Name:_____________
Folding Conics
Gather wax paper, a toolkit and a writing utensil.
Figure 1:
• Draw a line with a straightedge.
• Draw a point not on the line. Label it F.
• Fold the paper so the point is on the line and make a crease. Unfold.
• Fold the paper so the point is on a different part of the line and make a crease. Unfold.
• Repeat many, many times until something interesting emerges.
• Trace over the interesting result so you don’t lose it when your paper is flattened.
1. What happened?
2. Label three points, A, B and C, on the curve, then measure the distance to point F and the
distance to the line. (The distance from a point to a line is the distance to the nearest point on
the line.)
Point Distance to F
A
B
C
Distance to Line
What do you notice?
3. If you moved Point F closer to the line and folded another curve, describe how you think the
curve’s shape would change. What if you moved Point F further from the line?
4. How does this paper folding construction work? (Why did following these instructions result
in the rule you found in 2?)
Name:_____________
Folding Conics
Figure 2:
• Draw a circle (nearly as large as the paper will allow). Label the center X.
• Draw a point, other than the center, in the interior of the circle. Label it Y.
• Fold the paper so the point Y is on the circle and make a crease. Unfold.
• Fold the paper so the point Y is on a different part of the circle and make a crease.
• Repeat many, many times until something interesting emerges.
• Trace over the interesting result so you don’t lose it when your paper is flattened.
5. What shape did you create?
6. Measure the radius of your circle.
r=
7. Label three points on your curve, F, G and H, then measure the distance to the center of
the circle (X) and the distance to point Y. Record your results.
Point Distance to X
F
G
H
Distance to Y
What do you notice?
8. Find the center of the shape, label it Z.
Fold creases on the major and minor axes, label the endpoints N, E, S, W (think N for North).
Measure:
Horizontal radius
Vertical radius
ZY
NY
9. How could you find ZY given the equation of this ellipse, but without drawing the picture?
10. How does this paper folding construction work? Why did following these instructions result
in the patterns you noticed?
Name:_____________
Folding Conics
Figure 3:
• Draw a circle (not too large, toward the middle of the paper). Label the center C.
• Draw a point in the exterior of the circle. Label it F.
• Fold the paper so the point is on the circle and make a crease. Unfold.
• Fold the paper so the point is on a different part of the circle and make a crease.
• Repeat many, many times until something interesting emerges.
• Trace over the interesting result so you don’t lose it when your paper is flattened.
9. What happened?
10. Label three points on your curves, A, B and C, then measure the distance to the center of
the circle and the distance to point F. Record your results.
Point Distance to F
A
B
C
Distance to C
What do you notice?
11. If you moved Point F closer to the circle and folded another curve, describe how you think
the curve’s shape would change. What if you moved Point F further from the circle?
12. How does this paper folding construction work? (Why did following these instructions
result in the rule you found in 10?)
Name:_____________
Folding Conics
Why is this paper title Folding Conics?
Take a guess, write it down.
Discuss with your neighbor, write something down.
Discuss with the teacher, decide if you need to write anything else down.
Name:_____________
Folding Conics
Figure 1:
A parabola is a set of all points equidistant from a fixed line, called the directrix, and a fixed point not
on the line, called a focus.
When we fold the sheet so that one point lies directly over another, the crease is along the line
equidistant from the two points (the perpendicular bisector of the segment joining the two points).
The crease is a tangent line to the parabola.
A light bulb placed at the focal point of a parabolic mirror will have its light reflected in parallel lines, a
property used in flashlights and automobile headlights. Light or other electromagnetic waves traveling
in lines parallel to the axis will reflect off a parabolic dish towards a collector at the focus, which is how
satellite dishes and reflecting telescopes work.
Figure 2:
An ellipse is the set of all points P in the plane such that the sum of the distances from P to two fixed
points is a given constant.
The constant is the length of the major axis of the ellipse. In the activity, one focus is the point F and
the other focus is the center of the circle. A crease is a line that is equidistant to the focus F and a
point G on the circle.
The crease is a tangent line to the ellipse.
Elliptical domes, such as the Mormon Tabernacle in Salt Lake City or the Capitol in
Washington DC, create “whispering galleries” where even a pin dropped at one focus can be heard
more than a hundred feet away at the other focus. The sound waves from the whisper all travel the
same distance to bounce off the ceiling to meet simultaneously at other focus. A non-surgical
treatment of kidney stones also uses the reflection property of the ellipse. In Extracorporeal Shock
Wave Lithotripsy, the patient is placed so that the kidney stone positioned at one focus of an ellipse. A
high energy sound wave is created at the other focus, and it reflects off all parts of an elliptical tank
wall to break up the kidney stone.
Figure 3:
A hyperbola is the set of all points P in the plane such that the difference between the distances from
P to two fixed points is a given constant.
In the activity, the foci are again the point F and the center of the circle, C. As with the ellipse, a crease
is a line that is equidistant to the focus F and a point G on the circle.
The crease is a tangent line to the hyperbola.
In the Cassegrain telescope design, the primary parabolic mirror reflects light towards a focal point. A
secondary, hyperbolic mirror is positioned so that one of its foci coincides with the focus of the
parabola and the other is the collector behind a hole in the primary mirror. The advantage of the
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Folding Conics
Cassegrain design over the traditional reflection telescope is that the Cassegrain telescope can be
much more compact.