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Definition of an Ellipse?8/8/17
I was nosing around on some math sites and read the definition of an ellipse as: "An ellipse is formed by cutting through a cone at an angle." Okay. Moving on. Then at another site, I read the same thing, word for word. I wondered if they copied off each other. Not very good mathematicians, I thought. Then it hit me. They are both wrong. Or one was wrong and the other copied it. Why are they wrong? Art of Problem Solving, Ellipse Define 8/8/17
Are you looking for a definition similar to a circle's "all points a certain distance from a center point"? I'm not even close to being a math enthusiast, and also writing this without a whit of googling, but I recall from geometry class that two cones, one on top of the other, can be used to define several different geometrical concepts depending on how you section it. Point, circle, ellipse, hyperbola, maybe line and plane also if you slice it right. It's probably not the actual definition of an ellipse, but a way to create one. 8/8/17
An ellipse is formed by slicing through a tube at an angle other than 90° 8/8/17
David, there are lots of ways to describe and ellipse, and I think the slicing through a cone is the hardest to get my mind around, even though I turned several on my lathe, then slices and rotated the parts 180º to see if it were so, and it is. Keith, change your tube to a cylinder and an angle, and I'll agree, since all tubes are not round. I used some square tube yesterday, if you will forgive my being picky. Just looking at a circle drawn on a flat surface other than straight on will appear to be an ellipse, defined relative to the viewing angle. 8/9/17
They are wrong, because when a cone is cut through it's "point", you won't get an ellipse. But, if cones are infinite, you can't cut them this way, anyhow. I like the second part...
This is how mathematicians say
8/9/17
Playing around at break.
8/9/17
David Sochar, et al: What I recall from geometry class is that an ellipse is defined as the set of points E, in one plane, such that the sum of the distances from E to the two fixed foci on the major axis is constant. The distance between the fixed foci is determined by the square root of the squared length of the major axis minus the squared length of the minor axis. I suppose as practical woodworking knowledge the conical slicing parameters of an ellipse is more of an academic argument, as I have yet to lay out an ellipse using anything other than an ellipse tracker. And I also have yet to construct a cone and slice it in order to obtain an ellipse. I'm just sayin'.
8/9/17
Mark, while your at it, why don't you bisect your cone and cylinder where the plane comes out the other side rather than the bottom end. Then maybe you can duplicate the images and spin one end for end to overlay the other if you want to prove or disprove this. 8/9/17
Keith Mathewson and Mark B get to share the cigar. A slice thru a cone will not, as I visualize it, produce a balanced ellipse. A slice thru a cylinder at other than 90 degrees will produce an ellipse. As I run this thru in my head, I take things to the extreme to see if that clears it up. So I think of a short, wide, pointed cone 6" tall at the point and 6" wide at the base. Starting my cut 1" down from the point of the cone and exiting 1" above the base of the cone, the resulting shape will not be an ellipse, but two parabolas. Bilateral symmetry. Not Quadrilateral symmetry like an ellipse. Mark B's handy dandy drawing shows a conical slice as a parabola. The cylindrical slice is a half ellipse. So, am I correct that "An ellipse is formed by cutting through a cone at an angle" is incorrect and should read 'Cylinder' instead of cone? Did I find a mistake on the Internet? 8/9/17
Shearing completely through the cone at an angle would only produce an egg shape. That one would have to be mirrored to make it symmetrical. I deleted the first file as soon as I exported the jpeg so poof.
8/9/17
Hmm. Maybe?
8/9/17
Last one heh. I need to bone up on my geeImatree.
8/9/17
I think Keith Newton actually wins the cigar.
