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Circle Evolute

x=\cos t\qquad x'=-\sin t\qquad x''=-\cos t
\end{displaymath} (1)

y=\sin t\qquad y'=\cos t\qquad y''=-\sin t,
\end{displaymath} (2)

so the Radius of Curvature is
$\displaystyle R$ $\textstyle =$ $\displaystyle {(x'^2+y'^2)^{3/2}\over y''x'-x''y'}$  
  $\textstyle =$ $\displaystyle {(\sin^2t+\cos^2t)^{3/2}\over (-\sin t)(-\sin t)-(-\cos t)\cos t} = 1,$ (3)

and the Tangent Vector is
\hat{\bf T}=\left[{\matrix{-\sin t\cr \cos t\cr}}\right].
\end{displaymath} (4)

$\displaystyle \cos\tau$ $\textstyle =$ $\displaystyle \hat{\bf T}\cdot\hat{\bf x}=-\sin t$ (5)
$\displaystyle \sin\tau$ $\textstyle =$ $\displaystyle \hat{\bf T}\cdot\hat{\bf y}=\cos t,$ (6)

$\displaystyle \xi(t)$ $\textstyle =$ $\displaystyle x-R\sin\tau=\cos t-1\cdot\cos t=0$ (7)
$\displaystyle \eta(t)$ $\textstyle =$ $\displaystyle y+R\cos\tau=\sin t+1\cdot(-\sin t)=0,$ (8)

and the Evolute degenerates to a Point at the Origin.

See also Circle Involute


Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 77, 1993.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 55-59, 1991.

© 1996-9 Eric W. Weisstein