If possible, could someone explain the major elements of the conic sections? By that I mean

  • focus
  • directrix
  • principal axis
  • vertex
  • eccentricity
  • and any other features of a conic section that are important.

My precalculus textbook is unclear.

[And here is where I (OP) should put what I do know]
$\sim$ a friendly message from a person who edited this question


closed as off-topic by Andrew D. Hwang, Théophile, pjs36, hardmath, Davide Giraudo Jun 8 '17 at 19:40

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Théophile, pjs36, hardmath, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ Welcome to MSE! Please be somewhat clearer. Give us your definitions so we can explain them to you. In this way, we can help you in the best way possible :) $\endgroup$ – user370967 Jun 8 '17 at 14:55


Take two points $F_1$ and $F_2$ in the plane and fix a number $L$ that's bigger than the distance between the two points. Then an ellipse is is the set of points $P$ such that the distance from $F_1$ to $P$ plus the distance from $F_2$ to $P$ is $L$.

enter image description here

$F_1$ and $F_2$ are called the foci of the ellipse. Besides $L$, we define a few other useful numbers associated with ellipses.

enter image description here

Notice that in general ellipses has a longer direction (horizontal in the image above) and a shorter direction (vertical).

  • The line segment through the center of the ellipse to each of the furthest points is called the major axis. Half of this (the line segment from the center to just one of the furthest points) is a semi-major axis.
  • The other axis is the minor axis and it has two semi-minor axes, as well.
  • Those two further points on the ends of the major axis are called the vertices (plural of vertex) and the end points of the minor axis are called the co-vertices (plural of co-vertex).
  • The two points we used to draw our ellipse are called the foci (plural of focus).
  • The focal distance is the distance from the center to one of the foci.
  • The eccentricity describes how elongated the ellipse is. It is calculated as the focal length divided by the length of the semi-major axis. So it is the fraction of the length of the semi-major axis that the focal length takes up (think percentage but in decimal form).
    • The smallest the eccentricity can be is $0$.The is when the foci are right at the center. In this case you just get a circle.
    • Going the other way, as the focal length gets closer and closer to the length of the semi-major axis -- i.e., the eccentricity gets closer to $1$ -- the less circular and more elongated the ellipse looks.


Estimated release date: 12/15/2017.


Plans are to release it as DLC sometime next year.


Not the answer you're looking for? Browse other questions tagged or ask your own question.