You're right. They didn't make any claim of explanation. Nice visual, though.
While we're on the subject of curves, I learned something very interesting and counter intuitive this weekend. How to change a catenary curve into a parabolic one? By loading it!
Look at the Golden Gate bridge for example. At the stage of construction when the towers were in place and the suspension cables were strung over the towers and secured in their anchors, they traced a catenary curve. Indeed, they displayed the textbood definition of a catenary. But if you look today, those cables are no longer catenary curves, they're parabolas!
Weird, I know. Just by placing weights evenly spaced on the cable (or chain) you change the shape. This was...unbelievable to me. I refused to believe it. My vector arithmetic isn't that weak. I reasoned that the links of the chain themselves were like weights and pulled downward, just as if the chain had weights! WTF?
Here's the catch. Look at the GG bridge so you can see the majestic sweep of the suspension cables. Hover in your helicopter so you're line of sight is perpendicular to the road way. Got it? Good. Notice the support cables, the ones that descend from the suspension cables to the deck are all parallel. Notice that they're spaced evenly. This was what bugged me. How is that load on the CATENARY cable different from no load at all. I could understand that the curve would be deformed if the load were asymetrical, say one honkin weight, not centered on the cable. *That* would change the shape, sure. But these neat, evenly spaced cables, each supporting the same amount of weight, the same load... how does that change it?
The catch is that the "weights", the load, is NOT evenly spaced on the suspension cable. Stay with me here. Notice that along the deck, the distance between each vertical support cable is the same. Notice that the horizontal distance between each vertical support cable is the same for each space from support cable to support cable. But, notice the distance *along the suspension cable* from support cable connection to support cable connection is different. The section of the suspension cable that is nearest the tower and angled the farthest from horizontal has connections that are farther apart when compared to the section of suspension cable that is near the center, close to the bridge deck, and more horizontal, where the connections of the support cables are grouped more closely together.
NOW I could see my mistake. The catenary suspension cable was NOT being loaded "evenly" despite my first impression. There was a higher concentration of the load along the center of the curve, compared to a lighter load at the ends of the curve/cable at the ends near the towers. It was being "pulled" harder in the center with this arrangement and it makes perfect sense (now) to see that this would deform the curve. D'oh! I'm happy now.
