2) + (sin(t) * cos((4 / 2 * t)))), ((2 * 5) + (cos(t) * cos((5 / 2 * t))), (2 * 2) + (sin(t) * cos((5 / 2 * t)))), ((2 * 6) + (cos(t) * cos((6 / 2 * t))), (2 * 2) + (sin(t) * cos((6 / 2 * t)))), ((2 * 7) + (cos(t) * cos((7 / 2 * t))), (2 * 2) + (sin(t) * cos((7 / 2 * t)))), ((2 * 8) + (cos(t) * cos((8 / 2 * t))), (2 * 2) + (sin(t) * cos((8 / 2 * t)))), ((2 * 9) + (cos(t) * cos((9 / 2 * t))), (2 * 2) + (sin(t) * cos((9 / 2 * t)))), ((2 * 10) + (cos(t) * cos((10 / 2 * t))), (2 * 2) + (sin(t) * cos((10 / 2 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 3 * t))), (2 * 3) + (sin(t) * cos((1 / 3 * t)))), ((2 * 2) + (cos(t) * cos((2 / 3 * t))), (2 * 3) + (sin(t) * cos((2 / 3 * t)))), ((2 * 3) + (cos(t) * cos((3 / 3 * t))), (2 * 3) + (sin(t) * cos((3 / 3 * t)))), ((2 * 4) + (cos(t) * cos((4 / 3 * t))), (2 * 3) + (sin(t) * cos((4 / 3 * t)))), ((2 * 5) + (cos(t) * cos((5 / 3 * t))), (2 * 3) + (sin(t) * cos((5 / 3 * t)))), ((2 * 6) + (cos(t) * cos((6 / 3 * t))), (2 * 3) + (sin(t) * cos((6 / 3 * t)))), ((2 * 7) + (cos(t) * cos((7 / 3 * t))), (2 * 3) + (sin(t) * cos((7 / 3 * t)))), ((2 * 8) + (cos(t) * cos((8 / 3 * t))), (2 * 3) + (sin(t) * cos((8 / 3 * t)))), ((2 * 9) + (cos(t) * cos((9 / 3 * t))), (2 * 3) + (sin(t) * cos((9 / 3 * t)))), ((2 * 10) + (cos(t) * cos((10 / 3 * t))), (2 * 3) + (sin(t) * cos((10 / 3 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 4 * t))), (2 * 4) + (sin(t) * cos((1 / 4 * t)))), ((2 * 2) + (cos(t) * cos((2 / 4 * t))), (2 * 4) + (sin(t) * cos((2 / 4 * t)))), ((2 * 3) + (cos(t) * cos((3 / 4 * t))), (2 * 4) + (sin(t) * cos((3 / 4 * t)))), ((2 * 4) + (cos(t) * cos((4 / 4 * t))), (2 * 4) + (sin(t) * cos((4 / 4 * t)))), ((2 * 5) + (cos(t) * cos((5 / 4 * t))), (2 * 4) + (sin(t) * cos((5 / 4 * t)))), ((2 * 6) + (cos(t) * cos((6 / 4 * t))), (2 * 4) + (sin(t) * cos((6 / 4 * t)))), ((2 * 7) + (cos(t) * cos((7 / 4 * t))), (2 * 4) + (sin(t) * cos((7 / 4 * t)))), ((2 * 8) + (cos(t) * cos((8 / 4 * t))), (2 * 4) + (sin(t) * cos((8 / 4 * t)))), ((2 * 9) + (cos(t) * cos((9 / 4 * t))), (2 * 4) + (sin(t) * cos((9 / 4 * t)))), ((2 * 10) + (cos(t) * cos((10 / 4 * t))), (2 * 4) + (sin(t) * cos((10 / 4 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 5 * t))), (2 * 5) + (sin(t) * cos((1 / 5 * t)))), ((2 * 2) + (cos(t) * cos((2 / 5 * t))), (2 * 5) + (sin(t) * cos((2 / 5 * t)))), ((2 * 3) + (cos(t) * cos((3 / 5 * t))), (2 * 5) + (sin(t) * cos((3 / 5 * t)))), ((2 * 4) + (cos(t) * cos((4 / 5 * t))), (2 * 5) + (sin(t) * cos((4 / 5 * t)))), ((2 * 5) + (cos(t) * cos((5 / 5 * t))), (2 * 5) + (sin(t) * cos((5 / 5 * t)))), ((2 * 6) + (cos(t) * cos((6 / 5 * t))), (2 * 5) + (sin(t) * cos((6 / 5 * t)))), ((2 * 7) + (cos(t) * cos((7 / 5 * t))), (2 * 5) + (sin(t) * cos((7 / 5 * t)))), ((2 * 8) + (cos(t) * cos((8 / 5 * t))), (2 * 5) + (sin(t) * cos((8 / 5 * t)))), ((2 * 9) + (cos(t) * cos((9 / 5 * t))), (2 * 5) + (sin(t) * cos((9 / 5 * t)))), ((2 * 10) + (cos(t) * cos((10 / 5 * t))), (2 * 5) + (sin(t) * cos((10 / 5 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 6 * t))), (2 * 6) + (sin(t) * cos((1 / 6 * t)))), ((2 * 2) + (cos(t) * cos((2 / 6 * t))), (2 * 6) + (sin(t) * cos((2 / 6 * t)))), ((2 * 3) + (cos(t) * cos((3 / 6 * t))), (2 * 6) + (sin(t) * cos((3 / 6 * t)))), ((2 * 4) + (cos(t) * cos((4 / 6 * t))), (2 * 6) + (sin(t) * cos((4 / 6 * t)))), ((2 * 5) + (cos(t) * cos((5 / 6 * t))), (2 * 6) + (sin(t) * cos((5 / 6 * t)))), ((2 * 6) + (cos(t) * cos((6 / 6 * t))), (2 * 6) + (sin(t) * cos((6 / 6 * t)))), ((2 * 7) + (cos(t) * cos((7 / 6 * t))), (2 * 6) + (sin(t) * cos((7 / 6 * t)))), ((2 * 8) + (cos(t) * cos((8 / 6 * t))), (2 * 6) + (sin(t) * cos((8 / 6 * t)))), ((2 * 9) + (cos(t) * cos((9 / 6 * t))), (2 * 6) + (sin(t) * cos((9 / 6 * t)))), ((2 * 10) + (cos(t) * cos((10 / 6 * t))), (2 * 6) + (sin(t) * cos((10 / 6 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 7 * t))), (2 * 7) + (sin(t) * cos((1 / 7 * t)))), ((2 * 2) + (cos(t) * cos((2 / 7 * t))), (2 * 7) + (sin(t) * cos((2 / 7 * t)))), ((2 * 3) + (cos(t) * cos((3 / 7 * t))), (2 * 7) + (sin(t) * cos((3 / 7 * t)))), ((2 * 4) + (cos(t) * cos((4 / 7 * t))), (2 * 7) + (sin(t) * cos((4 / 7 * t)))), ((2 * 5) + (cos(t) * cos((5 / 7 * t))), (2 * 7) + (sin(t) * cos((5 / 7 * t)))), ((2 * 6) + (cos(t) * cos((6 / 7 * t))), (2 * 7) + (sin(t) * cos((6 / 7 * t)))), ((2 * 7) + (cos(t) * cos((7 / 7 * t))), (2 * 7) + (sin(t) * cos((7 / 7 * t)))), ((2 * 8) + (cos(t) * cos((8 / 7 * t))), (2 * 7) + (sin(t) * cos((8 / 7 * t)))), ((2 * 9) + (cos(t) * cos((9 / 7 * t))), (2 * 7) + (sin(t) * cos((9 / 7 * t)))), ((2 * 10) + (cos(t) * cos((10 / 7 * t))), (2 * 7) + (sin(t) * cos((10 / 7 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 8 * t))), (2 * 8) + (sin(t) * cos((1 / 8 * t)))), ((2 * 2) + (cos(t) * cos((2 / 8 * t))), (2 * 8) + (sin(t) * cos((2 / 8 * t)))), ((2 * 3) + (cos(t) * cos((3 / 8 * t))), (2 * 8) + (sin(t) * cos((3 / 8 * t)))), ((2 * 4) + (cos(t) * cos((4 / 8 * t))), (2 * 8) + (sin(t) * cos((4 / 8 * t)))), ((2 * 5) + (cos(t) * cos((5 / 8 * t))), (2 * 8) + (sin(t) * cos((5 / 8 * t)))), ((2 * 6) + (cos(t) * cos((6 / 8 * t))), (2 * 8) + (sin(t) * cos((6 / 8 * t)))), ((2 * 7) + (cos(t) * cos((7 / 8 * t))), (2 * 8) + (sin(t) * cos((7 / 8 * t)))), ((2 * 8) + (cos(t) * cos((8 / 8 * t))), (2 * 8) + (sin(t) * cos((8 / 8 * t)))), ((2 * 9) + (cos(t) * cos((9 / 8 * t))), (2 * 8) + (sin(t) * cos((9 / 8 * t)))), ((2 * 10) + (cos(t) * cos((10 / 8 * t))), (2 * 8) + (sin(t) * cos((10 / 8 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 9 * t))), (2 * 9) + (sin(t) * cos((1 / 9 * t)))), ((2 * 2) + (cos(t) * cos((2 / 9 * t))), (2 * 9) + (sin(t) * cos((2 / 9 * t)))), ((2 * 3) + (cos(t) * cos((3 / 9 * t))), (2 * 9) + (sin(t) * cos((3 / 9 * t)))), ((2 * 4) + (cos(t) * cos((4 / 9 * t))), (2 * 9) + (sin(t) * cos((4 / 9 * t)))), ((2 * 5) + (cos(t) * cos((5 / 9 * t))), (2 * 9) + (sin(t) * cos((5 / 9 * t)))), ((2 * 6) + (cos(t) * cos((6 / 9 * t))), (2 * 9) + (sin(t) * cos((6 / 9 * t)))), ((2 * 7) + (cos(t) * cos((7 / 9 * t))), (2 * 9) + (sin(t) * cos((7 / 9 * t)))), ((2 * 8) + (cos(t) * cos((8 / 9 * t))), (2 * 9) + (sin(t) * cos((8 / 9 * t)))), ((2 * 9) + (cos(t) * cos((9 / 9 * t))), (2 * 9) + (sin(t) * cos((9 / 9 * t)))), ((2 * 10) + (cos(t) * cos((10 / 9 * t))), (2 * 9) + (sin(t) * cos((10 / 9 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 10 * t))), (2 * 10) + (sin(t) * cos((1 / 10 * t)))), ((2 * 2) + (cos(t) * cos((2 / 10 * t))), (2 * 10) + (sin(t) * cos((2 / 10 * t)))), ((2 * 3) + (cos(t) * cos((3 / 10 * t))), (2 * 10) + (sin(t) * cos((3 / 10 * t)))), ((2 * 4) + (cos(t) * cos((4 / 10 * t))), (2 * 10) + (sin(t) * cos((4 / 10 * t)))), ((2 * 5) + (cos(t) * cos((5 / 10 * t))), (2 * 10) + (sin(t) * cos((5 / 10 * t)))), ((2 * 6) + (cos(t) * cos((6 / 10 * t))), (2 * 10) + (sin(t) * cos((6 / 10 * t)))), ((2 * 7) + (cos(t) * cos((7 / 10 * t))), (2 * 10) + (sin(t) * cos((7 / 10 * t)))), ((2 * 8) + (cos(t) * cos((8 / 10 * t))), (2 * 10) + (sin(t) * cos((8 / 10 * t)))), ((2 * 9) + (cos(t) * cos((9 / 10 * t))), (2 * 10) + (sin(t) * cos((9 / 10 * t)))), ((2 * 10) + (cos(t) * cos((10 / 10 * t))), (2 * 10) + (sin(t) * cos((10 / 10 * t))))}}
  2. Sequence(Text[k, ((2 * k), -0.5)], k, 1, 10) → Tw
  3. Sequence(Text[k, (-0.5, (2 * k) - 0.3)], k, 1, 10) → Ts
  4. CurveCartesian(30 + ((7 * cos(t)) * cos((a / b * t))), 12 + ((7 * sin(t)) * cos((a / b * t))), t, 0, (120 * pi)) → c

AI 绘图提示语参考

绘制一个极坐标曲线，公式为 r = cos(a/b · t)，其中 a 和 b 是可调节的参数，用于控制曲线的频率和整体形状。曲线由参数 t 从 0 采样到 20π 生成，形成一个连续的平滑图形。曲线的周期性结构随 a/b 的变化而改变，振幅也随之变化。图中可添加文本标签显示公式和参数含义，帮助理解曲线的数学意义。
以上提示语为根据原始绘图命令整理而来，仅供参考。