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Old Nov 14th 2008, 10:08 PM   #1
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"The Music" (angles, focal length, lenses, etc.) URGENT

The well known 17th Century Dutch painter Johannes Vermeer only painted 30 paintings during his life.
However, they were created with incredible, almost photographic-like detail. The use of perspective was ahead of his
time. Recently, many have come to wonder how Vermeer created such detailed paintings with sophisticated rendering of
perspective. Several art historians now believe that Vermeer used a Camera Obscura, basically a large camera of sorts, to
copy the image onto canvas.
A Camera Obscura consists simply of a convex lens on one side of a large box in which the artist would sit.
Detailed studies of Vermeer’s paintings (most were done in the same room shown below) give the dimensions of the
room and of the lens he possibly used. In this problem we will explore how this might have been done. The following
painting, “The Music Lesson” painted in about 1662 gives the most geometric information about this room.
The Music Lesson, 1662-65 73.3 cm x 64.5 cm, the image to the right
is a 1/10th scale version, (7.3 cm x 6.4 cm).
We can use perspective geometry to find several relationships between objects
and distances in the real room. However we require one piece of data about an
object in the room to determine the actual scale.
The chair that appears has distinctive Lion heads on top. These exact
chairs were found in a museum and measured. The height of the chair is 108.5
cm (to the top of the back support) and the height to the seat is 48.5 cm. Thus it
is 60 cm from the seat to the top of the back support. (There are many other
markers which can be used, for example the dimensions of known paintings,
size of musical instruments, etc.).
a) With the above fact and using the outlined 1/5th scale image (14.7 x 12.9 cm)
on the next page, determine the magnification of Vermeer’s lens.
b) Now we will use perspective geometry to find the length of the room. First,
on the image on the next page verify the location of the vanishing point (VP) as
shown in the image to the right. The horizontal line is the horizon. The lens of the camera
(or the eye of the artist) is at this level. Second, with the pattern of tiles on the floor find one
of the distance points (D). The distance point is where the diagonal lines of the floor tile
pattern intersect the horizon. (Note: there is only room on the page to plot one, if you want
to plot the other, attach another piece of paper, although it is not necessary for what is to
follow). On the next page draw the appropriate lines and locate VP and D.
c) Now consider the room with the subjects (consider at front wall), lens, and painting
(image). Draw (two) triangle(s) formed by the distance point(s), vanishing point, and the
position of the lens, V (viewpoint). By examining the angles formed by the tiles, find the
angles of the triangle (V-VP-D).
d) From the result in part a above, and the plot you have made in part b, and assuming that Vermeer’s camera used a lens
of focal length f with an image that was projected on the back wall of the room (and the object is on the far wall),
determine the total length of the room.
e) What is the focal length of the lens used?
f) There is another intriguing aspect to this painting. If you notice in the mirror there is a
reflection of the woman and above that there is a reflection of what appears to be the feet of an
easel (was Vermeer trying to fool us?). Upon closer inspection you can see the base of the back
wall reflected in the mirror. The intersection of the back wall and floor in the mirror is marked
on the outline on the next page. We could use this as a second method to determine the length
of the room, however let’s ask a more subtle question; What is the angle from the wall at which
the mirror hangs? To find this you should sketch a side view of the room on another piece of

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