8/9/17
I'll get my lathe going on this over the weekend. Enough talk, the real world will tell us. But in the meantime, if you can draw that short fat cone and start a 'cut' up near the point, and run it almost to the base, it can't produce a symmetrical ellipse. Right? Will it still make a decent ellipse if we have a 6" high cone and 30" diameter? Get extreme. I mean, I just can't see it. I do know that of the several ways I can draw an ellipse, they will all vary a bit. Not enough to look odd, but enough that one method is not compatible with another. They will not be concentric. Back to the quote that got me here  "An ellipse is formed by cutting through a cone at an angle" Why not describe it as a cylinder? I know that works. At least I thought it did. My world could be crumbling. Is this that infinite realities stuff I've been reading about? 8/9/17
It does, It just makes the ellipse longer and narrower. So the height of the cone is proportionate to the length of the ellipse. And the base diameter of the cone is proportionate to the width. Running your example of a 30" diameter base and a 6" overall height (looks like one of those oriental hats you see in vietnam movies) yeilds an ellipse that looks like a submarine or missle but nearly pointed at both ends (symmetrical). I dont know if this was one of the sites in the thread, but fiddling around at lunch found this: http://www3.ul.ie/~rynnet/swconics/planes_cutting_coneA.htm 8/9/17
Clarification, If the plane is completely on one side of the cone you are right. It wont be an ellipse. It will be Parabola/Hyperbola/combination (like my first example where the shearing plane is all on one side of the centerline of the cone). The ellipse comes in when you shear the tip of the cone off if that makes sense. 8/9/17
David, I know this is hard to get your mind around, because I've already been there. Over the last 35 years or so, I've been building a series of tables that I call my "Arborescent series", most of which had elliptical glass tops to represent the green of the canopy. Whenever I would call my supplier down in Dallas to order the top, I would ask for an ellipse with the long and short diameter . Almost every time, the salesman would respond, Now you want an Oval? No I would say, and oval can alternately be called an ellipse, but I don't want any straight sides with semicircular ends, like a track around a football field. This same sales jerk for at least 15 years would always come back with an ellipse being a cone sliced through at an angle. I'll bet I've wasted 10 hours of my life over the years trying to get him off that definition that I'm sure he has no capacity to understand. It just seems to me that it would produce an ovoid, which is an egg sliced longitudinally, another shape that I've used for some of my tops when the bases were asymmetrical. Mark, if you can enlarge and print two of those in your last image to flip end for end, to compare, I think you'll find them to be pretty close, if not symmetrical. As I stated before, I had to turn some cones on the lathe to play with, which led to more intrigue. It seemed that if I played with mixing some of the parts where the cone parts were glued back together opposite of the way they were cut, then maybe mitered again, and mixed with some cylindrical parts cut to render the same elliptical plane as the cone, that I could end up with some M.C. Escher mind bending works. http://https://www.flickr.com/photos/7391658@N03/30958169482/in/album72157672763602394/ 8/9/17
David, An ellipse is a locus of points around two points of origin where the combined distance from each point is a constant length. In other words, put two nails in a board and attach a length of string loosely to the nails. Stretch the string taut with a pencil and move your pencil while keeping the string taught and you will have drawn an eclipse. 8/10/17
Okay. I think I'm starting to see that it is an ellipse when a cone is sliced. But I will turn a few things to see how it actually happens. I think it violates the rules of an otherwise orderly universe. Einstein said "God does not play dice with the universe" but I think the cone making an ellipse is an example of probability in the universe. It is probably true. Therefore proving quantum mechanics exists in the woodshop. This will not help my level of insecurity when laying out large or complicated projects. Read and others, I understand and use the two locus method with nails and a string. I also use a mathematical method involving connecting dots laid out on a grid. And have resorted to two radii or three radii for half ellipses when making moldings on the shaper. Thanks for the good dialogue.
8/10/17
Cutting across a cone perpendicular to the axis produces a circle. If the slice is not perpendicular and its slope is less than the cone's the intersection is an ellipse. If the slice's slope equals that of the cone the intersection is a parabola, if greater than the cone's slope a hyperbola is produced. 8/11/17
David Sochar: Einstein also said that things should be as simple as possible and no simpler. While I have no real knowledge of higher math, I can't help but wonder if the conical slicing produces a shape that is elliptical, rather than an ellipse. I offer as an example a previous discussion of constructing a window casing in the shape of an ellipse. It is known that of the lines that define a casing of uniform width, one of them will be an ellipse and one of them will be elliptical, by which I mean "of or like an ellipse", but not a true ellipse, as defined by the geometric definition. Have the conical elliptical shapes been put through the test of seeing if they are in fact ellipses, or merely elliptical? If God throws dice then I hope he rolls a 7 or 11, because Momma needs a new pair of shoes. TonyF 8/11/17
I'm waiting to hear from one of the MIT rocket scientists that quit working for NASA, so he could satisfy his urge to smell the wood dust. They occasionally talk about elliptical orbits, so they should have an educated opinion. 8/11/17
Interesting reading at Dr. Math. A conic (or conic section) is a plane curve that can be obtained by intersecting a cone with a plane that does not go through the vertex of the cone... If no line of the cone is parallel to the plane, the intersection is a closed curve, called an ellipse. 8/11/17
It's not rocket surgery.
