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


albert said...

hi i have a hard question if you help me i will be very happy; In a country, there are four types of banknotes: 5, 10, 20 and 25 unit values. Each banknote has a serial number with more than four digits. A banknote is chosen randomly. What is the probability that the sum of the four last digits of the serial number is bigger than the banknote unit value?

Note:Four banknote types, and all the serial numbers (...0000 - ...9999) can be chosen with equal probability.

Rum Tan said...

If the larger number is under the x, then the ellipse is horizontal. If it is under the y then it is vertical.