Dynamic Unfolding of a Cylinder: Lateral Area and Shortest Path
2020-12-27 03:41
This GeoGebra applet dynamically demonstrates the unfolding process of a cylinder's lateral surface through 3D animation, visually revealing the geometric principles behind calculating lateral area and finding the shortest path on a cylindrical surface. Using parametric equations and surface commands, it constructs a cylinder model and employs sliders to control the unfolding angle, transforming t
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<expression label="text1" exp=""1、控制滑动条 α=Slider(0,2 π ,0.1) 为展开的角度\\
 圆柱半径参数 r=1，高度h=2\\
2、XOY平面上展开圆的曲线方程写法是最关键的。\\
 设圆心O=(α,0,0) 在X轴上运动，区间为[0,2 π ]\\
 用分段函数表示：动态分割的两部分曲线\\
 (1)当变量t<α, 展开部分为直线段 (t, y(O)-r,0)\\
 (2) 当t>=α,t<=2 π 时，展开部分为圆弧，\\
 该圆弧表示方式有多种\\
 最精简为 圆心+极坐标，即 O+(r;t-α- π /2)\\

 合并为：\\
 e(t) = Curve(If(t<α, (t, y(O)-r,0), O+(r;t-α- π /2)\\

3、接点C可以用e(α)表示，终点B可以用e(2 π )表示。\\
4、圆柱面可以表示为：\\
 Surface(e(t)+(0,0,b), t,0,2 π ,b,0,h)\\
5、操作滑动条α可以看到展开具体情况。
 ""/>
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