8/12/17
I believe that the confusing part is that in a circle, you cut the the cone. It matches the graph of that mathematically. The difference with the ellipse is that when you make your cut, it gives you specific x,y coordinates that you can graph. The graph describes the math. It does not match the actual physical section when you cut the cone. In effect it does not create egg sections. It creates the points required to graph it.
8/12/17
" things should be as simple as possible and no simpler"
And just so you know, I do have great respect for those that can understand and solve complex math. 8/14/17
Isn't an ellipse an isometric circle? 8/14/17
I draw an ellipse by connecting the dots. Last out the half minor and half major axes of the desired ellipse. Dived teach line into so many equal parts. The example has 18 parts on each line. Then connect the dots. 1 to 1, 2 to 2, etc.You will see the elliptical form coming up as you draw. Connect them all and you have an accurate, 1/4 of an ellipse.
8/14/17
They also have ellipses on all CAD programs, Dave 8/15/17
I have used various methods of drawing ellipses on the computer, but at some point, I need to do it full size in the shop. The ellipse above is for a drawing of a window that is 18' wide. When the job comes thru the shop, We will be able to recreate the ellipse accurately and reliably. Actually, there will be several concentric ellipses. I have used this method for 50 years to draw a proper ellipse of any size or scale. It is extremely versatile because it is the same process whether on 81/2 x 11 or 18' wide. 8/15/17
No disrespect, you clearly have great skills Dave, but that is not how I would do it. 8/17/17
A true ellipse may be generated by either an inclined plane bisecting a cylinder or cutting a cone. This includes an obliquely inclined plane. The reason they both yield ellipses in their crosssections, is that the cone is only part of a cylinder. In other words, the long length of the bisected cone, becomes the major axis of the ellipse of the inclined plane, which also bisects the cylinder. Confused?
8/17/17
David, Your spiderwebcurves were common more than 50 years ago (as you've already confirmed.) More commonly; elliptic curves of case and millwork, were just approximated by two radii.
Milling and fabrication of "elliptical" stairs in particular, was much easier from radii, while their handrail facemolds were elliptical. All of this, as well as your latticewebs, are now obsolete (as I'm sure you're aware.) CNC and CAD has made such drawings and patternwork, a dying practice. Today guys make CAD drawings and CNCmachines, make parts. These machines don't care whether they're cutting circles or ellipses and don't need fullsize patterns either. Your observations however, are still good, as we still need to understand what we're asking machines to do. 8/17/17
Cut a cone with an inclined plane. Let the long length of the bisected cone, be the hypotenuse of a right angle. The baselength of this angle, is the plandiameter of the imaginary cylinder, as well as the minoraxis of the crosssection ellipse image. When we slice the cone, we're actually cutting a smaller, imaginary cylinder. Both the cone and cylinder, share circular, plan views (progressively smaller circles in the cone.) Since they both stand upon circular plans with imaginary cylinders, any crosssection (other than perpendicular,) is going to be an ellipse. I know it seems counterintuitive but I believe this is a correct answer. Another question:
8/17/17
Perspective? 8/17/17
Spiral? Nautilus? Vertigo?
8/17/17
Since we're on the topic of Cones and Ellipses I don't know why I didn't think about sharing this project with you gents before now.
Here is a coffee table that I did incorporating both cone and ellipse, from Wenge and glass.
https://www.flickr.com/photos/7391658@N03/albums/72157630034448404 8/17/17
Keith, Nice art piece. I like the custom brass hinges and the ingenious, clamping fixture. Yes David,
Thanks. Your questions got me thinking (ouch!) 8/18/17
Thanks Jim, I had forgotten that sketch was in that file, and had to go take a look to see what you were talking about. I drew that to share with someone on another forum. However left off one critical trick that I used on this piece. Rather than having to figure out what the miter angle would be for however many staves it would take to keep the angles tight, I used a trick that I'd used on building strip plank canoes. I cut a half round on one edge of each, then cut a flute of the same size on the other. This acts like a hinge so there isn't any miter angle to worry about. Since I chose QS Wenge, which was all well matched, and used a thickened epoxy, blackened to blend with the fine grain line, the glue lines just don't show. I was trying to show that detail in the shot looking down on the edge of the bottom door, but it doesn't even show up there. There were 3 stations or forms in my gluing jig, to match the radius at that distance of separation. I was caught a bit off guard when I started clamping the first one, in that the epoxy is very slippery, so when the clamp pressure was applied, due to the wedge shape, they all wanted to slip out sideways, so I had to add an endstop to prevent that. Back then, I only had a little Powermatic 45, which I had beefed up with 1" thick steel endplates on the base, and 3 hp motor for turning hollowforms. However I mounted the cone outboard and then used a gearmotor salvaged from a conveyor system to drive a large pulley screwed to another faceplate inboard. I was able to drop down to about 1 rpm, and used my Makita 6" powerplane on a full length 2x8 toolrest. That part all went GREAT. The tricky part followed when I started cutting the basic parts, then trying to build the piece, so it would all not only matched up after being bisected by the glass top, but the wood could still move seasonally. Funny how trying to make something look so simple could be so hard. I may have been smarter back thirty years ago. ha 8/18/17
Cone containing cylinder at bisection. Bisection of cone is also bisection of cylinder.
8/21/17
This is an interesting discussion in that it brings together the formal definition and the practical methods for drawing an ellipse. The latter is of more consequence for most of us. I have used the pencil and string method, David's projection and interpolation technique and a trammel device with blocks on an arm travelling in a cross pattern. The last place I worked we would have plotted and tiled sections of the curve on paper. Now that I have a cnc router I would cut and tile thin sheet material and be a happy man. David's 6'x18' ellipse could probably be done in 6 sections in less than an hour including layout, toolpaths, cutting and cleanup, with a true curve as the result. Still, the old ways work even with the power off and we use the tools on hand to get it done. I like to see the different approaches shown here, and especially Keith's imaginative conical table. 8/21/17
Keith Was the outside of the cone veneered? You mention the joint that you used on canoes, but I don't see that in the pictures. Just wondering. 8/22/17
Hey Pat, Sorry that photo is of such poor quality that you can't see the joints, but they are there and hard to see. No, although since you ask,,,, it reminds me that I did another one after this one which was veneered with Brazilian Rosewood, which was a poor choice because it is so brittle. I did manage to pull the project off, but was wishing I had never gone that way. Like this cone, I divided it into 3 120º arc sections, but only used two of them, where they formed sort of a football in plan view, but open more on the front side.
Speaking of jigs, I'm still using the set that I made back in the mid eighties when CAD was not so easy to program these kinds of shapes. I could knock out a mdf template quicker than they could get it plotted. I made my jig set with a bunch of plywood scraps all about 10" wide of various lengths. They snap together either end to end, or at right angles, using two little butterfly keys at each joint, lining up the Tslots down the middle of their length, and or across the short way. The little T traveler cars are made of UHMW plastic, with a hole tapped in the middle for a screw to attach the trammel arm. The length of the track only needs to be as long as the DIFFERENCE between the long and short axis, and when plotting or making long narrow ellipses you need to get half of the template out of the way, because the elliptical path crosses under the template. The trammel arm has a hole which matches a router guide bushing, so rather than drawing, and sawing outside the line, then sanding up to it, there is a perfect template each time, without chasing the little humps and dips that are inevitable otherwise. 8/22/17
David Is the wood in this picture solid stock?
8/23/17
Yes, it is solid Wenge, although back then Paxton Lumber called it Panga Panga. Since you ask, I did do another poplar cone which I divided into 3 120º arc sections, but only used two of them to make a sort of football shape in plan view. I think I veneered those two parts with Brazilian Rosewood, which was not a great choice since it is so brittle and easy to split around the tight radius near the pointy end. However I did manage to pull it off. I wish I had some digital images of that piece, but it would probably take me half a day to find the slides. I would highly recommend choosing a solid wood and not dealing with the veneering process, unless you just like to torture,,,,,, I mean challenge yourself. 8/23/17
Yes, solid stock seems to be the consensus. Beautiful work by the way. 8/23/17
Its pretty obvious that if you cut a cone at the tip you wind up with a point, which is not an ellipse. 8/24/17
You also endup with a cone which is no longer a cone but a funnel. In math or geometry, "points" have no dimensional qualities,.. neither width nor length. Points are only specific spots or places of origin or lineintersections. The small end of an opencone or funnel, should either be a circle or an ellipse. I believe this is correct. Mathematicians have been working on this stuff for millenniums but I'm certainly not one of them. . 6/16/18
[(x^2)/(a^2)]+[(y^2)/(b^2)]=1
