Solution
Center of curvature is denoted by P
Lets determine the coordinates of the center of curvature at the point P on the astroid
Let θ=6π
ρ=δψδs=12 sin ψ cos ψ=6 sin ψAt θp=6π,ψp=−6π,ρp=6 sin (−3π)=6 (−23)=−33
Now
(X,Y)=(xp,yp)+ρn
So written
θ=6π, ψ=−6π,xp=233,yp=21,ρp=−33np=(−sin(−6π)cos(−6π))=(sin(6π)cos(6π))=(21,23)Thus;⟹(X,Y)=(233,21)+∣−33∣(21,23)⟹(X,Y)=(233,21)+(−233,29)∴ (X,Y)=(33, 5)
Hence, the center of curvature of astroid is (33, 5)
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