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There are many ways to draw an ellipse. Check out FHB issue 5 pg 25, 8 pg 50-51, 34 pg 58, 35 pg 14 and 54 pg 57. So let’s see a list of ways to draw a parabolic curve. I know of no easy way, only parallel line development.
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Before anyone suggests it, let me say that a string hanging between two nails forms a catenary curve which is not a parabola.
"parallel line development"? Educate me, please. What do you mean by that?
What points must your parabola match? Vertex, and what else?
*You could try this...http://graphics.cs.ucdavis.edu/CAGDNotes/Quadratic-Bezier-Curves/Quadratic-Bezier-Curves.html
*Joe's drawing shows a piece of a parabola, but because the "legs" are not the same length (the vertical leg appears to be shorter), the axis of symmetry is at some odd angle (not horizontal, vertical, nor even 45 degrees). This may or may not be desirable depending on what you are doing with the curve.Attached is an elaboration of the technique suggested by Joe's graphic. This shows construction of a parabolic arch which matches known endpoints and vertex.
*Here's the simplest description of the differences between an ellipse and a parabola. http://www.geom.umn.edu/docs/reference/CRC-formulas/node26.html#SECTION01700000000000000000I should explain why I use both an ellipse and parabolic curve in my work. I am a roofer. I do slate and architectural copperwork. This includes a lot of copper round top dormers. The parts for the valley are an ellipse, which is the part that sets on the pitched roof. The part for the round top dormer is a parabolic curve. I currently use a sheet metal pattern drafting book which uses parallel line development to draw the parabolic curve. It draws the intersection in 3D. I appreciate youse guys's input and will study your links for answers.
*Ted,Great post.A very simple and very clear explanation. I'm definitely saving that for future use.Terry,Are you saying that the copper that covers the top of the dormer, when i lying flat,is the shape of a parabola? I know that it's very close to a parabola, but I don't think it actually i isa parabola. I may be wrong on this one Terry. I'm pretty sure the answer is in your reference material. If it isn't, I'll bet that Ted LaRue has an answer.Ken
*Joe,Not so. The directrix is below the line PP' in your sketch (it would cut through the top of the grid. The point P is not on the directrix.You are correct that the distance from the focus to the vertex is equal to the distance from the vertex to the directrix, but the directrix isn't the line PP'.
*Can we agree, or am I wrong? 1) Is the valley piece that rest on top of the round-top dormer a parabolic curve?2) Is the distance from the focus to the vertex equal to the distance from the vertex to the directrix?
*Terry,If you pass a plane through a cylinder on an angle (other than 0º or 180º), the resulting curve is an ellipse. So when a roof plane intersects a circular dormer, the intersection is an ellipse.When cutting the copper however, you are undoubtedly cutting it while the copper is i lying flat.Correct?If so, the curve that you are looking for is i notan ellipse, in my opinion.As a matter of fact, I don't believe that it is an ellipse, or a parabola.Let me make myself clear. If you lay a piece of copper flat on the floor, and trace out a parabola on it, that parabolic shape, will i notturn into an ellipse, when wrapped around a cylinder.Also, if you trace out an ellipse on the copper while it is lying flat, that ellipse, when wrapped around a circular cylinder, will not turn into an ellipse.Ken
*KenSo if I agree with your interpretation, then I'm stuck drawing this shape using my parallel line method.This discussion has been very helpful in drawing a parabola. Now I just have to work with the information more until it sinks in.One last question:Can any of the methods I described to draw an ellipse, be used to draw a parabola?Thanks Joe, Ted and KenTight linesTerryCatch the answer next week, Be safe.
*Terry,Regarding your comment in your last post"So ( Ken ) if I agree with your interpretation, then I'm stuck drawing this shape using my parallel line method."I would say that statement is correct Terry.There may be other ways, however. But I'm fairly confident that it can be demonstrated, that wrapping either an ellipse, or a parabola, that has been traced out on a flat plane, will not result in the curve that you are after.I may be wrong, but I don't believe that there even exists a i mathematical namefor the curve that is formed, by unwrapping a slice though a cylinder into a flat plane.Ken
*Ken: You are right on. When a cylinder is intersected with a plane, the intersecting points do make up an ellipse. This ellipse then has to be projected down by parallel lines to develop the curved line on the flat stock, in this case copper. This line is not an ellipse, and what it actually is I am not aware of. You may be right also in that there is not a name for this curve, but my gut feeling is that there is a name for it. Ted?
