*
I agree with Mike. The tool you are considering is laying out an arch.
An ellipse is another word for oval. An ellipse has a precise mathematical formula, which I don’t know, where an oval is anything egg shaped, but usually symmetrical.
Mike also describes a string technique, but doesn’t quite give enough information.
If you have a given width and height of your ellipse, for a window, say, or for the top of an arch (in this case an ellipse is far more interesting than a simple arc or half circle),you need to mark and bisect a center line. On the bisecting line, call it A, measure out half the width of the ellipse, or the height of the arch, and put in a nail at B. Next lay out the length of the ellipse, or width of the arch, half on either side of line A, call it CAD, where I and I are the outside points. Now measure in from I and I half the length of A, and put a nail at the two new points, which are the foci. Now tighten the loop of string around the too new points and B. Pull the nail at B and put a pencil in the loop pulling tight to one side to point A. Keeping the string tight draw the ellipse, passing through B all the way out to D.
If I knew how to put a fancy drawing on the computer I would, it would help.If I could show you on a sheet of plywood,you could see it once and never forget it. Words have a way of getting muddled when describing mathematical procedures.
But if you try this and see how it works, you’ll be able to draw any ellipse given width and hieght.
There are other eqaully good ways to lay out ellipses using a framing square. Hopefully someone else can describe those ones.
Another trick, if you want to draw smaller ellipses is to get plastic pipe, commonly available up to 10 inches and cut it at an angle. Place the cut on the paper or wood and trace it.
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*
I made a mistake in the description of how to draw an ellipse with a string. In determining the foci, the distance you come in from the outside points, C and D,is half the total width of the desired ellipse,AB, not half AB as earlier described.
As for Joe's notion that a circle has two foci that share the same place, I'd like to get a mathemetician in on this to really help us out.
*
Two books I'd recommend for understanding and practical applications of this and other tricky geometry. Both are excellent in their ways.
*1. Ernest Joyce. The Technique (US Dictionary) of Furniture making. UK ISBN. 07134 0217 2. In the US I think it's 0 8069 6440 5. In the UK this book is regarded as "The Bible" of cabinetmaking. They say that if Joyce doesn't mention it at all, you'll probably have to invent it yourself.
*2. George Ellis. Modern Practical Joinery. ISBN 0 85442 039 8. 99 years old and still a font of excellent information! This book may only be available from an UK supplier, GMC Publications, but it's published by Stobart Davies.
You might try the online booksellers. They certainly can supply Joyce, but as before, I'm not sure about Ellis. I could scan relevant info out of Joyce for you if you like. Let me know. I Just don't have it in jpeg format to hand on my hard drive.
*geoff murray,
View Image © 1999-2000"Whenever, therefore, people are deceived and form opinions wide of the truth, it is clear that the error has slid into their minds through the medium of certain resemblance's to that truth." Socrates
*hey , you bozos, u made me go out to the garage & get the diagram i nailed up the last time i laid out an ellipse...pg. 662 Arch. Graphic Std's (6th ed.)i'm a graphics guy, not a math guy, so.....to get a TRUE ellipse they list 4 methodsAuxiliary CirclesCardParallelogramand the last is the one i use..STRING METHOD.... the major axis is the baseand the minor axis is the vertical, so, if the distance between the 2 ends is say 8' lay out a line that is 8' long and construct a minor axis at its midpoint perpindicular to the major axis...locate 2 pins ON the major axis an equal distance from the minor axis... the further they are from the center, the shallower will be the ellipse, and the closer they are to the center (and each other), the higher the ellipse...locate a 3d pin ( a temporary pin )at the point where the major and minor axis intersect.. (i.e. at the top)...the 3 pins form a triangle ....... and the 3 legs of this triangle are the total length of the string... tie a string together and check it against your pins... now pull the top pin and insert a pencil into the string loop..construct the ellipse by sliding the pencil from one end of the major axis tracing it along the top ... to the other end .. trace it lightly until you are sure you are keeping equal tension on the string...the pencil remains trapped in the string loop the entire time...there's your ellipse ... get out your Bosch jig saw and cut it out..i'm going to go nail this back up in the garage so i know where it is........i have to go thru this with my signs all the timeif you want a diagram of how to do this, email me & i'll fax it to you......Kermit
*Mike,
Big Bird View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe,This board wouldn't be half as interesting as it is if everyone restricted themselves to answering the original poster's question(s).Rich Beckman
*geoff,You don't need a mathemetician (and God knows I am not one). just keep bringing in those two foci until they are at the same spot. Take your pencil and string and what do you get?Rich Beckman
*I just bought Ellis from Chapters, a Canadian on-line seller. Amazon definitely has it also.Cheap at any price.
