My wife wanted me to make a new table top for a small table that lived on our rear patio. This was originally an indoor, café-type table with a flake-board top. This of course soon disintegrated in the outdoors. The table was also a little small. Although it came with four chairs, the round table-top meant that four people sitting there to eat did not have a lot of elbow room. I suggested an elliptical table. This would provide a little more elbow room without monopolizing the small patio. The only problem was that I couldn't remember how to draw the ellipse. I knew it had something to do with a piece of string attached to a couple of screws, but I couldn't remember the details. A search of the library proved fruitless so I decided to experiment.
I laid out a piece of plywood and drove in a couple of dry-wall screws. I tied a piece of string between them with some slack in the middle. Then I took a pencil, looped it into the string and traveled it around. Worked perfectly. I found that the distance between the two screws and the amount of slack in the string directly translates into the shape and size of the ellipse, Now, I'm sure that there's a formula to determine the size and shape of the ellipse mathematically, but I don't have a clue what it is! If you need to draw one of these, just do as I did; I changed the position of the screws and the length of the string a couple of times until the pencil line came out to what I wanted. One thing I did find out, the string and its attached pencil MUST pass BEHIND the screws otherwise the ellipse will not be true and you will have a difficult time tracing a fair line. Also, you must remember to take the pencil out of the loop and make a new loop each time you pass one of the screws. Otherwise, passing the screw puts another turn into the loop, thus shortening the string a little and making the line fail to meet the start properly.
In the drawing, it is fairly evident how this works. You can see that moving the screws farther apart would make the ellipse skinnier, while making the string longer would make it fatter and more circular. Both moving the screws and lengthening and shortening the string have these effects. Just fool around until you get it the way you want it!
It is probably possible to use a router with this providing you can devise a collar of some kind that can ride in a snug but loose fit around a long-shaft straight bit, thereby eliminating the need to use a saber saw and the potential humps and bumps that this usually contributes.
The picture shows the elliptical table-top that I made for the patio.