Segment of an Ellipse
Date: 09/06/2001 at 03:37:34 From: Martin Braaksma Subject: Segment ellipse Dear Dr. Math, I am a Dutch technical aid worker (volunteer) in Africa looking for the formulas to calculate the segment of an ellipse. The reason is that we often use horizontal oval tanks for storing drinking water and fuel, and we would like to be able to calculate the contents. I found your formulas for calculating the segment of a circle, so that solved the same question with regard to round tanks. I got stuck with calculating the segment of an ellipse, though, because in this case I only seem to be able to know d and not c/2 or the hypotenuse, and I need to know 2 sides of the triangle in order to calculate theta, don't I? In the case of a circle, the hypotenuse is known as well because it equals r, so there I did not have a problem. Could you help us out? Thank you in advance, Martin Braaksma Doctors without Borders
Date: 09/06/2001 at 13:23:37 From: Doctor Peterson Subject: Re: segment ellipse Hi, Martin. I'm not sure what dimensions you know. Assuming you are using the diagram in our FAQ: http://mathforum.org/dr.math/faq/formulas/faq.circle.html#segment I would expect h and c (depth and width) to be the easiest numbers to get, and d one of the hardest. In any case, the ellipse is somewhat different. You will need three numbers to determine the exact shape, since it takes two numbers to define the ellipse itself. I will use these pictures, showing your ellipse, together with a circle obtained by compressing the ellipse: ************* ***** ***** | ***** *** | *** ** |b ** * |b * * | a * * | b * * +--------------* * +------* * |d c'/2 * * |d c/2 * **-----------+-----------** *-----+-----* ***** |h ***** *** |h *** ************* ***** The area of the segment of the ellipse will be a/b times the area of the corresponding segment of the circle, since the ellipse is stretched by a factor of a/b horizontally. The hard part is to determine the value of b, which will be used in place of the radius r in the formula. I have written c' for the chord of the ellipse, to distinguish it from c in the circle; c'/c = a/b. If you use an angle in the formula, it will be the angle in the circle, not in the ellipse. If you can tell me what you can measure (and, in fact, how you know the shape is an ellipse), I can help work out the details. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 09/06/2001 at 19:26:09 From: Martin Braaksma Subject: Re: Segment ellipse Hi, Doctor Peterson, First of all, thank you for your quick response. Since this is about a real-life case, I guess that we would only be able to measure the height, width, and length of the tank and the height of the fluid. We would measure the height of the fluid with a measuring stick from the top of the tank, probably using a cm-scale, or through a transparent parallel hose. So d, for example, could be derived. An oval tank (ellipse) could have any shape, I guess, as long as b < a. I hope this information is sufficient for you to provide us with some more calculation details. Looking forward to your response again. Thank you, Martin Braaksma
Date: 09/06/2001 at 22:06:06 From: Doctor Peterson Subject: Re: Segment ellipse Hi, Martin. I just realized that I was picturing a slightly harder problem, in which you only have the segment of the ellipse, so you have to deduce the axes from the height, width, and something else about what you have. Since you presumably have the whole ellipse, or at least the bottom half of it, it's not so hard. Referring to my diagram, and my earlier comments, I'll use the following formula from the FAQ for a segment of a circle: K[circle] = r^2 arccos(d/r) - d sqrt(r^2-d^2) Applying this to the circle I drew, r is actually b, the vertical semiaxis of the ellipse. The area of the segment of the ellipse, as I pointed out, is a/b times this area, or K[ellipse] = (a/b)[b^2 arccos(d/b) - d sqrt(b^2-d^2)] = ab arccos(d/b) - ad sqrt(1-(d/b)^2) If you know the semiaxes a and b and the depth, you just need to find d = b - h and plug that, together with a and b, into this formula to find the area, then multiply by the length of the tank to get its volume. Note also that none of this really depends on b being less than a. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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