*Rich,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*hey BigBird... little inflated view of yurself , eh? speakin of birds in gilded cages....don't pull my string....it screws up the ellipse...b just want u to know , this frog wouldn't taste good anyway... too old and tough for you...Kermit
*Kermit,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*This is a very interesting discussion. As you all appear to know, things may look like something, but not be that something. This is the case of ellipses. All ellipses are ovals, but not all ovals are ellipses. An ellipse is the set of all points whose sums of distances to two fixed points, the foci, is a constant. Thus, the loose string attached to two nails forms an ellipse since the sum of distances is exactly the length of the string. It also turns out that a pipe cut at an angle describes an ellipse for the same reason, but the proof is a lot more complex. A framing square riding on to nails will produce a circle, since its two legs act as perpendicular chords and the diameter of the circle will be the distance between the two nails. With this in mind, the gizmo in question will not produce an ellipse, although for a short distance the figure produced will appear elliptical in shape. There are several arguments that can be used for this, but one is that the tangent line (line touching the curve at one point)to the curve will not be vertical at the end points, but rather angled equal to the angle between the legs of the gizmo. There are other shapes that appear to be famous curves. For example, a cable hanging over two points of support, like a phone wire over two telephone poles, looks like a parabola, but it is not---it is called a catenary! Another is a piece of paper wrapped around a tube and drawn on using a compass. The resulting curve looks just like an ellipse, but it is not. Why not? It can be shown that it does not satisfy the definition of the ellipse that was described above. Now, who said that math is not fun???
*Howard.. i like the string method cause of it's ease of use.. but Architectural standard claims that the other three methods produce true ellipses also.....i think that the CARD method is basically the string method , and also the same as the device that Ryan was asking about... it looks to me like the AUXILIARY CIRCLES & THE PARALLELOGRAM methods are constructions, that wind up being differential equations...i.e. what i am trying to say is that the more contructions you do the closer to a true ellipse you make..but the string... and .. card .. never change the total length , thus complying with your proof...b but , hey whadda i know, im just a graphic kinda guy....
*Howard Love,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*There is a special place in hell for the users of colored script that is pale enough to make these 44 year old eyes hurt.
*Howard Love,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Howard,good points sans one. A hanging phone wire will be a parabola. As long as the wire is of uniform weight along the length of the wire it will be a parabola. It becomes a catenary (sp?) when the weight supported by the wire is uniform along the x-axis instead of along the curve. The cable in a suspended bridge supports the deck that is uniform along the length of the deck, not the length of the cable and this is a catenary.
*Ryan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Here's a definition of a catenary, straight from Merriam.the curve describing the shape taken by a homogeneous cable that is suspended by its endpoints. Any freely hanging cable or string assumes this shape, also called a chainette, if the body is of uniform mass per unit of length and is acted upon solely by gravity.
*I think we're saying the same thing. Assuming gravity to be constant and also assuming a body at rest (no inertial properties), mass and weight can be interchanged most of the time.A wire with unlimited flexibility that has a uniform mass per unit length supported at both ends will be in the shape of a parabola.The same wire supporting a mass that is uniform along the x-axis, not along the length of its own curve, will form a catenary. I suppose this could happen with a wire of varying density but I was thinking of a suspension bridge where the cable supports a deck that has a uniform mass per unit lenght along the x axis, not along the curve of the cable.Now, like I said, it was along time ago that I actually studied this stuff and my first "engineering" jobs after college were all in facilities and plant maintenance so I haven't done any real engineering since senior year final exams so don't ask me to show my math.
*Correction: I am wrong.I went back and checked out one of my old texts. It turns out that Howard is right. A cable supported at both ends will form a catenary.The formula is y = a cosh(x/a)It turns out that Galileo also thought it would be a parabola so I guess I'm in good company. Now If I can just convince everybody that the sun is actually revolving around the earth...damned flat planet.