*Joe,In my construction, C was the vertex. I made no claims about P or Q except that C was the midpoint of PQ. That doesn't make Q the focus. In fact, if you keep the length of PQ fixed, and keep C as its midpoint, the position of the focus will change if you vary the length of AB. If you let h = length(QB) and k = length(QC), then the focus will be h*h/(4k) units below the vertex and the directrix will be the same distance above the vertex.The fact that the vertex was 1/2 the length of PQ is simply a result of the construction technique. The midpoint of one leg can be connected to the midpoint of the other as a horizontal line segment, and the midpoint of that segment is the vertex of the parabola.
*Hey Ken...Can you offer a convincing argument that the flattened copper won't be a parabola? I've got some extra milkbones I'd like to unload...Stan, I don't know of a name for the curve, but it isn't "parabola". I'm betting Ken will go for the milkbones and show us why.
*Trivia:If you put some epoxy resin in a circular pie pan which has vertical walls, and then spin the pie pan on a turn-table while the epoxy sets up, you'll have formed a paraboloid (parabolic dish). Centrifugal force throws the resin up along the sides in the shape of a parabola. This technique can be used to make parabolic microphones.
*FWIWhttp://www.xahlee.org/SpecialPlaneCurves_dir/Parabola_dir/parabola.htmlThen click your way around the links. WAY over my head. But interesting.Half a milkbone for a cool site???????
*Ralph,Thanks for providing the cool site. The Milkbone Distribution Committee is sending you the 1/2 milkbone, as you requested. ( Please allow 7 dog days for delivery)Ted,I may find some time in the next few days to provide a proof that the "unwrapped" copper, when lying flat, can't be a parabola. I'm starting a new roof today and am very busy, so if you can provide a proof, go ahead and do so, before this thread heads off to the archives.I hate to give up the opportunity to get those milkbones tho.....If blue eyed devil hears about this, he'll be after them. Try to keep it quiet.Ken
*To proof or to roof, that is the question... :)
*The premise was that we were given the ends of the arch A and B, and also the vertex C. If you construct QP to be any length other than twice QC, then your parabola won't pass through C. That's a consequence of the geometry of the construction, as I explained two posts ago. A point 50% of the way down one leg is connected to a point 50% down the other (similar to the way a point 10% down one leg is connected to a point 90% of the way down the other leg). That puts the vertex at the midpoint of QP. If you want that vertex to be C, then you have to make QP twice as long as QC.This has nothing to do with the focus or directrix.
*1) You can draw a parabula using a pencil/string construction much like an ellips may be drawn. 2) Intersection problems are covered in most sheet metal books.3) Most intersection curves are not named.
*Folks,Here's another way to generate a parabola. Refer to the enclosed diagram. Figure out how far you want the focus to be from the vertex (length "a" in the picture). Draw perpendicular lines as shown. The points on the parabola have the property that the distance from any point to the focus is the same as the perpendicular distance from that point to the line at the bottom. Hope this helps...
*Terry,I don't agree with Joe's last post. I don't see any way that you can wrap an ellipse around a cylinder and have it match the ellipse that it formed on the flat surface of the main roof.I still believe that neither an ellipse nor a parabola will work, although I would say that a parabola would come closer than an ellipse.As I said before, I don't believe there is a name for the curve that results when you "unwrap" the copper from the cylinder and lay it flat. But I'm almost positive that it isn't a parabola or a ellipse.Ken
*This sounds similar to descriptive geometry. Too bad it's been 25 years since I've used it! If you can find an old textbook take a look. It is a graphical technique to fine true size and shape of a plane or intersection given 2 views.
