Perspective and Projective Geometry

WHERE THE TRAIN TRACKS MEET

Parallel lines don’t meet. That’s what makes them parallel.

But parallel lines sometimes appear to meet—think of a pair of train tracks, alone in an empty landscape, the two rails seeming to converge as your eyes follow them closer and closer to the horizon. This is the phenomenon of perspective: when you try to depict the three-dimensional world on your two-dimensional field of vision, something has to give.

The people who first figured out what was going on here were painters. The moment, early in the Italian Renaissance, at which painters understood perspective was the moment visual representation changed forever.

How exactly Florentine artists like Filippo Brunelleschi came to develop the modern theory of perspective has occasioned a hundred quarrels among art historians. What we know for sure is that the breakthrough joined aesthetic concerns with new ideas from mathematics and optics. A central point was the understanding that the images we see are produced by rays of light that bounce off objects and subsequently strike our eye. The theory of vision by reflected rays was worked out in great detail by the eleventh-century Cairene mathematician Abu ‘Ali al-Hasan ibn al-Haytham (Alhazen). His treatise on optics, the Kitab al-Manazir, was translated into Latin and taken up eagerly by philosophers and artists.

The main point is this: a point P on your canvas represents a line in three-dimensional space. Thanks to Euclid, we know there’s a unique line containing any two specified points. In this case, the line is the one containing P and your eye. Any object in the world that lies on that line gets painted at point P.

Now imagine you’re Filippo Brunelleschi standing out on the flat prairie, the canvas on an easel in front of you, painting the train tracks. The track consists of two rails, R1 and R2. Each one of these rails, drawn on the canvas, is going to look like a line. And just as a point on the canvas corresponds to a line in space, a line on the canvas corresponds to a plane. The plane P1 corresponding to R1 is the one swept out by the lines joining each point on the rail to your eye. Similarly, the plane P2 corresponding to R2 is the one containing your eye and R2. Each of the two planes cuts the canvas in a line, and we call these lines L1 and L2.

The two rails are parallel. But the two planes are not. They meet at your eye, and parallel planes do not meet anywhere. But planes that aren’t parallel have to intersect in a line. In this case, the line is horizontal, emanating from your eye and proceeding parallel to the train tracks. The line, being horizontal, does not meet the prairie—it shoots out toward the horizon, never touching the ground. But it meets the canvas at some point V. Since V is on the plane P1, it must be on the line L1 where P1 cuts the canvas. And since V is also on P2, it must be on L2. In other words, V is the point on the canvas where the painted train tracks meet. In fact, any straight path on the prairie that runs parallel to the train tracks will look, on the canvas, like a line through V. V is the so-called vanishing point.

The conceptual shift that Brunelleschi made here is the heart of what mathematicians call projective geometry. Instead of points in the landscape, we think of lines through our eye. The difference is that there are horizontal lines, which don’t intersect the ground at all. These correspond to the vanishing points on our canvas, the places where train tracks meet. You might think of this line as a point on the ground that is “infinitely far away” in the direction of the tracks. And indeed, mathematicians usually call them points at infinity. When you take the plane Euclid knew and glue on the points at infinity, you get the projective plane.

In the projective plane, the two ends of the y-axis meet at the point at infinity, and the axis is revealed to be not really a line but a circle. If you travel infinitely far to the right, until you arrive at R, and then keep on going, you find yourself still traveling rightward but now heading back toward the center from the left edge of the picture.

This kind of leaving-one-way-and-coming-back-the-other enthralled the young Winston Churchill, who recalled the one mathematical epiphany of his life: "I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus—or even the Lord Mayor’s Show, a quantity passing through infinity and changing its sign from plus to minus."

In fact, point R is not just the endpoint of the horizontal axis, but of any horizontal line. If two different lines are both horizontal, they are parallel; and yet, in projective geometry, they meet, at the point at infinity. David Foster Wallace was asked in a 1996 interview about the ending of Infinite Jest, and he replied: "There is an ending as far as I’m concerned. Certain kinds of parallel lines are supposed to start converging in such a way that an ‘end’ can be projected by the reader somewhere beyond the right frame."

The projective plane has the defect that it’s kind of hard to draw, but the advantage that it makes the rules of geometry much more agreeable. In Euclid’s plane, two different points determine a single line, and two different lines determine a single intersection point—unless they’re parallel, in which case they don’t meet at all. In the projective plane, you don’t have to make any exceptions to the rule that two lines meet at a point, because parallel lines meet too. Any two vertical lines, for instance, meet at P, and any two lines pointing northeast to southwest meet at Q. Two points determine a single line, two lines meet at a single point, end of story. It’s perfectly symmetrical and elegant.

Mathematical elegance and practical utility are close companions. One thing the projective plane is good for is representational painting. Another is picking lottery numbers.