Ellipses

An ellipse is a plane curve surrounding two focal points, such that, for all points on the curve, the sum of the distances to the focal points is constant.

Test the definition yourself by moving the point on the ellipse, notice how the sum of the distances is always $6$.

The formula of an ellipse centered at the origin $(0,0)$ is

where $a$ and $b$ are the horizontal and vertical diameter lengths respectively.

For example, the following ellipse has formula:

The focal points of an ellipse centered at $(0,0)$ are $(\pm c,0)$ if $a>b$ (horizontal ellipse) and $(0,\pm c)$ if $a>b$ (vertical ellipse), where $c$ is calculated as $c = \sqrt{|a^2-b^2|}.$

Let's see how the focal points affect the shape of the ellipse for a fixed horizontal diameter. What shape do we get when $c=0$?

And what happens if we change the center?

If we move the ellipse horizontally or vertically, we need to aplly the corresponding transformations to the variables $x$ and $y$.

• Horizontal shift of $h$ units: Change the value of $x$ for $x-h$.

• Vertical shift of $k$ units: Change the value of $y$ for $y-k$.

Together with the variables $x$ and $y$, the other elements affected by the translation are the focus points, these are now $(\pm c + h, k)$ for horizontal ellipses and $(h, \pm c +k)$ for vertical ellipses.

Perfect! Now let's perform this transformations on the formula of the ellipse.

The formula of an ellipse centered in $(h,k)$ is:

let's see what this looks like. The fomula:

gives us the following ellipse:

To sum up, we've learnt how to plot a general ellipse centered at $(h,k)$, with horizontal and vertical diameters $a$ and $b$ and focus points $(h+c, k)$ and $(h, k+c)$.