*
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*So much hyperbole... being strung around here.... in this thread...1975 Texas Instruments calculator and the hyperbolic function key...to be used with the trig keys in the case of figuring loads on utility wiring and such way back in the 101 days...near the stream,aj
*OK here's one....If you drove one nail in Fusco's head, one in Lugano's, tied a string around them, then drive one in mine...now as I run around these two would I be marking out an oscillating ellipse as Joe tries to catch me and slap the daylights out of me and Fred just goes back and forth between joining in to ...to....to...well, he keeps changing his mind...or his name...or...still near the stream, and maybe having had one to(o) many, but I'm laughing,ajSorry but I just had to add my own hyperbole!!!!!
*hey jac, yur full a it .... TI dint have no calc wit whatever u just said in 1975.....
*Got mine from an engineer at GE...Yep...They did dude...Had it and the slide rule for the last zillion years and I do know what year in school I picked it up!!!near the stream, sorry you missed out on the inside deal!!aj
*alls i member is paying $69 bucks for a desktop in '75 that could add.. sub.... multiply & divide... period....guess u got me there... i still got all my sticks too, ever try n use em?... is it just like ridin a bike?
*Remember Mike...The slide rule is capable of so much...but you should remember the basics of multiplying and dividing....log scales...knowing approximately the answer...putting the decimal point in the right place...extrapolating an extra digit or two... or three!..(as my buddy and I used to drive our teacher crazy!!)near the stream, trying to remember pie...3.141593...for some reason I memorized a hundreds digits...once...aj
*Guys,I have sent a lot of proofs into math journals and I have never seen the level of insight, humility and intelligence that has appeared in this column. Just one more insight into the catenary--let's suppose that the catenary is a frictionless wire and let's call its high point A and its low point B. If a bead is placed any where on the wire between A and B, it will take the same amount of time, when released, to reach point B. Now the question is, who first discovered this? Who says math is not fun????
*Hey guys! Would you go thru that one more time??
*Off subject, but what the heck...In one of the science workshops I ran for my daughter's 4th grade class, one of the problems I presented to them was how to get a wagon with square wheels to track, or ride, smoothy. The only thing that couldn't be meddled with was the wagon itself.The answer? The roadbed needs to be a series of inverted caternaries.
*I'm always impressed how much knowledge this group of dumb carpenters actually has. My wife looked at me like I was nuts when I told her that ten carpenters were comparing the mathematical properties of caternaries and parabolas and the various ways to describe an elipse.So how about that mc² thing huh?
*Not in my wildest dreams did I ever imagine that one day I would be in Breaktime discussing catenaries.But as long as we are on the subject, here's a quote from a fairly modern Calculus text that I have on the shelf."It can be shown that clothelines, chains, telephone lines, and electric power cables that are strung from one support to another always hang in the shape of a hyperbolic cosine curve.In contrast, the cables of suspension bridges, which do not hang freely but support a uniform load per horizontal foot, hang in parabolas. Hyperbolic cosine curves are sometimes called chain curves or catenaries, the latter term deriving from the Latin "catena" for "chain"."
*Ken,Does your text have a proof for the suspension bridge parabola? For anyone interested in how curves are proved, check out dandelion spheres or conic sections for proving the properties of parabolas, ellipses and hyperbolas.
*ROFL!
*Howard,No, there's no proof offered for the suspension bridge parabolas. It's presented as an observation, which I assume is based on fact. It seems logical to me, however, since the caterary takes its shape only when it isn't used to support other weights, which would be the case in suspension bridges.
*
I recall in a high school Algebra textbook a problem which stated that the cables on the Golden Gate Bridge hang in a parabola. I remember that because I remember trying to tell the teacher that I thought they hung in a catenary. My father had a PhD in math so I had heard about the catenary at an early age. The teacher didn't know what a catenary was and didn't appreciate my concerns...
In the late 60's I saw a "tip" in Popular Mechanics magazine on drawing a circle.....the tip was to hang a string between two nails in the plywood. I was quite surprised that such an error was published. The string would form a catenary rather than an arc of a circle (or an arc of an ellipse, for that matter).
