Determine whether an ellipse intersect a horizontal or vertical line

Assume an ellipse of width \(\sigma\) and length \(\kappa \sigma\) is centered at \((x_0, y_0)\), and has angle \(\theta_0\) with the \(x\)-axis. How do we determine whether it intersects a horizontal line or a vertical line?

It turns out the criteria is very simple

• The ellipse intersects a horizontal line \(y = y_1\) if and only if the following equation hods: \[ \triangle_1 = \sigma^2 \left(\cos^2(\theta_0) + \kappa^2 \sin^2(\theta_0)\right) - (y_1-y_0)^2 \geq 0 \]
• The ellipse intersects a vertical line \(x = x_1\) if and only if the following equation hods: \[ \triangle_2 = \sigma^2 \left(\sin^2(\theta_0) + \kappa^2 \cos^2(\theta_0)\right) - (x_1-x_0)^2 \geq 0 \]