Description
1. In addition to the similarities and difference from Class 01, list 5 other ways in which eyes and cameras are similar. List 5 other ways in which they are different.
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2. Prove that for a thin lens, the image is in focus when |
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− |
+ |
= |
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z
=
leaving the object at
Reason as follows: A ray
parallel to
the
axis
= (
,
)
(0, )
⃗
. If the image is in focus, then
before hitting the
image plane at
passes through the lens, then bends to pass through focal
point
, )
(
(0, − )
⃗
similarly, a ray leaving the
will be bent parallel to the z
object at
passing through the negative focal point
⃗
.
axis and
hit the image plane at the same point
⃗
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Hint: As discussed in class, consider similar triangles from the lens to the focal point and the |
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focal point to the image plane. There are 2 pairs of similar triangles, one for the positive and |
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negative focal point. Then show that |
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− += |
′ |
= |
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3. Suppose that, in the imaging geometry above, the image plane is located distance |
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has |
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from the lens, so that the image is out of focus. Show that the blur circle |
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diameter |
, where d is the lens diameter. |
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= |∆ | |
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+ ∆ |
. |
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Hint: Consider rays coming from the top and bottom of the lens that would be in focus at |
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What happens when they hit the image plane at ′? |
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A typical human eyeball is 2.4 cm in diameter and contains roughly 150,000,000 receptors. Ignoring the fovea, assume that the receptors are uniformly distributed across a hemisphere (it is actually closer to 160°).
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How many receptors are there per mm2?
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Mars has a diameter of 8,000 km and an average distance from Earth of 225,000,000 km. Using a value of f equal to the eye’s diameter, on how many receptors does the image of Mars fall?
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Define world ray |
= ⃗ |
| ⃗ |
= ⃗ |
+ |
,0≤ ≤ |
∞� |
. Show that it projects to camera |
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5. Show that a ray in the world projects to a line segment in the image as follows: |
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= ⃗ |
| ⃗ |
= (1 − ) ⃗ |
+� |
⃗ |
⃗ |
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line segment |
→ ∞ |
onto the |
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⃗ |
⃗ |
where |
is the projection of |
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image plane and |
is the projection of ray |
in the limit as |
. You should find that |
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ranges from 0 to |
1 and is related non-linearly to . |
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⃗ |
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I.P.
