Elliptograph (?)

[dlbristol]

Thanks, all, for the info, sources and links. The definitions alone didn't do it for me.

In the "Learning Something New" Department:

Ellipse: the path of a point that moves so that the sum of its distances from two fixed points (foci) {if you change these the originals become a bunch of old focis} is constant]

Now, translating all this into a physical machine is the fun part.

Good pun there keakap! I wish I had used that in class! Your definitions are accurate and as you point out not real helpful with lay out. I have not done this with a woodworking project, but we did it many times as a manipulative method to study the " conic sections" in math class.
Once you have the rectangle from which to cut the ellipse, you lay out the major or long axis and the minor or short axis by drawing a line that bisects the long and short sides of the rectangle. The foci ( new or old:D ) are on the major axis and @ equal distances from the intersection of the axis lines. As long as the string is long enough to go to the edge of the major axis, you can draw an ellipse. ( it may not fit on the minor axis however) As long as the total distance from each foci to the end of the minor axis is longer than the distance between the foci you can draw an ellipse. ( in may not fit on the major axis)The trick is to move the foci around to get the fit you need and the curve you want. ( a good fit may not yield a pleasing curve) My objective with this was not to get a specific oval, but to give the kids a feel for what was happening and to " see" and "feel" the shape. We were able to get a pretty close fit on rectangular paper sometimes. Generally the closer the foci get to the axis intersection, the closer to round the figure becomes. If you hold the foci and vary the string length, the curve will change as will the axis length. This is why I used this exercise to introduce the concept of multiple variables, any change in any one dimension changes the others.

I did help a friend do a table top long ago, and we started with a big sheet of cardboard and just moved foci and axis measurements until we got a good fit. ( typical math teacher approach) The art teacher at our school, free handed a quarter of the curve and by moving it and flipping it came up with a very pleasing form. My friend used that one!!:rolleyes: There are specific formulas that will produce an ellipse given the axis measurements, but they would still require trial and error to locate the foci for a specific curve. With as many spreadsheets as we see for things like compound miter cuts and the like, I would bet there is some one some where who has a spread sheet to fit up a ellipse. I would also bet that a lot of the tops we see are not true ellipses, but something like my art teacher's free hand curve. ( an Oval, as your definition states. More than one kid tried to use a french curve to draw an ellipse on one of my tests! If you play with the string long enough, you get a feel for how the Eliptograph works.

[keakap]

...The trick is to move the foci around to get the fit you need and the curve you want.This challenge was the first to present when I first saw the elliptograph. ... ...This is why I used this exercise to introduce the concept of multiple variables, any change in any one dimension changes the others. Indeed-- what better way to put sight to feel with mathematical mental gymnastics. ...

It is easy enough to think through the dynamic multiple variables, but a little harder making it all physically live in harmony. I made the "device" in CAD just "imagining" what slider B was doing while slider A slid up its groove, etc., and with the two sliders drawn, with the arm attached, "thought" through an entire ellipse until the sliders didn't hit, then stepped it thru with the program. Fascinating.
What a SHAME that I've come to appreciate the fun aspects of simple math after all this time, and was bored to tears by it when in school.
I ate up every word of your message. Thanks for the reply, and the detail.
['spread sheet to fit up an ellipse'-- I just realized, that .jpg of the CAD drawing was in effect a graphical spread sheet of a half ellipse.]

I believe I will finish that little project, just for the heck of it. Size should be about the same as my mortising gauge (5-1/2x8).

[Btw, did you see how the OLD HOUSE guys did theirs, with daul-ing carpenter squares? Piece o cake.]

