9.2.3. Path¶
9.2.3.1. Bulge¶
Let \(\mathbf{P}_0\), \(\mathbf{P}_1\), \(\mathbf{P}_2\) the vertices and \(R\) the bulge radius.
The deflection \(\theta = 2 \alpha\) at the corner is
\[\mathbf{D}_1 \cdot \mathbf{D}_0 = (\mathbf{P}_2 - \mathbf{P}_1) \cdot (\mathbf{P}_1 - \mathbf{P}_0) = \cos(\pi - \theta)\]
The bisector direction is
\[\mathbf{Bis} = \mathbf{D}_1 - \mathbf{D}_0 = (\mathbf{P}_2 - \mathbf{P}_1) - (\mathbf{P}_1 - \mathbf{P}_0) = \mathbf{P}_2 -2 \mathbf{P}_1 + \mathbf{P}_0\]
Bulge Center is
\[\mathbf{C} = \mathbf{P}_1 + \frac{R}{\sin \alpha} \mathbf{Bis}\]
Extremities are
\[\begin{split}\begin{align}
\mathbf{P}_1' &= \mathbf{P}_1 - \frac{R}{\tan \alpha} \mathbf{D}_0 \\
\mathbf{P}_1'' &= \mathbf{P}_1 + \frac{R}{\tan \alpha} \mathbf{D}_1
\end{align}\end{split}\]