Drawing or Constructing an Ellipse or Oval
Date: 02/22/2006 at 10:33:09 From: Ken Subject: How can I lay out a perfect oval? As a woodworker I have an occasional need to lay out ovals for some projects. Many years ago a friend gave me a formula for laying out an oval and I can't remember it. I know it involved setting two pins at a calculated distance from the long point and tying each end of a piece of string cut to a pre-determined length. Then a pencil was placed inside the string and the string was pulled taught. Then the pencil was run around and created an oval. How do I calculate the distances to set the points and the length to cut the string to create different sized ovals? As I recall the method was fairly simple which was perfect for me as a layman.
Date: 02/22/2006 at 11:31:35 From: Doctor Rick Subject: Re: How can I lay out a perfect oval? Hi, Ken. Mathematically, the shape to which you refer is an ellipse. The string-and-tack method is well known; here is a rather brief explanation that I found in the Dr. Math Archive by searching for the words "ellipse string": Tracing an Ellipse http://mathforum.org/library/drmath/view/54949.html Let me fill in a few details. You will see there a formula d = sqrt(a^2 - b^2) Here, a and b are the major and minor semi-axes of the ellipse--that is, the distances from the center of the ellipse to the farthest and closest points on the ellipse--and d is the distance from the center of the ellipse to where you place one of the tacks. (The other tack is the same distance in the other direction from the center.) Then the instructions say to make the string the length of the triangle shown in the figure. This will also be twice the distance from the point farthest from the center to the more distant tack (focus). This length is easy to calculate: it is twice the sum of a and d. Length of string = 2(a + d) Here's an example. If the ellipse is to be 10" long and 8" wide, the semi-major axis (a) is half of 10", that is, 5", while the semi- minor axis (b) is half of 8", that is, 4". We calculate the distance from the center to a tack by d = sqrt(a^2 - b^2) = sqrt(5*5 - 4*4) = sqrt(25 - 16) = sqrt(9) = 3" Thus you start from where you want the center of the ellipse, draw a line through it along the longer axis of the ellipse, and place the tacks 3 inches on either side of the center point. Then make a loop of string with length L = 2(a + d) = 2(5" + 3") = 16" Put the loop around the two tacks, put a pencil in the loop and pull it taut, and draw your ellipse! Here is a different method of laying out an ellipse, that you may find more accurate: Accurate Drawing of an Ellipse http://mathforum.org/library/drmath/view/55085.html - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 02/22/2006 at 16:28:32 From: Ken Subject: Thank you (How can I lay out a perfect oval?) Dr. Rick, Thank you for the timely answer to my question. It was very helpful and it worked wonderfully. Thanks again.
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