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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.
Associated Topics:
High School Conic Sections/Circles
High School Constructions

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