What Is a Hyperbolic Structure in Conic Sections?

[peter.ell]

I'm trying to understand how structures based on conic sections work. For example, when people speak of a parabolic mirror or a parabolic orbit for a satalite, I know what they mean, but when they speak of a hyperbolic mirror or a hyperbolic orbit, what does that actually mean? Is a hyperbolic structure, with the exception of shapes like nuclear power steam vents, just a wider parabola?

Also, what really is the difference between a parabola, half a hyperbola, and a semi-cirlce? If a semi-circe has equal curvature all the way around, and it has a definite center, then why doesn't a circular mirror bring all light to a single focus, while a parabolic mirror can? What's the difference between a center of curvature and a focus?

Thank you so much for helping me. It's been years since I took algebra and none of this was ever explained to me.

 

[tiny-tim]

welcome to pf!

hi peter.ell! welcome to pf!

Also, what really is the difference between a parabola, half a hyperbola, and a semi-cirlce? If a semi-circe has equal curvature all the way around, and it has a definite center, then why doesn't a circular mirror bring all light to a single focus, while a parabolic mirror can? What's the difference between a center of curvature and a focus?

a conic section has two foci, and light from one focus will be reflected onto (or, for a hyperbola, away from ) the other focus

for a parabola, one focus is at infinity, so light from infinity (ie parallel light) will reflected onto the focus

for a circle, the foci are in the same position (the centre), and light from infinity won't be reflected there

… when they speak of a hyperbolic mirror or a hyperbolic orbit, what does that actually mean? Is a hyperbolic structure, with the exception of shapes like nuclear power steam vents, just a wider parabola?

no, it means either a one-sheet hyperboloid (a cooling tower), or a two-sheet hyperboloid (basically, two shallow dishes separated and facing away from each other)

only the two-sheet hyperboloid can be a mirror, because it's obtained by rotating the hyperbola so the the focus stays still: so it is still a point, and can still be a focus! (in making the one-sheet hyperboloid, the focus becomes a circle, which is no good)

place a real object at one focus of a two-sheet hyperboloid, and there will be a virtual imag at the other focus