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Dear Editors,
Galloway's article on perspective is, unfortunately, not really up to the usual
Math Horizon standards. There are a couple of out-and-out mistakes
in it, and there are also a number of claims that, while technically true, are
misleading or poorly presented.
Here are two mistakes in a row in paragraph 3:
"For example, the further an object is away from the observer, the smaller it appears in a painting and the higher it appears in the painting. A circle drawn in perspective will appear as an ellipse."
If we assume that the canvas is vertical (as Galloway does,
although he doesn't say so explicitly), then things above the artist's eye will
appear LOWER, not higher, in the painting as they recede into the distance.
This is true, for example, for the ceiling tiles and tops of doors in Figure 8.
If the objects are all at the same height as the artist's eye (such as the heads
of the disciples in the same painting), their images in the painting will all
likewise appear at the same height -- that is, neither higher nor lower.
Moreover, sometimes the perspective image of a circle is actually a
parabola. It's an exceptional case, I know, but since Galloway brings up
anamorphosis at the end of this essay, he ought to acknowledge that case.
Galloway later claims that one-point perspective "is appropriate when the
orthogonals represent lines parallel to the line of sight . . . " Parallel lines
don't intersect. If your line of sight is parallel to a line L, then you
can't see L, so you don't draw it. But Galloway correctly notes that
we do see these orthogonals; it is more correct to say that one-point
perspective happens when one set of lines is orthogonal to the plane of the
canvas (hence, the name orthogonals),
and that the other sets of lines (the transversals) are parallel to the canvas.
Finally, Galloway is vague and misleading when he talks about viewing distance.
He claims (correctly, in a technical sense), "For a picture drawn in one point
perspective, there is one and only one correct viewing point from which to
observe the painting," (and by this, he means, "to get a sense of three
dimensions popping out at you.") He then goes on to say, "The illusion can
collapse if the viewer veers away even slightly from this observation point."
Well, it can collapse, but does it? Galloway proceeds to draw a
checkerboard in the style of Alberti, and he (correctly) constructs it to have a
viewing distance of 1.25 inches. Are ANY of his readers looking at
the drawing from that close? No! Does it still look pretty good?
Yes! But there's no real discussion of why either side of this argument is
correct: that is, why Alberti's construction gives a correct viewing
distance (it's a subtle argument, really), why the picture would look better if
you could actually stick your eye 1.25" from the paper and still focus, nor why
it still looks like a reasonable drawing from 2 feet away even though Galloway
just implicitly claimed it wouldn't.
In fact, there are two other subtle confusions here. One is that
Alberti's construction--in particular, his viewing distance--assumes that the
checkerboard is square. But in Figure 2 ("Pavimento to be projected"),
Galloway draws the checkers as rectangles, not squares. The second is that
Galloway's claiming that the "O" represents the observer is downright
misleading; the "O" is on the plane of the canvas, and the viewer should be in
front of the canvas.
Sincerely,
Annalisa Crannell
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Dr. Annalisa Crannell
Faculty Don of Bonchek College House
Box 3003, Department of Mathematics
Franklin & Marshall College
Lancaster, PA 17604-3003
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