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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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