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

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.

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.

Dr Math - About Ellipses

It's not rocket surgery.

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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.

" things should be as simple as possible and no simpler"
Sage and simple advice. I never try to define what I layout with a three point trammel or nails and string beyond it's what I know as an ellipse. As to suppliers - I just send a template.
Rather than trying to wrap my head around seemingly simple but very complex geometry, I need to focus on how to get that kid to come to work on time.....

And just so you know, I do have great respect for those that can understand and solve complex math.

Isn't an ellipse an isometric circle?

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.

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They also have ellipses on all CAD programs, Dave

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 8-1/2 x 11 or 18' wide.

No disrespect, you clearly have great skills Dave, but that is not how I would do it.

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 cross-sections, 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?

David,

Your spiderweb-curves were common more than 50 years ago (as you've already confirmed.) More commonly; elliptic curves of case and mill-work, were just approximated by two radii.
Conjoined parts from two radii, in most cases, remain nearly indistinguishable from the true ellipse.

Milling and fabrication of "elliptical" stairs in particular, was much easier from radii, while their handrail face-molds were elliptical.

All of this, as well as your lattice-webs, are now obsolete (as I'm sure you're aware.) CNC and CAD has made such drawings and pattern-work, a dying practice.

Today guys make CAD drawings and CNC-machines, make parts. These machines don't care whether they're cutting circles or ellipses and don't need full-size patterns either.

Your observations however, are still good, as we still need to understand what we're asking machines to do.

Cut a cone with an inclined plane. Let the long length of the bisected cone, be the hypotenuse of a right angle.

The base-length of this angle, is the plan-diameter of the imaginary cylinder, as well as the minor-axis of the cross-section 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 cross-section (other than perpendicular,) is going to be an ellipse.

I know it seems counter-intuitive but I believe this is a correct answer.

Another question:
What is the plan-view or projected image of a regularly ascending, spiral line, drawn upon a cone?

Perspective?

Spiral?

Nautilus?

Vertigo?

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.
When I get tired of working with, or trying to be creative with flat stock my creative side likes to challenge the craftsman side.

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

Keith,

Nice art piece. I like the custom brass hinges and the ingenious, clamping fixture.

Yes David,
The word I'm looking for is "volute."

Thanks. Your questions got me thinking (ouch!)

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 end-stop 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 hollow-forms. However I mounted the cone outboard and then used a gear-motor 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" power-plane 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

Cone containing cylinder at bisection.

Bisection of cone is also bisection of cylinder.

Click the link below to download the file included with this post.

biisection_of_cone.pdf

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.

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.

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.
Rather than glass, it had a wooden boat shape table surface, with 3 Honduras Rosewood drawers under it, and a door above. Sorry I don't have any images of it in my digital files. That was long before I got my first digital camera in ~2000, and would be lost in hundreds of boxes of unorganized slides.

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 T-slots 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.

David

Is the wood in this picture solid stock?

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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.

Yes, solid stock seems to be the consensus.

Beautiful work by the way.

Its pretty obvious that if you cut a cone at the tip you wind up with a point, which is not an ellipse.

You also end-up 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 line-intersections.

The small end of an open-cone 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. .