The equation of the evolute is
r(t)=x(t)⋅i+y(t)⋅j→⎩⎨⎧X(t)=x(t)−x′(t)y′′(t)−x′′(t)y′(t)y′(t)⋅(x′(t)2+y′(t)2)Y(t)=y(t)+x′(t)y′′(t)−x′′(t)y′(t)x′(t)⋅(x′(t)2+y′(t)2)
(more infotmation : https://en.wikipedia.org/wiki/Evolute )
In our case,
{x(θ)=acos3(θ)y(θ)=asin3(θ)→⎩⎨⎧x′(θ)=−3acos2(θ)⋅sin(θ)x′(θ)=2−3acos(θ)sin(2θ)x′′(θ)=2−3a(−sin(θ)sin(2θ)+2cos(θ)cos(2θ))y′(θ)=3asin2(θ)cos(θ)y′(θ)=23asin(θ)sin(2θ)y′′(θ)=23a(cos(θ)sin(2θ)+2sin(θ)cos(2θ))
Now we transform the expressions that enter the equations for the evolute :
x′(θ)2+y′(θ)2=(2−3acos(θ)sin(2θ))2+(23asin(θ)sin(2θ))2=49a2⋅sin2(2θ)⋅(cos2(θ)+sin2(θ))≡49a2⋅sin2(2θ)x′(θ)2+y′(θ)2=49a2⋅sin2(2θ)x′(θ)y′′(θ)−x′′(θ)y′(θ)==2−3acos(θ)sin(2θ)⋅23a(cos(θ)sin(2θ)+2sin(θ)cos(2θ))−−2−3a(−sin(θ)sin(2θ)+2cos(θ)cos(2θ))⋅23asin(θ)sin(2θ)==4−9a2sin(2θ)⋅(cos2(θ)sin(2θ)+2sin(θ)cos(θ)cos(2θ)++sin2(θ)sin(2θ)−2sin(θ)cos(θ)cos(2θ))==4−9a2sin(2θ)⋅(sin(2θ)⋅(sin2(θ)+cos2(θ)))==4−9a2sin2(2θ)x′(θ)y′′(θ)−x′′(θ)y′(θ)=4−9a2sin2(2θ)
Substitute the found expressions into the equations for the evolute, which were indicated at the very beginning :
X(θ)=x(θ)−x′(θ)y′′(θ)−x′′(θ)y′(θ)y′(θ)⋅(x′(θ)2+y′(θ)2)==acos3(θ)−4−9a2sin2(2θ)23asin(θ)sin(2θ)⋅49a2sin2(2θ)==acos3(θ)+23asin(θ)sin(2θ)=acos3(θ)+3asin2(θ)cos(θ)==acos(θ)(cos2(θ)+3sin2(θ))≡acos(θ)(1+2sin2(θ))X(θ)=acos(θ)(1+2sin2(θ))Y(θ)=y(θ)+x′(θ)y′′(θ)−x′′(θ)y′(θ)x′(θ)⋅(x′(θ)2+y′(θ)2)==asin3(θ)+4−9a2sin2(2θ)2−3acos(θ)sin(2θ)⋅49a2sin2(2θ)==asin3(θ)+23acos(θ)sin(2θ)=asin3(θ)+3acos2(θ)sin(θ)==asin(θ)(sin2(θ)+3cos2(θ))≡asin(θ)(1+2cos2(θ))Y(θ)=asin(θ)(1+2cos2(θ))
ANSWER
{x(θ)=acos3(θ)y(θ)=asin3(θ)→{X(θ)=acos(θ)(1+2sin2(θ))Y(θ)=asin(θ)(1+2cos2(θ))
Comments