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.
1
1
1
2. Prove that for a thin lens, the image is in focus when
−
+
=
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
⃗
Hint: As discussed in class, consider similar triangles from the lens to the focal point and the
focal point to the image plane. There are 2 pairs of similar triangles, one for the positive and
negative focal point. Then show that
− +=
′
=
3. Suppose that, in the imaging geometry above, the image plane is located distance
has
from the lens, so that the image is out of focus. Show that the blur circle
diameter
, where d is the lens diameter.
= |∆ |
+ ∆
.
Hint: Consider rays coming from the top and bottom of the lens that would be in focus at
What happens when they hit the image plane at ′?
4. 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°).
a. How many receptors are there per mm2?
b. 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?
Define world ray
= ⃗
| ⃗
= ⃗
+
,0≤ ≤
∞�
. Show that it projects to camera
5. Show that a ray in the world projects to a line segment in the image as follows:
= ⃗
| ⃗
= (1 − ) ⃗
+�
⃗
⃗
line segment
→ ∞
onto the
⃗
⃗
where
is the projection of
image plane and
is the projection of ray
in the limit as
. You should find that
ranges from 0 to
1 and is related non-linearly to .
⃗
I.P.
