Download Comment on “Anamorphic images,” by J. L. Hunt, B. G. Nickel, and

Euler-Lagrange equation for ␾.兴 For ␪ = ␾ Eq. 共11兲 reduces to
Eq. 共2兲 if x = 共1 / 3兲H, that is, at the center of percussion. It is
not difficult to show that ␾ ⬎ ␪ for x ⬍ 共1 / 3兲H and ␾ ⬍ ␪ for
x ⬎ 共1 / 3兲H. This result illustrates the varying acceleration
known as the free-fall paradox.
This apparatus make an interesting demonstration. A side
view shows that the pendulum bob lags the frame for 3x
⬍ H and leads it for 3x ⬎ H 共as shown in Fig. 1兲. For 3x
= H the frame and pendulum move together. The sounds
made by the frame and pendulum bob striking the table are
not simultaneous except for 3x = H, and hence the relative
accelerations can be determined audibly. This demonstration
is not one of the variations that Varieschi referenced,5 nor
does it appear on the many physics demonstration websites
that he has searched.
Fig. 1. A pendulum of mass m is suspended by a light string from a frame
of mass M, which is pivoted at its base. This falling-pendulum arrangement
may be used to demonstrate the variation of the vertical acceleration along
the length of a falling chimney.
is H and pendulum length is x. Because M Ⰷ m, the frame
equation of motion is the same as that for the falling rod 关see
Eq. 共2兲兴 which integrates to
g
␪˙ 2 = 3 关1 − cos共␪兲兴,
H
共10兲
g 3
␾¨ =
sin共␪兲cos共␪ − ␾兲 − 3共1 − cos共␪兲兲sin共␪ − ␾兲
x 2
册
− sin共␾兲 .
The author is grateful to Professor Varieschi for his efforts
and encouragement.
a兲
Electronic mail: [email protected]
Gabriele Varieschi and Kaoru Kamiya, “Toy models for the falling chimney,” Am. J. Phys. 71, 1025–1031 共2003兲.
2
This result is derived in Keith R. Symon, Mechanics 共Addison-Wesley,
Reading, MA, 1961兲, 2nd ed., Chap. 5, p. 241.
3
A. S. Ramsey, Dynamics 共Cambridge University Press, Cambridge,
1938兲, 2nd ed., Vol. 1, Chap. 16, pp. 237–238.
4
S. L. Loney, Dynamics of a Particle and of Rigid Bodies 共Cambridge
University Press, Cambridge, 1926兲, Chap. 19, pp. 345–346.
5
Gabriele Varieschi 共personal communication兲, April 2005. Professor Varieschi has also tested our apparatus and confirms that the pendulum bob
leads the frame for 3x ⬎ H and lag behind for 3x ⬍ H.
1
the same equation as Eq. 共4兲 of Ref. 1. The pendulum equation is
冋
ACKNOWLEDGMENT
共11兲
关To derive Eq. 共11兲 we substituted Eqs. 共2兲 and 共10兲 into the
Comment on “Anamorphic images,” by J. L. Hunt, B. G. Nickel,
and Christian Gigault †Am. J. Phys. 68 „3…, 232–237 „2000…‡
Alan J. DeWeerda兲 and S. Eric Hillb兲
Department of Physics, University of Redlands, Redlands, California 92373
共Received 29 July 2005; accepted 9 September 2005兲
关DOI: 10.1119/1.2117148兴
Hunt, Nickel, and Gigault1 concluded their paper on anamorphic images with a brief discussion of the ambiguous
location of an image in a convex cylindrical mirror. They
also made some observations of the effects of this ambiguity
on viewing the image. In a recent paper2 we explored the
same ambiguity for a concave cylindrical mirror and the resulting peculiarities associated with viewing. We draw on
that work to clarify the nature of the convex cylindrical mirror’s image and to explain some observations of anamorphic
images.
