## Wednesday, December 5, 2012

### 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: $\triangle_2 = \sigma^2 \left(\sin^2(\theta_0) + \kappa^2 \cos^2(\theta_0)\right) - (x_1-x_0)^2 \geq 0$