(No, this isn’t for homework. If it were homework, I would have a textbook and maybe could figure it out myself.)
Our house is old. So old it has (decaying) ornamental plaster work on the ceiling of major rooms. We’re going to fix that up now, by making molds of the subunits and casting new pieces. It looks like a lot of fiddly work, but not really more challenging than ornamentally icing a cake. Sorta.
Anyway, in the dining room there are many gaps in the basic oval, and one missing section is at least a quarter of the oval. A previous owner seems to have chipped off/sanded down that area and then painted the ceiling flat white.
I know we can redraw the oval by putting nails in the foci and using a loop of string of the proper size, but…well, it’s been a looong time since geometry.
The longest ‘diameter’ of the oval is 14’ 8.5" and the ‘shortest’ one (at least, going side to side perpendicularly at the 7’ 4.25" mark) is 9’ 2"
Is that enough data to calculate where the foci should be, and how long a loop of string I need? If not, what else should I measure?
If your “oval” is an ellipse (as your statement about the string suggests) then you can find information on the geometry of ellipses in a lot of places on the internet. I just bpoint out that not all ovals ARE ellipses – there are other forms.
If it is an ellipse, then you should know that there are a lot of ellipses, with foci ranging from right on top of each other (that’as a circle, but technically a limiting case of an elipse) to really far apart.
If your ellipse has its shortest dimension along the y direction – call this distance 2 * b – and the longest dimension along the x-direction – call its measurement 2 * a, then the equation is ((x/a)^2 + (y/b)^2) = 1.
The foci are each located a distance ((a^2) - (b^2))^0.5 from the center (so they’re twice that distance apart.
And your string should be 2*a in length, and looped around a nail or pin stuck in each focus. In other words, make your string equal in length to the long dimension of your ellipse.
In other words, you need a 14’ 8.5" long string, tied into a loop (allow for the knot). Each focus ought to be 69.015’ (5’ 9.015") from the center, or 10’ 18.03" apart from each other
Wonderful! I knew there must be an equas ion, I might even have known it once, for a few hours before a test.
So I bisect the ceiling both ways, then measure out each way from the center point 5’ 9" along the long axis and that’s where the foci should be.
It occurs to me that once I get nails into those places I can find the correct length of the string simply by looping it around the nails and my pencil and make the loop taut when it’s touching the remaining plasterwork at any or, maybe better, several points?
Thanks for your help!
(Now I just have to round up a couple of helpers to keep the string from falling down off the nails.)
You sure about that, Cal? I thought the length should be 2a + 2(focal distance from center) – in other words, it should reach from the edge of the ellipse, along the major diameter to the farthest focus, and back.
Ah, you’re right – I was thinking of the sum of the distances from each focus, which has to be 2 * a = major diameter. (That’s easy to see if you go from one focus across the center to the opposite end along the line between the foci, then back to the other focus. One leg is longer than a by the same distance the other is sdhorter than a )
You have to add the distance from one focus to the other, which adds 2 * (a^2 - b^2)^0.5
This makes the string length – lemme see – 26’ 2.53", if I did that right. Longer than I thought – that distabnnce between the foci adds up.
Ah; I see. The string length would be 2*a if you tied each end to the nails rather than looped it. That might be an easier method for StarvingButStrong: no helpers required.
StarvingButStrong, it mihgt be safer to copy one of the other four quarters of the existing. If you can find the center line of the oval, you could measure the distance to the remaing part of the oval from the center line, at several points, and mirror those measurements across the centerline. At a minimum, you ought to do this at a few points, and make sure the curve you get using the string is passing through those points.