*
Ted,
This past December I worked for a framer for a few weeks helping him cut the roof and make about 60 arches that were called for in the house.
He allowed me to use my own methods to make the arches but mentioned one day that the method he normally uses to define the circular arches is to allow a thin piece of chain to hang from two points on the plywood giving it enough slack until it has the proper arch height. He then uses fast drying spray paint to spray the chain. He then cuts that curve left behind when the chain is removed. He says its quick and doesn't require and calculation.
Apparently you can get reasonably good results from this method as long as the arches don't have much arch height. I don't think the reults would be very good however if the arch were close to a semicircle.
I mentioned to him that the curves that were being formed were catenaries not circular arcs as he thought, but I'm sure he is still using this method as we speak.
I never actually watched him make an arch using this method, as he let me do things my way on the project, as I mentioned earlier.
*That's facinating. Your framer's method dosen't procuce a circular arch, of course, but probably produces a curve that is much more visually interesting, depending on his eye.
*
I saw an ad in a magazine (JLC?) for a tool for marking out eliptical arches.
This thing is like a carpenters square except that the two legs pivot and can be locked into any angle.
So what you do is, drive a nail at the ends of your major axis and one nail at the tip of your arch (end of minor axis).
You then set the angle on this tool so that it's flexible virtex is touching the top nail and the two legs are resting against the two side nails.
You then move this tool so that the two legs are always in contact with the side nails and the virtex is supposed to strike out an elipse.
Here's a bad sketch to explain what I'm talking about.
Does this thing really strike out an elipse or just some other type of arch? And for somebody who's better at math than I am, what is this shape?
Before my brain started to atrophy, I could have done this myself. I really miss my mind.
*Ryan , that tool you were talking about really does make true ellipses...but the diagram you show is just a simple arch..an ellipse has two foci, an arch only one radius..an ellipse can be laid out with a loop of string and a pencil secured around two nails driven into the foci...the tool you describe replaces the string loop...Mike
*You can make an on site version of the new tool.Draw the curve you want on a sheet of ply. Tack a nail at the top/center point. Tack 2 more nails on each side/end point. Take 2 pieces of furring/any similar scrap and lay them up each leg, crossing at the top point and tack them together. Drill a hole centered at the top of the crossing pieces for a pencil. Using the nails as guides, slide the pencil down one side then down the other.Should mark out your elipse. Or of the new tool is cheap enough, stop screwing around with scraps, and go buy it! Jeff
*I agree with Mike. The tool you are considering is laying out an arch.An ellipse is another word for oval. An ellipse has a precise mathematical formula, which I don't know, where an oval is anything egg shaped, but usually symmetrical.Mike also describes a string technique, but doesn't quite give enough information. If you have a given width and height of your ellipse, for a window, say, or for the top of an arch (in this case an ellipse is far more interesting than a simple arc or half circle),you need to mark and bisect a center line. On the bisecting line, call it A, measure out half the width of the ellipse, or the height of the arch, and put in a nail at B. Next lay out the length of the ellipse, or width of the arch, half on either side of line A, call it CAD, where I and I are the outside points. Now measure in from I and I half the length of A, and put a nail at the two new points, which are the foci. Now tighten the loop of string around the too new points and B. Pull the nail at B and put a pencil in the loop pulling tight to one side to point A. Keeping the string tight draw the ellipse, passing through B all the way out to D.If I knew how to put a fancy drawing on the computer I would, it would help.If I could show you on a sheet of plywood,you could see it once and never forget it. Words have a way of getting muddled when describing mathematical procedures. But if you try this and see how it works, you'll be able to draw any ellipse given width and hieght.There are other eqaully good ways to lay out ellipses using a framing square. Hopefully someone else can describe those ones.Another trick, if you want to draw smaller ellipses is to get plastic pipe, commonly available up to 10 inches and cut it at an angle. Place the cut on the paper or wood and trace it.
*
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Mike,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Thanks Joe, That was what I was looking for. Don't mean to be a pain but could you show your work, or at least explain your logic.The last math class I had was Advanced Calc for engineers (calc 5), I sat here disgusted with myself that I can't do this stuff anymore. Point me in the right direction so I feel like I figured it out.
*Ryan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle