Exploratorium Math: The real-life hyperbola

This still amazes me every time I see it:

How incredible is it that you can move a straight line and it makes a curve?! I guess that's what math is all about. But I like this exhibit because if you just showed me the tilted bar spinning around without the plexiglass, then asked me what kind of hole I should cut in the plexiglass that will allow the tilted bar to pass through... I would never guess the hyperbola. 

Questions I want to ask (myself or my students): 

• How does the angle/slope of the tilted bar relate to the shape/equation of the hyperbola?
• How does the angle/slope of the tilted bar relate to the slopes of the hyperbola's asymptotes? 
• What's up with the intersection of the hyperbola's asymptotes? Are they perpendicular (I can't tell at all from this picture)? What changes can you make to the tilted bar or the way its rotated that will either (1) keep the asymptotes perpendicular (if they already are)? (2) make them perpendicular if they're not already?
• Where are the foci of the hyperbola? How do they relate to the position of the tilted bar? 
• If you make the bar longer, but keep it at the same angle, what will need to change about the hyperbola and the way it's cut out of the plexiglass? 

I pretty much just want to make kids build this.