球的 pedal 曲面

几何画板及绘图内容加载中，请稍等……

2018-06-10 06:52

立体几何参数方程轨迹

所属专题 surface 3D /3D-Oberflächen 1

这个范例演示了球的 pedal 曲面。涉及的核心数学概念包括参数方程和轨迹。通过使用 GeoGebra 命令如 OrthogonalLine, Intersect, Sequence，可以构造出球的 pedal 曲面。适合高中数学教学和学习。

GeoGebra 命令及参数

OrthogonalLine(B, a_1) → f
Intersect(f, a_1) → C
Sequence(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°)), a, 1, t) → 리스트1
Sequence(Intersect[OrthogonalLine[B, (x(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))) * (x - x(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))))) + (y(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))) * (y - y(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))))) + (z(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))) * (z - z(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))))) = 0], (x(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))) * (x - x(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))))) + (y(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))) * (y - y(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))))) + (z(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))) * (z - z(((cos(Mod[(a * 2), 180]° - 90°) * cos((Div[(a * 2), 180]° * 5))), (cos(Mod[(a * 2), 180]° - 90°) * sin((Div[(a * 2), 180]° * 5))), sin(Mod[(a * 2), 180]° - 90°))))) = 0], a, 1, t) → 리스트2

AI 绘图提示语参考

绘制一个球面的垂足曲面。先确定球外一点作为固定点，再在球面上取一系列点，这些点按参数方程分布在球面上。对每个球面上的点，作过固定点且垂直于该点与球心连线的平面，并求出该平面与球面的交点。随着参数变化，这些交点形成一条空间曲线，最终由这些曲线构成整个垂足曲面。图形应展示球面、固定点以及由垂足关系生成的曲面，体现参数化与轨迹特征。

以上提示语为根据原始绘图命令整理而来，仅供参考。