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.
 A: First, I just want to mention an experiment you can try yourself.  If you take a flashlight, the beam of light which comes out of the end is roughly conical, and you can cast the beam onto a wall or other surface.  See what conditions are necessary to get a circle, ellipse, parabola, and hyperbola.  The next part of the experiment is a bit of a stretch, but imagine that all of the light that leaves your flashlight has been traveling in a straight line for all time, even before the light left the flashlight (it may help to think of the light traveling backwards in time for this).  Circles, ellipses, and parabolas are what you get when that light didn't come into contact with the surface in the past, and a hyperbola is when this backwards beam did.
One way to think about conic sections is by projective geometry, and one way to think about projective geometry is by imagining a point light source casting light in all directions with a sphere around it which blocks light in a certain pattern.  The rule for this sphere is that if it lets light out at some point, it must let light out at the exact opposite point, and vice versa.  We interact with this contraption by putting a screen near the light and seeing the patterns cast on the screen, and where you place this screen is arbitrary.  If you want to cast a line on the screen, you want there to be a great circle on the sphere which lets light through, and great circles always cast lines (so long as any light from the circle meets the screen, but one can always move the screen to see some light).  This is the reason for the two-sides rule: if it looks like an infinitely long line from some perspective it had better look like an infinitely long line from all perspectives.
What casts a circle? A non-great circle on the sphere does.  What does the light pattern look like which radiates from the sphere? A double cone.  What does one get when after moving the screen? Circles, ellipses, parabolae, and hyperbolae.  A parabola is a circle reprojected so one point is infinitely far away.  A hyperbola is a circle reprojected so two points are infinitely far away, the two branches being the two halves of the circle.
