生成讲解动画 公式（参数方程）： { x=r(cost+tsint) y=r(sint−tcost) ​ 运动特点： 将一根绕在圆上的绳子逐渐拉直时，绳端的轨迹，呈现螺旋向外延伸的形态，如齿轮齿廓曲线。 约束条件： r>0 （基圆半径） t≥0 （参数 t 为展开角度，单位为弧度）， t=0 时起点为 (r,0) 。

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Welcome to the study of involute curves. An involute is a special curve created when we unwind a string from a circle. Imagine a string wrapped around a circular base, and as we pull it straight, the free end traces a beautiful spiral pattern. This curve has important applications in engineering, particularly in gear design. The involute curve is mathematically described by parametric equations. The x-coordinate is r times cosine t plus t sine t, and the y-coordinate is r times sine t minus t cosine t. Here, r is the radius of the base circle, and t is the unwinding angle measured in radians. When t equals zero, we start at point (r, 0). As t increases, the curve spirals outward. Now let's observe the string unwinding process in detail. As we unwind the string, several key geometric properties emerge. The string always remains tangent to the circle at the contact point. The length of the unwound string equals the arc length, which is r times t. The string stays perfectly straight as it unwinds, and the free end traces the involute curve. Watch how the curve develops as the parameter t increases. Here we see the complete involute curve extending through multiple rotations. The curve spirals smoothly outward from the base circle, creating an elegant pattern. Notice how the curve never intersects itself and continues to expand as the parameter t increases. Key points are marked at intervals of pi radians, showing the curve's progression. This beautiful mathematical form has practical applications in mechanical engineering, particularly in gear tooth profiles. The involute curve finds its most important application in gear design. Involute gear teeth provide smooth power transmission with constant velocity ratios. The special geometry ensures that gears mesh properly even with slight variations in center distance. Watch how the involute-shaped teeth maintain contact as the gears rotate. This mathematical elegance makes involute gears the standard choice in modern mechanical engineering, from watches to industrial machinery.