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we replace Ω with the disk whose boundary is the osculating circle of ∂ Ω at γ ( s ) (call its radius R ). We have the following expression for the new normalized nonasymptotic density (see Figure 4.16(a)): 

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where x = p is the positive solution to √$_{r}$ 2 − x 2 = R − √$_{R}$ 2 − x $^{2}$. Differentiating with respect to r and then taking the limit as r goes to 0 gives us − 1 3 $_{πR}$. That is, for the case where Ω is locally a disk, the curvature at γ ( s ) is given by − 3 π lim$_{r}$$_{→}$$_{0}$ ∂ ∂r g ( s,r ) πr 2 . 

If the curvature of ∂ Ω at γ ( s ) is negative, we can set up a similar surrogate (see figure 4.16(b)) and again obtain that κ ( γ ( s )) = − 3 π lim$_{r}$$_{→}$$_{0}$ ∂ ∂r g ( s,r ) πr 2 . 

Lastly, this calculation gives the right result in the curvature 0 case when ∂ Ω is locally a straight line (so g ( s,r ) πr 2 = 1 πr 2 ∫ r − r √$_{r}$ 2 − x 2 dx = 1 2 for sufficiently small r and − 3 π lim$_{r}$$_{→}$$_{0}$ ∂ ∂r g ( s,r ) πr 2 = 0). 

For the case where ∂ Ω is not locally a circle or straight line, the corrections to the integrals are of order O ( x $^{3}$) as r goes to 0 and have no impact on the final answer so 

