Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone. Depending on the angle between the plane and the cone, four different intersection shapes can be formed. Each shape also has a degenerate form. There is a property of all conic sections called eccentricity, which takes the form of a numerical parameter
Types of conic sections
This figure shows how the conic sections, in light blue, are the result of a plane intersecting a cone. Image 1 shows a parabola, image 2 shows a circle (bottom) and an ellipse (top), and image 3 shows a hyperbola.
Parabola
A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane. Every parabola has certain features:
- A vertex, which is the point at which the curve turns around
- A focus, which is a point not on the curve about which the curve bends
- An axis of symmetry, which is a line connecting the vertex and the focus which divides the parabola into two equal halves
All parabolas possess an eccentricity value
Non-degenerate parabolas can be represented with quadratic functions such as
Circle
A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle. All circles have certain features:
All circles have an eccentricity
where
The degenerate form of the circle occurs when the plane only intersects the very tip of the cone. This is a single point intersection, or equivalently a circle of zero radius.
Conic sections graphed by eccentricity
This graph shows an ellipse in red, with an example eccentricity value of
Ellipse
When the plane's angle relative to the cone is between the outside surface of the cone and the base of the cone, the resulting intersection is an ellipse. The definition of an ellipse includes being parallel to the base of the cone as well, so all circles are a special case of the ellipse. Ellipses have these features:
- A major axis, which is the longest width across the ellipse
- A minor axis, which is the shortest width across the ellipse
- A center, which is the intersection of the two axes
- Two focal points—for any point on the ellipse, the sum of the distances to both focal points is a constant
Ellipses can have a range of eccentricity values:
where
The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle.
Hyperbola
A hyperbola is formed when the plane is parallel to the cone's central axis, meaning it intersects both parts of the double cone. Hyperbolas have two branches, as well as these features:
- Asymptote lines—these are two linear graphs that the curve of the hyperbola approaches, but never touches
- A center, which is the intersection of the asymptotes
- Two focal points, around which each of the two branches bend
- Two vertices, one for each branch
The general equation for a hyperbola with vertices on a horizontal line is:
where
The eccentricity of a hyperbola is restricted to