Image that you have 2 cylinders- one large circular cylinder with diameter ‘D’ and one small circular cylinder with diameter ‘d’. You position the small cylinder so that its axis is perpendicular to the axis of the large cylinder and butt the 2 cylinders together, trimming the small cylinder until it fits. The intersection of the 2 cylinders is saddle-shaped.
You then remove the small cylinder once it has been trimmed and unroll it into a flat plane. The saddle shape now looks like a sine wave, but is it? What is the formula for this curve in terms of ‘d’ and ‘D’??
Thanks.
Just to clarify–we’re talking about arranging the cylinders to look like a +, and then considering the intersection, right?
I’m thinking more like a ‘T’, with the larger diameter cylinder forming the top of the ‘T’ and the smaller cylinder forming the base of the ‘T’.
Sorry if that wasn’t clear.
Then you can solve the problem in two steps. First, the curve in space where the two cylinders intersect is the set of points (x,y,z) where y[sup]2[/sup]+z[sup]2[/sup]=d[sup]2[/sup] and x[sup]2[/sup]+y[sup]2[/sup]=D[sup]2[/sup]. (This assumes the small cylinder is lying along the x-axis while the large one is lying along the z-axis.)
Second, re-write the above in cylindrical coordinates corresponding to the small cylinder. That is, let y=rcos(t) and z=rsin(t). We get r[sup]2[/sup]=d[sup]2[/sup] and x[sup]2[/sup]+r[sup]2[/sup]*cos(t)[sup]2[/sup]=D[sup]2[/sup]. Unrolling the small cylinder corresponds to throwing away the “r” coordinate and getting a function for x in terms of t; if we do so (and if we assume x is positive) we get x=sqrt(D[sup]2[/sup]-d[sup]2[/sup]cos(t)[sup]2[/sup]).
So no, it’s not quite a sine curve.
This sounds like a homework problem.
No, I’m not a student.
I was in the process of trying to fit-up some piping when it occured to me that I could save myself a heap of time if I could just make myself a template that wraps around the pipe and then scribe off where to make the cut.
When I first looked at it, I thought it looked just like a simple sine-curve, so I made a sine-template and cut the pipe. Then when I went to fit it up I noticed there were some fairly large gaps- it didn’t work. Now I know where I went wrong!
Thanks, Math Geek!