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Q: equation to determine foci position to draw ovals ( No Answer,   7 Comments ) Question
 Subject: equation to determine foci position to draw ovals Category: Science > Math Asked by: mxnmatch-ga List Price: \$5.00 Posted: 23 Oct 2006 16:02 PDT Expires: 22 Nov 2006 15:02 PST Question ID: 776205
 ```I often need to draw large ovals on fabric. I always use the method of attaching a string to two points and then swinging the marker around to make the oval. To make the oval wider I move the foci down from the center line. To make the oval taller I move the foci closer together. However, I just experiment until I get the oval I want. I'd like to just use equations to determine, given a desired width and height for the oval, the horizontal distance between the foci and the vertical distance from the center line.``` Clarification of Question by mxnmatch-ga on 25 Oct 2006 22:28 PDT ```Ok, I guess I had two problems then. That information is exactly what I needed to do ellipses. But, when I say move the foci down from the center line I'm talking about drawing a certain type of curve. In particular, I recently needed to create what you might call "orange slices" which, when sewn together, make a sphere. Those slices aren't elapses. I draw them using the same method except I move the foci off the axis. For instance, let's say that I want to create a 1 foot diameter sphere. By experimentation I found that if the axis goes from (-1.5,0) to (1.5,0) (that's 3 feet long) then I can draw one side of the slice by attaching a string at (-1.25,-1) then running it to (-1.5,0) then to the other foci at (1.25,-1). I can then draw the curve starting at (-1.5,0) then going around to (1.5,0) just like I would with an ellipse. To draw the other side I just fold the cloth along the y axis at 0 and cut it along the line I just drew. Take six of those, sew them together, and I get a sphere with a 1 foot diameter. I'd like to know what the equation is that would tell me, given a desired sphere diameter, what the locations of the foci and the length of the axis are. Even better would be if I could choose how many slices make up the sphere. If I could find out how to draw the slices to make an egg shape then that would be fantastic. But, that's probably getting outside the initial scope of the question.``` There is no answer at this time. Subject: Re: equation to determine foci position to draw ovals From: badgerw-ga on 24 Oct 2006 01:47 PDT
 ```I am not quite sure what you mean by "mov[ing] the foci down from the center line." Moving the foci up or down from the center line will just move the ellipse up or down. BTW an "oval" can mean lots of shapes; the shape we're talking about here is more specifically an ellipse. The "center line" (by which I mean the line drawn between the two foci) can be either vertical or horizontal, and this will give different shapes for your ellipse. If the foci are in line horizontally, the ellipse will be wide; if the foci are in line vertically, the ellipse will be tall. If the foci are coincident (i.e. they are in the same place) then you will draw a perfect circle, i.e. an ellipse that is neither tall nor wide. This discussion will assume a horizontal center line, i.e. the two foci are horizontally in line with each other. This will give results that you can use to create wide ellipses, and if you need a tall ellipse then you can just turn the fabric sideways (which is the same as switching "width" for "height" in the following discussion). Let's say you want an ellipse that will be T tall and W wide. We need to figure out L, the length of the string, and D, the distance between the foci. The length of the string is simply the same as the desired width of the ellipse. L=W. Easy enough. The distance between the foci is more complicated: D = 2*SQRT((L/2)^2+(H/2)^2) where (just to be clear) * is multiply, SQRT is square root, / is divide, and ^2 is square. So take the length, divide it by two, and square it. Call this "A". Do the same for the height, and call it "B". Add A+B. Take the square root of this sum, and then multiply that by two. This is the distance between the two foci. Again remember that this whole discussion is based on the foci being horizontally next to one another.```
 Subject: Re: equation to determine foci position to draw ovals From: myoarin-ga on 24 Oct 2006 05:07 PDT
 ```Perhaps that is correct, but the length of the string is: 1) double the distance from one focus to the most distant point on the ellipse beyond (in line with) the other focus; 2) the distance between the foci plus the distance from each focus to the point of the ellipse's maximum width, which is the point that a perpendicular from the midpoint of the line between the foci would intersect the ellipse. Since I can't solve that this morning, here is the formula: http://www.analyzemath.com/EllipseProblems/EllipseProblems.html "The length of the major axis is 2a, and the length of the minor axis is 2b. The two foci (foci is the plural of focus) are at (� c , 0) or at (0 , � c), where c^2 = a^2 - b^2." Just plug in your a and b amounts and solve for c, which will be the distance of each focus from the center point of your ellipse, that is the distance between the foci will be 2c.```
 Subject: Re: equation to determine foci position to draw ovals From: barneca-ga on 25 Oct 2006 09:21 PDT
 ```myoarin is correct. badgerw is close to correct, with different terminology, except for a sign error and inconsistent notation. using badgerw's terminology, L=W is correct. badgerw uses both H and T for the height of the ellipse; let's use H. the plus sign in badgerw's equation for D should actually be a minus sign. also, it can be simplified down to: D = sqrt (W^2 - T^2) you can check this by using a circle as a special case. W=H=DIAM, L=DIAM, and D=0. -cab```
 Subject: Re: equation to determine foci position to draw ovals From: barneca-ga on 25 Oct 2006 09:35 PDT
 ```um... the tiny little insignificant error in my post above was intentional, just to see if anyone was paying attention. yeah, that's it, intentional. also, badgerw confused me with his T vs. H thing. so really, if you think about it, my error is badgerw's fault. because otherwise, the slightly superior tone of my comment would have made me look like an ass. anyway, to be clear, the "T" in my equation should be "H". now, where is that "Edit Comment Before Anyone Sees It" button? -cab (temporarily humble)```
 Subject: Re: equation to determine foci position to draw ovals From: myoarin-ga on 25 Oct 2006 11:00 PDT
 `With unusual modesty, let me say that I just found a good website. :-) Myo`
 Subject: Re: equation to determine foci position to draw ovals From: badgerw-ga on 27 Oct 2006 10:55 PDT
 ```Oops! Yeah. I accidentally switched the sign (should be a -, not a +) and used H instead of T. As for the clarified question: It took me a while to figure out what you meant, but now I get it. You are talking about (basically) the panels on a beach ball. Each of these panels is roughly the shape of a flattened football, with two pointy tips and two uniform curves. So your question is you're making a beach ball with 6 panels and X diameter, how do you cut the panels? That's quite a bit harder, and I doubt I can answer that without some kind of software. Ok, now we're going to say that R is the radius of the desired sphere (X/2). I can tell you that the long dimension of the panels (from pointy tip to pointy tip) will be pi*R in length (you came close by saying that your 3-foot shape would make a 1-foot diameter beach ball) and the short dimension of the panels (the widest part of the curves) would be one-third of that, or pi/3*R. To make a sphere with a different number of panels, change the denominator. 6 panels = 3; 8 panels = 4; 10 panels = 5; etc. Of course, this is ignoring the extra that you will need to leave in order to sew the panels together!```
 Subject: Re: equation to determine foci position to draw ovals From: myoarin-ga on 28 Oct 2006 05:11 PDT
 ```HI, I am sure there is a mathematical formula for your "orange slices", but if someone can't find it, you could make a pattern from a "beach ball", either finding one with those panels and make several measurements of the "latitudes" (you would just need several between the "equator" and the pole, since the other half would be identical); OR find a large globe, and work from that. Once you have the dimensions and your pattern, you can draft it to the necessary scale by multiplying the dimensions. Cheers, Myoarin``` 