Objective: To
demonstrate how a carpenter can draw an ellipse on wood or a sheet of wall
board using simple tools.
Level: This
demo can be used as a hands on activity for anyone with a basic knowledge of
geometry and/or shapes that had some dexterity--4th/5th graders on up to
junior high. (See, Using this
Demo with Young Students.) It also would be appropriate for elementary
education and middle grades education majors in their geometry courses,
in an introductory modeling class, or possibly a general mathematics class
for industrial arts.
Prerequisites:
Familiarity with basic geometry and shapes. For students beyond junior
high some background relating to the shape of an ellipse is appropriate. A
detailed understanding of the mathematical equation of an ellipse is not
necessary. 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.)
Using this
Demo with Young Students:
One of our reviewers suggested the following approach. Use
basic office or art supplies to make a jig. On a narrow flat piece of wood or
stiff cardboard measure mark the distances between points P, Q, and R. On a
sheet of paper mark a set of axes. Through the wood or cardboard make a small
hole at point P so that a pencil can inserted. (For young children the hole
can be predrilled.) You could try to anchor a pencil through the hole
with tape or a rubber band or just insert a pencil tip to mark points. If have
anchored the pencil, then with a bit of concentration you can trace an ellipse
using the directions given under Figure 8. If you just are marking points,
then you can later return to connect then with a smooth arc. If this were done
in an art room or where markers or crayons are available, then after they made
their half or full ellipse, let then decorate it. This is a great opportunity
to introduce the concept of "geometric construction" to a young audience
without ever saying the word!
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
http://www.trendmachinery.co.uk/ellipsejigs/
A
search engine will find lots of other products that are available.
