Find the evolute of the involute of a circle, x=2a(cos…

Find the evolute of the involute of a circle, $x=2a(\cos t+t\sin t)$, $y=2a(\sin t-t\cos t)$.

Step 1: Derivatives

\texttt{sympy.diff} gives $x'=2at\cos t$, $y'=2at\sin t$, $x''=2a(\cos t-t\sin t)$, $y''=2a(\sin t+t\cos t)$, matching the source exactly.

Step 2: Curvature-formula components

\texttt{sympy.simplify} gives $x'^2+y'^2=4a^2t^2$ and $x'y''-y'x''=4a^2t^2$ — both confirmed independently, matching the source.

Step 3: Center of curvature

Since the ratio $(x'^2+y'^2)/(x'y''-y'x'')=1$, \texttt{sympy.simplify} collapses $X=x-y'$ and $Y=y+x'$ to \[X=2a\cos t,\qquad Y=2a\sin t,\] matching the source exactly.

Step 4: Eliminate the parameter

\texttt{sympy.simplify}$(X^2+Y^2)=4a^2$ identically for all $t$, confirming the evolute is the circle $x^2+y^2=4a^2$.