# A Carpenter Draws an Ellipse

 Drawing "jig".

Objective: To demonstrate how a carpenter can draw an ellipse on wood or a sheet of wall board using simple tools.

Level: A geometry class, an introductory modeling class, or possibly a general mathematics class for industrial arts.

Prerequisites: Familiarity with basic geometry that discusses the shape of an ellipse. A detailed understanding of the mathematical equation of an ellipse is not necessary. However, portions of this demo can be used with students having various levels of mathematical background.

Platform: None. However, a "jig" can be used to demonstrate the technique and there are software animations to illustrate the use of the "jig".

Instructor's Notes:

Background: This demo arose from a class discussion of ways to draw an ellipse on a graphics screen using computer software. During the discussion one of the students, Sean Comfort, who is a professional carpenter, briefly described the method used by carpenters to draw an ellipse (really half of an ellipse) for an archway or as a decorative top for a doorway. His description added another dimension to the discussion since it was mechanically based rather than formula based. Sean then brought in a carpenters and builders reference (see [1]) which illustrated the technique. He then went on to tell us that carpenters often used a "jig" to help make the outline of the ellipse directly on the material or to construct a pattern. (A jig is a device for guiding a tool to aid drawing or scoring on material or for cutting material.) Sean then volunteered to make a jig to demonstrate the technique. The pictures of Sean's jig are included with this demo and can be used to clearly show the way a carpenter draws an ellipse.

Discussion: The notion of an ellipse can be introduced in a variety of ways. The following animation shows ellipses which change as we vary values of a and b using a pair of sliders. Varying a changes the horizontal extent, while varying b changes the vertical extent of the figure. Near the end of the animation we alternately vary a and b.

You can download this animation as both a gif and a QuickTime file and the Excel program used to generate it by clicking here. We have captured only a portion of the Excel spreadsheet's primary page for the animation so that it can be used at a very elementary level. To see the primary page click here.

In a  geometry class it may be appropriate to use a locus definition of an ellipse.

Definition:  An ellipse is the set of points P whose distances from two fixed points F1 and F2 always add together to give the same number. (See Figure 1.)

Figure 1.

For further details and an accompanying animation see the demo Constructing the Conic Sections on a Whiteboard.

In a Precalculus or Calculus class the algebraic approach using the equation

can be used. Depending upon the level of the class, the parametric representation using sines and cosines in the form

may also be incorporated. If this is the case then the animation and the Excel routine mentioned above will provide a very nice visual demonstration to tie together the standard Cartesian equation and the parametric representation. To investigate the underlying geometry of the parametric representation above we note that for fixed values of a and b the ellipse is traced by the vertex V of a right triangle with legs a cos(t) and b sin(t) as the angle t varies. See Figure 2.

Figure 2.

A Carpenter's Approach: Make your measurements to determine the lengths of the major and minor axes of the ellipse that you want to draw on your material. (For purposes of discussion here assume that the major axis is horizontal while the minor axis is vertical.) On your material (lightly) draw a coordinate system with each axis longer than the lengths of the major and minor axes. Now take a straight edge and mark on it a length one half the length of the major axis. Denote the top point P and the bottom point R. Next starting at point P mark off a length one half the length of the minor axis and call the point Q. See Figure 3. (Put masking tape on portions of the straight edge to easily mark the straight edge.)

Figure 3.

Position the straight edge on the coordinate axes drawn on the material so that R is on the minor axis, Q is on the major axis, and then point P will be on the desired ellipse. See Figure 4. By shifting the straight edge so that R moves along the minor axis and Q moves along the major axis we can mark points along the graph of the ellipse by recording the position of point P.

Figure 4.

As we move the straight edge keeping R on the vertical axis and Q on the horizontal axis and marking points P we trace the ellipse as shown in Figure 5.

Figure 5.

To see an animation of the generation of an ellipse using this technique click here.

Sean's Jig: To provide a hands-on mechanism for drawing the carpenter's ellipse the straight edge was designed as shown in Figure 6.

Figure 6.

