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hbghlyj
Posted at 2023-2-20 07:12:48
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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