K=[r2+(r′)2]3/2∣r2+2(r′)2−rr′′∣
r′=3cos3θ,r′′=−9sin3θK=[(sin3θ)2+(3cos3θ)2]3/2∣(sin3θ)2+2(3cos3θ)2+9(sin3θ)2∣
=[1+8(cos3θ)2]3/210+8(cos3θ)2
θ=0
r(0)=0,r′(0)=3,r′′(0)=0
K=[(0)2+(3)2]3/2∣(0)2+2(3)2−0(0)∣=32
The radius of curvature of a curve at a point is called the inverse of the curvature K of the curve at this point:
R=K1=23
xC=x−y′′y′(1+(y′)2)
yC=y+y′′1+(y′)2
x=rcosθ=sin(3θ)cosθy=rsinθ=sin(3θ)sinθ
yx′=xθ′yθ′=r′cosθ−rsinθr′sinθ+rcosθ
(yx′)θ′=(r′cosθ−rsinθ)2(r′′sinθ+r′cosθ)(r′cosθ−rsinθ)
+(r′cosθ−rsinθ)2(r′cosθ−rsinθ)(r′cosθ−rsinθ)
−(r′cosθ−rsinθ)2(r′′cosθ−r′sinθ)(r′sinθ+rcosθ)
−(r′cosθ−rsinθ)2(−r′sinθ−rcosθ)(r′sinθ+rcosθ)
yxx′′=(r′cosθ−rsinθ)3(r′′sinθ+r′cosθ)(r′cosθ−rsinθ)
+(r′cosθ−rsinθ)3(r′cosθ−rsinθ)(r′cosθ−rsinθ)
−(r′cosθ−rsinθ)3(r′′cosθ−r′sinθ)(r′sinθ+rcosθ)
−(r′cosθ−rsinθ)3(−r′sinθ−rcosθ)(r′sinθ+rcosθ)
θ=0
x(0)=0,y(0)=0
yx′(0)=3(1)−0(0)3(0)+0(1)=0
yxx′′(0)=(3)3(0+3)(3−0)
+(3)3(3−0)(3−0)
−(3)3(0−0)(0+0)
−(3)3(0−0)(0+0)=32
xC=0−320(1+(0)2)=0
yC=0+321+(0)2=23 C(0,23)
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