圆的渐伸线是内、外旋轮线的极限情形

青青子衿

证明：圆的渐伸线是内、外旋轮线当动圆半径趋于无穷大的情形

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Mathcurve.com

Involutes of a circle
For a circle with parametric representation $(r\cos(t), r\sin(t))$, one has
$$\vec c'(t) = (-r\sin t, r\cos t)$$
Hence $|\vec c'(t)| = r$, and the path length is $r(t - a)$.

Evaluating the above given equation of the involute, one gets
\begin{align*}
X(t) &= r(\cos t + (t - a)\sin t)\\
Y(t) &= r(\sin t - (t - a)\cos t)
\end{align*}
for the parametric equation of the involute of the circle.

The $a$ term is optional; it serves to set the start location of the curve on the circle.  The figure shows involutes for $a = -0.5$ (green), $a = 0$ (red), $a = 0.5$ (purple) and $a = 1$ (light blue). The involutes look like Archimedean spirals, but they are actually not.

The arc length for $a=0$ and $0 \le t \le t_2$ of the involute is
$$L = \frac{r}{2} t_2^2.$$

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Epi/Hypocycloid (外/内旋轮线) Parametric Formula :

1. {(a + b)*Cos[t] + b*Cos[(a + b)/b*t], (a + b)*Sin[t] + b*Sin[(a + b)/b*t]}

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$a$ is the radius of the fixed circle. $b$ is the radius of the rolling circle.
If $b$ is positive, this formula gives epicycloid. If $b$ is negative, this formula gives the hypocycloid.

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应该是在Epi/Hypocycloid方程中, 把参数$t$取为$b$的函数, 求极限
具体是怎样的?

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