[dlbristol]

Glad you appreciated the post. Don't feel to bad about being bored by some of the math class stuff, you got enough from that to be able to "make it yours" later. As a teacher, I'd consider that a success. I have seen a guy use squares to create a curve, but I can't recall how it was done. If I recall correctly, he plotted a lot of points and fitted the rest of the curve. One of the things I know about math is that there are lots of ways to do things and they are often different. Like the string and focus method and the squares. I once had an exchange student from England who showed me an entirely new( to me) method for solving a quadratic equation, it took me a week to figure out why it worked. Carpenters will often tell me they never used the math they had in school and then lay out a set of stairs or roof pitch in 30 seconds using a square!! I am interested in your CAD program, because I would like to know how it works. It may use some sort of " curve fitting" method. Let me know if you figure that out.
Just thought of this!!
The dualing squares method can be approximated with what I call a " string art" method. After laying out both axis, you can divide the sides of your rectangle ( from axis to corner) into an equal number of parts.By connecting the points beginning on the end major axis to the points beginning at the corner of the minor axis side you get an approximate elipse. The more points, the " smoother" the curve. I know this method was used by carpenters to lay out arches in houses. You can have great fun changing the spacing on one or both sides. Again I used this to " illustrate to kids several concepts in calculus. I won't go into that, just the mention of " Delta/ Epsolon " proofs can induce sever psychotic reactions in most of us.

[keakap]

Glad you appreciated the post. Don't feel to bad about being bored by some of the math class stuff, you got enough from that to be able to "make it yours" later. As a teacher, I'd consider that a success.
Good point! I shall hereafter consider my teachers successful (if I didn't already). I have seen a guy use squares to create a curve, but I can't recall how it was done. [below] ...I am interested in your CAD program, because I would like to know how it works. It may use some sort of " curve fitting" method. Let me know if you figure that out.-- Will do!
Just thought of this!!...

The Old House guys (Tommy?) drew a long axis line (H) mid-crossed by a short axis line (V). The squares went long arms along the short axis about 1/8 apart, short arms on top, one going left, one right, top along long axis. (A big "T", split down middle)
Drawing arm: {1/2 x 3/4 x (long axis/2 +1)} nail near bottom, far enough to slide in groove separating squares]e[/color] (half in this case) drawn by pushing pencil (first one quarter then next) while nail 1 slides between squares, nail 2 slide along top square arms. (This is almost easier to envision mentally than to do physically.)

In other words, a simpler construction (and one that would be a lot harder to carry around) of an "elliptograph", the only "data" needed being the length of the axes, and great for people with three hands.

(But I wanna build one of beech and brass, for kicks.)

Yes, indeed, there are often many different ways to do the same thing. And I'm curious now to work on making that spreadsheet, or having the cad prog make one for me {copy the calculator data into Excel}.

But alas for now, back to work-- a "gift event" looms...

[JPG]

The Old House guys (Tommy?) drew a long axis line (H) mid-crossed by a short axis line (V). The squares went long arms along the short axis about 1/8 apart, short arms on top, one going left, one right, top along long axis. (A big "T", split down middle)
Drawing arm: {1/2 x 3/4 x (long axis/2 +1)} nail near bottom, far enough to slide in groove separating squares]e[/color] (half in this case) drawn by pushing pencil (first one quarter then next) while nail 1 slides between squares, nail 2 slide along top square arms. (This is almost easier to envision mentally than to do physically.)

In other words, a simpler construction (and one that would be a lot harder to carry around) of an "elliptograph", the only "data" needed being the length of the axes, and great for people with three hands.

(But I wanna build one of beech and brass, for kicks.)

Yes, indeed, there are often many different ways to do the same thing. And I'm curious now to work on making that spreadsheet, or having the cad prog make one for me {copy the calculator data into Excel}.

But alas for now, back to work-- a "gift event" looms...

Stick/nails/pencil - I be confused! Does the following make sense???

Stick length = (long axis)/2 + 1"

Nail 1 1/2" from one end of stick.

Pencil 1/2" from opposite end. Thus nail 1 to pencil spacing = one half long axis.

Place stick with nail 1 in groove formed by the long legs of the squares. Place pencil (near end of stick) at end of short axis above the squares short legs. Locate nail 2 at the intersect of the axis.

The stick now has three points located as follows: Nail 1 near one end of the stick. Pencil spaced from nail 1 half length of long axis. Nail 2 spaced half length of short axis from pencil.

Start the 'drawing' by placing nail 1 in groove(squares long legs), nail 2 at corners of squares(and intersection of the axis). This places pencil at end of short axis. Slide the nails along groove and short legs of squares until pencil arrives at end of long axis.