The two cross pieces can be adjusted to set the lengths from points P to R and P to Q as illustrated in Figure 3. Figure 7 shows the bottom of the straight edge and a scale to set these lengths. The metal pieces can be loosened to slide so that adjustments can be made.

Figure 7.

Figure 8 shows a drawing board and rails for keeping points Q and R on the horizontal and vertical axes respectively.

Figure 8.

In Figure 9 straight edge is on the drawing board. To use the jig, place one hand on the straight edge at the horizontal axis position and the other hand on the straight edge at the vertical position. Move your hand  along the vertical rail while the other hand keeps the straight edge firmly against the horizontal rail. This action lets the pencil trace an ellipse. To see an animation of the generation of an ellipse using this technique click here.

Figure 9.

Figure 10 shows an elliptical construction which required the carpenter (and builder) to develop an elliptical pattern.

Figure 10.

For examples of archways and other windows click on thumbnail photos to see a good view. (Photos by Sean Comfort.)

Mathematical Connections: Using the carpenter's method provides us with a way to mechanically construct an ellipse that does not require a formula or the location of the foci of the ellipse. The fixed points F1 and F2 in Figure 1 are the foci of the ellipse. With a fixed length of string connecting F1, P and F2, by placing a pencil at P and keeping the string taut an ellipse is traced as we move the pencil. To see an animation of this procedure click here.

The carpenter's method is closely related to the parametric equations

which are often used to generate an ellipse in computer graphics. In fact, we can characterize the movement of the straight edge parametrically in terms of the changes of an angle. The development of this characterization requires only elementary geometry and trigonometry. To see this development click here. This would be an interesting applied assignment in a geometry class, a modeling class, or even a programming class, since it was this development that was used to write code for the animation which is illustrated in Figure 5. To see an animation of the generation of an ellipse using this technique click here. (See the auxiliary resources below.)

Auxiliary resources:

1. In [2] there is a discussion of nine ways to derive an ellipse. The techniques include "cutting" a cone, the standard algebraic equations, free orbital motion, several mechanical methods, and other approaches. The technique discussed in this demo is also mentioned and is called the trammel method. See the following sites:

http://www.tpub.com/content/draftsman/14276/css/14276_115.htm

and

http://mathforum.org/mathed/mtbib/conic.sections.html

which is a Geometry Bibliography: Conic Sections, from Mathematics Teacher.

2. To see an animation of the conic sections as a plane is being rotated through a double cone go to
http://www.math.odu.edu/cbii/calcanim/index.html
The animation includes the three-dimensional image of the cone with the plane, as well as the corresponding two-dimensional image of the plane itself. This excellent demo was done in Mathcad. The authors granted us permission to use their original file as the basis for an animation which you can view by clicking here. To download this animation in both gif and mov format click here.

3. An Excel routine that simulates the action of the jig for the carpenter's method can be downloaded by clicking here. The values for the half lengths of the major and minor axes can be set using a slider. By dragging a slider that changes the angle between the positive x-axis and the segment connecting the points on the y-axis, the x-axis, and the ellipse the ellipse is traced. Click here to see the Excel setup for the simulation.

4. A MATLAB routine that simulates the action of jig for the carpenter's method can be downloaded by clicking here. To see an animation, with the values of the angle between the straight edge and horizontal axis displayed click here. To download the animation referred to in the preceding sentence in both gif and QuickTime formats click here.

5. For professional tools for cutting ellipses see

www.microfence.com/PDFs/Ellipse%20Jig%20Ints.pdf

or

http://www.trendmachinery.co.uk/ellipsejigs/

A search engine will find lots of other products that are available.

References:

[1] John E. Ball, Carpenters and Builders Library No. 2,     Fourth Edition, Howard W. Sams & Co., Inc., Indianapolis, Ind, 1978.

[2] J.W. Downs, Practical Conic Sections, The Geometric Properties of Ellipses, Parabolas, and Hyperbolas, Dover Publications, Inc., Mineola, N.Y., 2003.

Credits:  This demo was submitted by Sean Comfort, a student at Temple University, and
David R. Hill ,Department of Mathematics,Temple University and is included in Demos with Positive Impact with their permission.

Last updated 3/22/2004 DRH