Since the sharpness of the corners is caused when

\frac{ⅆ r}{ⅆ θ}

is different between the two segments, and

\frac{ⅆ r}{ⅆ θ} = 0

for each end of the semicircle I decided to experiment with quadratic spirals where

r = a + b {(θ - θ_{2})}^{2}

since

\frac{ⅆ r}{ⅆ θ} = 0

θ_{2}

I am not pleased with the result.