Instructor's
Notes:
This is a good activity to get students
out of their seats and doing something physical and mathematical.
It illustrates that the "locus definitions" of the conic sections can actually
be used to construct the conic sections.
The word locus may not be familiar
to some students. Hence for the prerequisite mentioned above you can use
the following.
The word locus is used
in geometry to mean the set of all points
which satisfy a given geometric condition.
It is often convenient to think of the locus as the path traced out
by a point which moves in such a way as always to satisfy the given
condition.
A particularly simple locus is the set of
all points a fixed distance r > 0 from a fixed point. It has been
helpful to have students construct this locus on paper and provide the
common names associated with the fixed distance and the fixed point. (Of
course this locus is a circle. Click on the thumbnail sketch below to see
the animation.)
Generating a Circle
Although students may have already seen the
construction of an ellipse, they may not have seen how the definitions
may be used to construct a parabola and hyperbola. We can also justify
that the shapes produced by the constructions are actually the conic sections.
This takes some thinking to produce the correct relationships.
If the construction of the ellipse is well
known to the students, then it can be demonstrated at the board with the
help of a student. The instructor can provide the justification that the
points drawn by the pen actually satisfy the locus definition of the ellipse.
The construction and justification of the
parabola are a bit more difficult. Most students have not seen this construction,
and most are a bit surprised that the construction actually creates a parabola.
It is very rare that a student knows how
to construct a hyperbola. A survey of colleagues showed that most of them
didn't know either.
All of the constructions can be done by
pairs of students, but groups of three work better.
Materials:
* string
* tacks for attaching the
string to the wooden bars
* small suction cups (available
at craft shops for window hangings)
make good fixed
points or foci
* a large "T-square" (to slide
along the pen tray) can be easily made
by gluing a wooden
bar ( about 3 feet long) to a 6 inch piece of 4x4
* a wooden bar (3 to 4 feet
long) for the hyperbola construction
* colored dry erase marking
pens
(Note: a blackboard can be used when the
suction cups are replaced by tape and chalk replaces the dry erase markers.)
Demo Directions and Justifications
1.
Constructing an Ellipse
Attach suction cups to the whiteboard at
two points, F1 and F2.
Cut a piece of string and tie the ends
to F1 and F2.
Place the marking pen at the location
P, and, keeping the string taut,
slowly move the pen. The resulting figure
is an ellipse.
(An animation of this construction is available
at the beginning of this document.)
Definition: An ellipse is
the set of points P whose distances from F1 and F2 always add together
to give the same number.
Justification that the construction produces
an ellipse:
Suppose the length of the string
is S. Then the distance from P to F1 plus the distance from P to
F2 is simply the length of the string which is constant. For each
point P,
dist( P, F1 ) + dist( P, F2 ) = length of the string = S
so the collection of the points
P is an ellipse.
2.
Constructing a Parabola
Cut a string slightly longer than H and
attach one end to the whiteboard at F (suction cup) and pin the other end
to the top of the T-bar at A. (The "free"
length of the string should be approximately
equal to H, the height of the T-bar.)
Put the base of the T-bar on the pen tray
of the whiteboard. Place the marking pen at the location P, and, keeping
the string taut, slowly move the T-bar. The pen should stay adjacent to
the wood bar as the T-bar is moved. The resulting figure is a parabola.
(To see an animation of this construction
click on the thumbnail icon below.)
Generating a Parabola
Definition: A parabola
is the set of points P whose distance from a fixed point,
F, equals the distance from a fixed line,
L .
Justification that the construction produces
a parabola:
Suppose the length of the string
is S (and S = H). Then the distance from P to F plus the distance
from P to A is simply the length of the string, S:
dist( P, F ) + dist( P, A ) = S
We also know that the distance
from the line L to P plus the distance from P to A is just the height of
the T-bar, H:
dist( P, L ) + dist( P, A ) = H
But S = H
so dist(
P, F ) + dist( P, A ) = dist( P, L ) + dist( P, A )
and dist( P, F ) = dist( P,
L )
so the collection of the points
P is a parabola.
3.
Constructing a Hyperbola
Cut a string slightly shorter than H and
attach one end to the whiteboard at F2 (suction cup) and pin the other
end to the top of the wood bar at A. (The "free" length of the string
should approximately equal H, the length of the wood bar.) Put the
corner of the wood bar at F1. Place the marking pen at the location P,
and, keeping the string taut, slowly rotate the wood bar around the pivot
point F1. The pen P should stay adjacent to the wood bar as the bar is
rotated. The resulting figure is a hyperbola.
(To see an animation of this construction
click on the thumbnail icon below.)
Generating a Hyperbola
Definition: A hyperbola is the
set of points P whose difference of distances from two fixed points, F1
and F2, is constant .
Justification that the construction produces
a hyperbola:
Suppose the length of the string
is S (and S = H). Then the distance from P to F2 plus the distance
from P to A is simply the length of the string, S:
dist( P, F2 ) + dist( P, A ) = S
We also know that the distance
from the line P to F1 plus the distance from P to A is just the length
of the wood bar, H:
dist( P, F1 ) + dist( P, A ) = H
When we subtract the first distance
equation from the second, we get
H -
S = { dist( P, F1 ) + dist( P, A ) } -
{ dist( P, F2 ) + dist( P, A ) }
= dist( P, F1 ) -
dist( P, F2 )
so dist( P, F1 ) -
dist(
P, F2 ) is constant ( = H -S)
for all points P, and the set of points P is a hyperbola.
Notes:
1. A natural extension to the geometric
constructions is the derivation of the quadratic equations for the conics
from the locus definitions.
2. Each of the animations for constructing
the conic sections, parabla, ellipse and hyperbola, is available in a larger
form by clicking on big-conic. These are suitable
for a classroom monitor or display when using this demo.
Credits:
This demo was submitted by
Dale
Hoffman
Department of Mathematics
Bellevue Community College
and is included in Demos
with Positive Impact with his permission.
