Circle Evolute

$$x=\cos t\qquad x'=-\sin t\qquad x''=-\cos t$$ (1)

$$y=\sin t\qquad y'=\cos t\qquad y''=-\sin t,$$ (2)

so the Radius of Curvature is

$$R = {(x'^2+y'^2)^{3/2}\over y''x'-x''y'}$$

$$= {(\sin^2t+\cos^2t)^{3/2}\over (-\sin t)(-\sin t)-(-\cos t)\cos t} = 1,$$ (3)

and the Tangent Vector is

$$\hat{\bf T}=\left[{\matrix{-\sin t\cr \cos t\cr}}\right].$$ (4)

Therefore,

$$\cos\tau = \hat{\bf T}\cdot\hat{\bf x}=-\sin t$$ (5)

$$\sin\tau = \hat{\bf T}\cdot\hat{\bf y}=\cos t,$$ (6)

so

$$\xi(t) = x-R\sin\tau=\cos t-1\cdot\cos t=0$$ (7)

$$\eta(t) = y+R\cos\tau=\sin t+1\cdot(-\sin t)=0,$$ (8)

and the Evolute degenerates to a Point at the Origin.

See also Circle Involute

References

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.