For simplicity, consider a point object reflected from a
convex cylindrical mirror with its axis vertical. As illustrated
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Am. J. Phys. 74 共1兲, January 2006
http://aapt.org/ajp
in Fig. 1, the rays reflected from a linear, vertical cross section project to point a, while those reflected from a curved,
horizontal cross section project to point b. To understand the
nature of the reflection from the entire mirror, other vertical
and horizontal cross sections must be considered. Rays reflected from all horizontal cross sections project to points
that form a vertical line image passing through point a. On
the other hand, rays reflected from all vertical cross sections
project to points that form a curved line image3 in the same
horizontal plane as the point object. Because a single point is
projected to two perpendicular lines, the image is astigmatic.
The separation of the two image lines increases as the dis© 2006 American Association of Physics Teachers
83
Fig. 2. A ray entering an observer’s eye projects through the two line
images.
Fig. 1. Rays reflected by 共a兲 a horizontal or 共b兲 a vertical cross section of the
cylindrical mirror project to different points.
tance between the object and mirror increases. A consequence of this worsening astigmatism is that parts of an anamorphic image viewed higher in the mirror appear less
sharp.
The two line images are helpful for understanding how an
astigmatic image is viewed. As shown in Fig. 2, a ray reaching an observer’s eye must project through both line images.4
When viewing the image with two eyes, the angle at which
their gazes converge provides a depth cue. If two eyes are
oriented so that the interocular axis is horizontal 共vertical兲,
then the two gazes converge on the vertical 共horizontal兲 line
image.5 Therefore, the image of the point object either appears to stand on the vertical line or to lie on the horizontal
one. This effect accounts for the observation in Ref. 1 that
the anamorphic image appeared to stand up when viewed
with the head held upright, but appeared to recline in the
x-y plane when viewed with the head tilted.
Hunt et al.1 also reported that the effect due to the head’s
orientation persisted when one eye was closed. This observation is intriguing because the above explanation only applies to binocular viewing. We explored two approaches to
monocular observations.
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Am. J. Phys., Vol. 74, No. 1, January 2006
When starting with one eye open, we did not notice any
difference as the head’s orientation was changed. When
viewing with one eye, the amount of accommodation necessary to focus gives a depth cue. For an astigmatic image, the
eye tends to balance the clarity along the two axes and focus
at a point between the two line images. Regardless of the
head’s orientation, this intermediate focus would give the
anamorphic image an inclination between the standing and
reclining inclinations observed with two eyes, which is what
we observed. While making these observations we were very
careful to keep the eye at the same location. There is a natural tendency to lower the eye while rotating it by tilting the
head sideways, which reduces the viewing angle and decreases the apparent inclination of the anamorphic image.
We also started with both eyes open and observed what
happened when one eye was closed without changing the
head’s orientation. In both orientations, we noticed the sensation of the one eye changing its focal length upon closure
of the other. However, closing one eye did not always create
the strong perception that the inclination changed. When the
interocular axis was vertical, the image no longer appeared to
lie horizontally after either eye was closed. In this case, the
viewing angles are not the same for the two eyes, but the
difference can be minimized by viewing from a large distance. On the other hand, with the interocular axis horizontal,
the inclination of the image did not appear markedly changed
when either eye was closed. An explanation of this final observation may require going beyond physical cues to psychological ones.
a兲
Electronic address: [email protected]
Electronic address: [email protected]
1
J. L. Hunt, B. G. Nickel, and Christian Gigault, “Anamorphic images,”
Am. J. Phys. 68共3兲, 232–237 共2000兲.
2
Alan J. DeWeerd and S. Eric Hill, “The dizzying depths of the cylindrical
mirror,” Phys. Teach. 43共2兲, 90–92 共2005兲.
3
The curve is the inner loop of a limaçon. See the Appendix to Ref. 2
which is available at the EPAPS homepage as document E-PHTEAH-43012502: http://www.aip.org/pubservs/epaps.html
4
Figure 13 and Eq. 共12兲 of Ref. 1 relate to the x-z plane of our Fig. 2. Their
s⬘ is the distance from the z axis to our vertical image line and their z is
the height at which the gaze crosses that image line.
5
If the eyes are at other orientations, then the gazes do not converge.
b兲
Notes and Discussions
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