At the risk of sounding like a fool, if the equation describing a circle is
x^2 + y^2 = r^2
then (let’s say the “joint angle” is perpendicular), substituting R sin(x/R) for x, isn’t the formula for the “mysterious curve” just
R^2 sin^2 (x/R) + y^2 = r^2
(obviously not an ellipse)?? No higher mathematics or whiteboards or even envelopes needed, just the back of an old grocery receipt in my case.
We could continue in this vein, but not sure that helps the OP who wants to actually draft such curves on a piece of metal, versus the practical suggestions already given. Though you could plot the curve on a computer and use a stencil cutter or something to make your own templates if you didn’t want to buy any or plot things by hand. A water-jet cutter would actually need some sort of CNC computer file, so you really need to use the CAD system anyway like people are saying.
I didn’t think the cite was that complex. It was covered in a 1-year high school drafting course along with pretty much every elementary drafting technique. It’s basically exactly what the OP wants, sheet metal pattern design.
Of course, if it’s being done in a big company for production purposes, it does need to be done right with formal drawings etc. I don’t think Ford or GM wants their car parts designed on the back of an envelope. They have enough recall problems as it is. But for one-off construction, the only question is - do you trust your personal drafting skills?
You can do the same on AutoCAD today, the drawing concepts don’t change. The drawing precision will be more exact, you can blow up certain areas to ensure accuracy; and the curve fitted to the points is also likely a better fit; and you can with enlargement do more points. As I mentioned the only guestimate is the length of the big cylinder segments between points - the AB, BC etc. Not sure if AutoCAD has a “how long is this curve” function.
Slight hijack of my own thread. The water jet cutting is a dream come true for home fabricators. Inexpensive and deadly accurate. I recently brought them a water buffalo horn and had them slice it in .008 thickness slabs I could use as fletchings on my flight arrows. I have to use primitive materials but it didn’t say anything about the technique I have to use to cut them.
It is shipped in from China and other Asian countries. Used in arts and crafts quite a bit. English long bows use the tips for making horn nocks on the bows.
A while back I was making a still and didn’t feel like cutting out all the copper by hand. I layed out all the parts and for $100.00 minimum charge everything was cut out perfectly.
For what it’s worth, laser cutters are becoming similarly mainstream nowadays (perhaps even more so), and they’re providing a similarly-great boon to hobbyists. My uncle’s fond of telling how, with exacting effort, he managed to cut a particular piece of etched window glass without damaging it, a feat that even the local professional glass-cutters considered likely beyond their skill… and all I can think of is that, if given the same task nowadays, I’d just take it down to the library’s makerspace and laser-cut it.
Yeah. At arts & crafts shows / festivals / sales one sees some amazingly intricate artwork cut from paper, metal, glass, or wood. The kind of thing that’d take man-months if done by hand as it was not so long ago.
The “artness” of it sorta falls apart when you know the “artist” downloaded the pattern from the 'net and after a few mouse clicks a machine in his basement churns out identical copies at a rate of 10 to the hour.
Actually, the OP requested a mathematical formula, insufficiently addressed above. We each presented an alternative…
Out in the garage, contemplating the intersection of two pipes, one can choose between utilizing a simple compass and progressing to the next project in short order, OR, spend $1500 on an AutoCAD license, **OR ** drag an unused drafting table and equipment out of storage to assemble a projection. (The latter two assuming you have the knowledge to utilize these tools). Lacking a mathematical solution, the choice seems obvious to me.
FWIW: Drafting a projection as described, wasn’t covered till third year in High School. YMMV:rolleyes:
I gave a formula for a special case ( (sin x)² + y² = k ). If you want a more general but messier-looking formula (messier as in an arbitrary ellipse positioned anywhere, versus a perfect circle centered at the origin), take the equation describing the intersection of the first pipe (cylinder) and a plane, which will be an ellipse (conic section), and for x substitute sin(x) or perhaps R² sin(x/R) depending on how you have your variables set up. So what is insufficiently addressed?
Think about what comes next, though. Say you want to plot a curve like (sin x)² + y² = 1 on your computer screen. The machine has to decide which pixels are on and which are off, a finite-precision numerical calculation. So you have left the ideal geometrical world anyway. Similarly, the water-jet file contains commands driving the cutting head to a series of coordinates (x,y), so the CAD program cannot work purely symbolically when it comes time to output G-code.
Is nothing but a “simple compass” sufficient even for an ellipse?
Anyway, is the OP’s problem a lack of access to AutoCAD? Or have her questions been answered?
My question was answered after the first post I indicated a mechanical solution would be preferable. My math skills are limited. I actually saw several methods I could use.
For what it’s worth, it’s not hard to qualify for a no-cost Autocad license, provided that you’re not using it for professional purposes. And given the OP’s hobbies, I wouldn’t be surprised if he has a copy already.
I do have a copy but have never taken the time to learn it. I have an older version on my old computer and haven’t even thought about it for several years.
My math skills are lacking, and I don’t “see” solutions to problems of this nature mathematically. Sorry if I didn’t recognize the fact that you solved the puzzle… I thought the discussion was still open, math-wise.
FYI: His questions.
Not meaning to pile on here but you keep speaking of ellipses. IMO …
Regardless of size or orientation there is no intersection between two idealized pipes of circular cross section that results in an elliptical line of intersection. If nothing else, the curve of intersection is 3D and ellipses are 2D figures. Further, IMO there is no 2D projection of that 3D line of intersection which is properly elliptical.
As demonstrated by the various figures linked in various posts, the path of intersection is always a more complex and irregular shape; never an ellipse.
Yes, a plane cutting a cylinder is an ellipse (or a point or a circle in the degenerate cases), but that’s not a meaningful starting point when there’s no plane involved in the actual problem.
I may well be confused about what case(s) you’re covering and what ellipse you’d describing. If so, please unconfuse me.
I imagined unrolling one pipe, oriented along the y-axis, into a flat strip tangent to the original cylinder keeping the line x = z = 0 fixed. The ellipse in question was supposed to be the intersection of the impinging pipe and the plane z=0. (The orthographic projection of the cylinder onto this plane then maps the line of intersection onto this ellipse.)
Of course, if you trace the intersection onto the cylindrical sheet and unroll it, it does not become an ellipse. What will it be? I claim that, after unrolling, a point (x,y,z) originally on the cylinder maps to (R sin^-1 (x/R), y, 0) on the planar sheet, where R is the radius of the big cylinder. Therefore, whatever your impinging cylinder [simple example: x² + y² = r²], first take the equation of the ellipse obtained by substituting z=0 [just x²+y²=r² in the simple example] and replace “x” by “R sin(x/R)”.
Fair enough but I, for one, would love to see the general solution. This is GQ in the Dope after all, is it not?
A general solution could be defined on the variables r₁, r₂, ∠θ, and ∠ψ described in post 20, but even those do not describe the full, general solution because they assume the two axes intersect, and they do not need to. That, too, was discussed.
OK, then besides the two radii, you need to write down the equation of an arbitrary line in 3D space. Decide what variables you want to use to do that (angles? a couple of vectors?), and proceed as in my post… (the general solution will be a distorted (general) ellipse as I explained)
(The link in #35 has nice sketches of what some of the curves look like.)
My apology, I quickly realized that my math skills would not be able to handle the formulas being presented. I am limited to very basic self taught geometry and trig. The symbols being used are meaningless to me. I decided at that point a mechanical solution might be better for me.
