Answer to Question #145441 in Differential Geometry | Topology for Dolly

Question #145441
Find the centre of curvature of astroid
x^2/3+y^2/3=a^2/3
1
Expert's answer
2020-11-24T17:13:57-0500
SolutionSolution


Center of curvature is denoted by PP

Lets determine the coordinates of the center of curvature at the point PP on the astroid


Let θ=π6\theta = \frac{\pi}{6}


ρ=δsδψ=12 sin ψ cos ψ=6 sin ψAt θp=π6,ψp=π6,ρp=6 sin (π3)=6 (32)=33\rho = \frac{\delta s}{\delta \psi}=12\ sin\ \psi\ cos\ \psi = 6\ sin\ \psi\\ At\ \theta_p=\frac{\pi}{6},\\ \psi_p=-\frac{\pi}{6},\\ \rho_p=6\ sin\ (-\frac{\pi}{3})=6\ (-\frac{\sqrt{3}}{2})=-3 \sqrt{3}

Now

(X,Y)=(xp,yp)+ρn(X,Y)=(x_p, y_p) +\rho n

So written

θ=π6, ψ=π6,xp=332,yp=12,ρp=33np=(sin(π6)cos(π6))=(sin(π6)cos(π6))=(12,32)Thus;    (X,Y)=(332,12)+33(12,32)    (X,Y)=(332,12)+(332,92) (X,Y)=(33, 5)\theta=\frac{\pi}{6},\ \psi= -\frac{\pi}{6}, x_p=\frac{3 \sqrt{3}}{2}, y_p=\frac12, \rho_p=-3 \sqrt{3}\\ n_p=(-sin(-\frac{\pi}{6})cos(-\frac{\pi}{6}))=(sin(\frac{\pi}{6})cos(\frac{\pi}{6}))= (\frac12, \frac{\sqrt{3}}{2})\\ Thus;\\ \implies(X,Y)=(\frac{3 \sqrt{3}}{2}, \frac12)+|-3 \sqrt{3}|(\frac12,\frac{\sqrt{3}}{2})\\ \implies(X,Y)=(\frac{3 \sqrt{3}}{2}, \frac12)+(-\frac{3 \sqrt{3}}{2},\frac{9}{2})\\ \therefore\ (X,Y)=(3 \sqrt {3},\ 5)


Hence, the center of curvature of astroid is (33, 5)(3 \sqrt {3},\ 5)



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