# Why are hyperbolas defined by two branches?

Why are hyperbolas defined by two branches, unlike a parabola which only have one? Geometrically, it looks like a slice. When plotted on a graph, it's two separate curves. Why?

We were never taught about conic sections in high school, so when I saw this image, I was curious. I understand that a parabola is the result of a planar intersection made parallel to the edge of the cone, but I don't see how making the plane near-parallel to the axis makes it split into two branches, nor how they correspond to the geometry.

I did read the page on PurpleMath, but it was more from an equation point of view. This table on Wikipedia seems to show it as a "reverse ellipse", which seems make hyperbolas have more in common with ellipses than parabolas.

• There's another side to the cone that you don't see in your picture. But you do in this one. Apr 24, 2015 at 23:24
• This image shows two cones though. Does a conic intersection actually rely on two cones, or is just an abstracted way of picturing it? Apr 25, 2015 at 2:31
• Depends how you define a cone. If you define a cone by revolving a straight line around an oblique axis that intersects it, both "cones" form one single entity. Try making a plot of $z^2=x^2+y^2$ for instance. Apr 25, 2015 at 6:18