*Joe, here is a repost of part of the first graphic I posted. I thought it was clear that A and B were endpoints of the arch. I added the focus and directrix in the correct locations to help you see that they have no relevance to the construction.
*I was mistaken. The curve does have a name. I just now had time to give this problem a serious look, and found out that the math isn't very complicated at all. The curve is not a conic section.Here is a plot of one half of the curve (the other half would be a mirror image). You'll notice that the concavity changes as x gets large. This part of the curve would not normally be used since the top of the dormer is only the top half of a cylinder. I showed a little more of the curve to help illustrate that it isn't a parabola.
*If you're planning to post that on your webpage, try centering the paper a little better on the scanner. :)
*Ted,If the curve indeed does have a name, what is it? the witch of Agnessi, or a catenary perhaps?Ken
*Ken, Would you believe, ArcCosine?Let m be the slope of the roof (e.g., 0.5 for a 6/12 roof).Let r be the radius for the dormer top.Let x be a distance along the "ridge" of the dormer from where it intersects the roof out toward the face. Then the distance from the center line to the curve (when the copper is lying flat) is y = r * ArcCosine(1-m*y/r)This formula will work well when you don't have a piece of paper big enough to place over the dormer for a template.The derivation is pretty simple. I'm trying to draw it up in AutoCAD to post, but it may not get posted until tomorrow.
*Ted,I find that fascinating. I'm signing off for the evening, but it will surely be fun to pursue this a little further.Very much looking forward to your posts tomorrow.i AwesomeKen
*"You can always have a piece of paper big enough." Back in the old days of WW-II, and earlier, when building ships they always had to lay things out full sized in a loft to cut the steel plates to shape. Used humongous flexible strips and weights called Ducks to hold the strips in place on the floor. Probably use CAD/CAM now. Can't get much bigger than a ship in my book.Don
*Ted,I know you'll probably be posting something interesting in the next few days. I'm looking forward to the weekend, so I can spend more time with this very interesting subject.Ken
*Joe,I think you're stealing my idea model making. But that's OK.As for the idea of using AutoCAD, I remember the days of BASIC, at least in the Commodore Plus +4 version when you could combine formulas and plot the results on screen. Perhaps you could work up something in Excel and chart the results. After all, it's a formula, isn't it? -Peter
*JoeIn order to make the pattern for the dormer top, you had to place the paper on the top. This works well in your model, which was site built, but on site there are at least two problems.1) Field conditions: The guy framing this doesn't have a good pattern and so his work is not true.2) You can't turn over a 10 foot radius dormer to make a clean pattern.In short I need to make a pattern that is true, and then use it to adjust the framing so that my valley piece is smooth and true.Terry
*Joe It takes at least 2 hours to make the ellipse and ?-curve pattern. But because my patterns are correct everything else that fits to them goes together better.Just one question for you. How do you cut the plywood for the dormer top, where's that pattern come from?TerryPS Are you serious with this Wong kid about getting in the trades?
*Lee Valley has a 36" Flexible Curve that may be of some help, or maybe somewhere you can buy a longer one, bend it to fit the curve and then trace that? May be easier than dragging tracing paper up there.
*Joe It takes at least 2 hours to make the ellipse and ?-curve pattern, and make sure they fit. Parallel line development is a slow process on a 10 foot dormer, that's why I posted the original question.I've seen your work on this site. I believe I could draw a pattern off of your dormer and it would work, but the irregularities in the framing that I have to work over don't allow this. I first have to fix the framing and go from there, so I need a pattern.Having seen your work, and read some of your comments, I'm surprised you'd steer people away from a career in construction. I admire your math and construction skills and wonder about your response to Alan.
*Sorry I didn't post more information about that formula earlier. I've been short on time.Below is a diagram showing the derivation of the formula. It could be simplified more, but I hope it is clear enough for anyone interested in it.
