Saturn 1B Scalloped Transition

[Microspeed]

Something I've been wondering for a while is how to mathematically describe the shape of the cut-outs on the Saturn 1B's scalloped transition (intersection of a cylinder with a cone with central axes parallel). If there are any mathematicians (differential geometricians?) that might be able to explain how to figure out the shape, that would be quite cool. The shape is very close to an ellipse (I've always used ellipses to approximate the shape when making the parts [minor axis being the diameter of the tube, with the major axis figured with trig based on the angle of intersection], and it's worked well enough) but I'd really like to know what that 'perfect' shape is.

 

[Chrisn]

A cone with elliptical cutouts?

 

[luke strawwalker]

Exactly... section of an off-center parallel axis cylinder intersecting a cone describes an ellipse.

Later! OL JR

 

[Microspeed]

But doesn't an ellipse only truly describe the intersection of a cylinder with an angled plane? For instance, had the tubes been moved closer to the center of the cone, you can see that the intersection looks quite a bit more dramatic. I think that the reason ellipses work so well on the Saturn is because the tubes are offset so far that the cone begins to approximate a plane fairly well (less curvature over the area that the tube intersects), but the upper bit should still be a bit narrower.

Here are a few RockSim exports showing what I mean. You can see how the shape starts out rather oddly pointed, then gets more and more elliptical as the tube moves outward.

You can also see it in how the transition itself gets laid out. The last image is a template for a scalloped transition approximation drawn in Corel with simple ellipses. Even though all the calculated dimensions are exact to the thousandth of an inch, the ellipses still overlap slightly on their upper end, but when folded, the tank tubes fit next to each other with no overlap, though the elliptical cutouts might have led you to believe otherwise.