*I've been trying to design a mechanical device to draw the curve, but haven't come up with anything I consider practical. I should point out that the curve is really nothing more than a sine wave in a different orientation. Perhaps a wheel with a hole in it for a pencil, rolled along a straight edge (sort of like the "Spirograph" toy) would work. If someone wants to try such a device, I can provide the formulas for the diameter of the wheel and the position of the hole. The big problem is that a different diameter wheel would be needed for different situations (I think it depends on the ratio of the roof slope to dormer radius). Gearing could be used to eliminate that problem, but now you're talking Rube-Goldberg. :)
*Somebody help!!!!Is this the curve? Whitch of Agnesi?Please look at tracing Witch of Agnesi once your in the site.http://www.xahlee.org/SpecialPlaneCurves_dir/WitchOfAgnesi_dir/witchOfAgnesi.htmlhttp://www.xahlee.org/SpecialPlaneCurves_dir/WitchOfAgnesi_dir/witchOfAgnesiGen.mov
*Terry,That's another curve that looks similar, but isn't. The piece which wraps over the roof of the dormer ends when you reach the vertical sides. Something unusual must happen to the curve that those points. The witch of agnesi has no such "special" points...it drifts off to infinity, gradually approaching zero. That kind of analysis also indicates that the curve isn't a parabola.The curve is a sine wave. The "special" points on that sine wave are the points where the curve changes from concave up to concave down.Consider this. Suppose instead of the conventional dormer, you simply had a cylinder stuck horizontally into the main roof and projecting several feet out. Now the piece which was wrapped over the dormer roof can continue to be wrapped around to the underside of the cylinder. If you flattened out that piece, what would its shape be? In a couple of minutes I'll post a picture to illustrate.
*Here is a sketch showing a sine wave and the portion which would be used to cover the roof of the dormer. It also shows the expanded portion which would be used if you wanted to wrap around a full cylinder (as described in my previous post). Notice that the concavity of the wave changes where the dormer roof portion ends.
*Terry,I agree completely with Ted on this. The curve he is showing is the curve that you are after.I've often thought that it was a sine or cosine curve myself, but I couldn't prove it. Ted has shown this to be the case in his derivation. Quite frankly, he made the problem look like childs play. The question now before you, is how do you apply these ideas, and how practical is it to use these formulas in real life situations.My feelings are that they could be very useful. I think you could take Ted's formula and use it to trace out the curve on a piece of copper (or plywood), while the copper (or plywood) was lying flat on the floor and get good, consistent results.I wouldn't mind picking a dormer width and a main roof pitch, and going through the process with you, if you'd like.Ken
*Ted Where did you learn this stuff?What's your educational background?What's your profession?Just curious, you don't have to answer or you can e-mail me direct. I'm very curious.My answers to these questions:I basically taught myself.High school geometry (80% gpa)Self employed slate roofer. and copper smith (21 years)
*Ken I appreciate the offer. I have a job I'm just starting with several different dormer situations. I'll contact you when I get ready to post it here. You guy's have been a great help. Right now I'm inspired to get a digital camera and post some pictures of my past projects for you guys.Thanks again for all the help! Terry
*Terry,I like your idea about getting the digital and posting some pictures. I have a friend that I borrow one from for such occassions, but I really want one also. ( just in case you find a buy one get one free deal ) It would be great if you posted a series of pics showing all stages of the process, right down to attaching the copper.I may post something before you get started, demonstating how to use Ted's formula for a specific situation, so keep a lookout for new posts in this thread.BTW, we're all very lucky to have Ted's continued participation in these threads. He's excellent with both math and computer, and has helped many solve their difficult problems here.Ken
*If I remember my basic geomtry a parabola is a curvethat approaches parrallel but never quite gets there. A hyperbola is a section of a cone that approaches the cones angle the same way. seems that you guys may be confusing the two. as to how to draw one its all greek to moi.My two cents Mr. T.
*Tony,It's not a parabola or a hyperbola. You need to read all the posts.Ken
*
There are many ways to draw an ellipse. Check out FHB issue 5 pg 25, 8 pg 50-51, 34 pg 58, 35 pg 14 and 54 pg 57. So let's see a list of ways to draw a parabolic curve. I know of no easy way, only parallel line development.