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The point at infinity

Infinity and strange math · level 8, lesson 7 of 7

A question to start

Stand between the rails of a long, straight railway and look along it. The two rails are parallel: they are the same distance apart all the way, and they never meet. Yet in your eyes, and in any photograph, they come together at a single point on the horizon. Photographs taken by rovers on Mars show the same thing: a rover's two parallel wheel tracks, stretching back across flat ground, seem to meet in the distance.

horizon vanishing point

Every painter since the Renaissance has used this point, the vanishing point, to make flat pictures look deep. Here is the mathematician's question: what if we took the picture seriously? What if we added to the plane a point where parallel lines do meet, a "point at infinity"? Would geometry break, or would it get better?

It gets better. The result, the projective plane, is a geometry with no exceptions: every two lines meet, exactly once. This last lesson of the topic turns infinity into a place you can draw.

The idea

Why parallel lines meet in a picture

Think of a painter looking at the world through a pane of glass, and marking on the glass where each object appears. A point on the ground far away is seen along a line of sight that is almost level, so it appears just below the horizon. The further away it is, the closer to the horizon it appears.

Follow one rail into the distance: its image creeps up towards one particular point of the horizon, the point you would see by looking exactly along the rail. The other rail points in the same direction, so its image creeps towards the same point. That is why the rails meet in the picture: parallel lines share a direction, and lines with the same direction vanish at the same point.

Lines in a different direction (the edges of a road crossing the railway at an angle, say) vanish at a different point of the horizon. And the horizon itself is where all the vanishing points of the ground lie. Renaissance painters and architects worked out these rules in the 1400s.

Adding points at infinity

Copy the painter. Start with the ordinary plane and add:

  • one new point at infinity for each direction: all lines with that direction (a family of parallel lines) pass through it;
  • one new line at infinity, which consists of all the points at infinity, and nothing else. It plays the role of the horizon.

The result is the projective plane. Notice that a line's two "ends" go to the same point at infinity (going left along the line or going right, you head in the same direction, up to reversing). So each projective line closes up into a loop. You saw this in level 7: bending an open interval into a half circle and projecting from the centre matched it with the whole line. Add one point to close the gap, and a line becomes a circle.

A geometry with no exceptions

In ordinary geometry two different lines meet in one point unless they are parallel. That exception spoils many theorems, which need extra cases. In the projective plane:

  • every two different points lie on exactly one line, and
  • every two different lines meet in exactly one point.

Check the cases. Two ordinary lines that cross meet where they always did. Two parallel lines meet at the point at infinity for their shared direction. An ordinary line and the line at infinity meet at that line's point at infinity. No exceptions remain.

The two rules above have the same form with "point" and "line" swapped. This duality means that every theorem of projective geometry has a twin with points and lines exchanged, proved for free.

The French engineer Girard Desargues used points at infinity in this way in 1639, and projective geometry was developed fully in the 1800s.

Coordinates for points at infinity

How can you calculate with a point that is infinitely far away? The trick is homogeneous coordinates. Write the ordinary point \((x, y)\) as a triple \([x : y : 1]\), and agree that multiplying all three numbers by the same nonzero number gives the same point. So

\[[2 : 3 : 1] = [4 : 6 : 2] = [-2 : -3 : -1] \quad\text{all mean the point } (2, 3).\]

Any triple \([X : Y : Z]\) with \(Z \ne 0\) is an ordinary point: divide by \(Z\) to get \(\left(\tfrac XZ, \tfrac YZ\right)\). The triples with \(Z = 0\) (not all three zero) are the points at infinity. Why? Take the points \((t, 2t) = [t : 2t : 1]\) running out along the line \(y = 2x\). Divide by \(t\): \([1 : 2 : \tfrac1t]\). As \(t\) grows, \(\tfrac1t \to 0\), and the point approaches \([1 : 2 : 0]\). So \([1 : 2 : 0]\) is the point at infinity in the direction "1 across, 2 up": slope 2.

There is a picture behind this. A triple \([X : Y : Z]\) and all its multiples make up one line through the origin in three-dimensional space: the painter's line of sight, with the eye at the origin and the glass as the plane \(Z = 1\). Each line of sight that hits the glass gives an ordinary point. The lines of sight parallel to the glass (\(Z = 0\)) never hit it; they are the points at infinity.

A line \(ax + by = c\) becomes \(aX + bY = cZ\) (put \(x = X/Z\), \(y = Y/Z\) and multiply by \(Z\)). The line at infinity is \(Z = 0\).

One curve, three costumes

Projective geometry also unifies the curves called conics: ellipses, parabolas and hyperbolas. In the projective plane they are all the same kind of curve, seen differently against the line at infinity:

  • an ellipse misses the line at infinity: it is closed and bounded;
  • a parabola touches the line at infinity at one point: its two arms go off in the same direction;
  • a hyperbola crosses the line at infinity at two points: the directions of its two asymptotes.

This is the old fact, known to the ancient Greeks, that all three are slices of one cone, seen in a new light: a circle drawn in perspective can appear as an ellipse, a parabola or a hyperbola, depending on where the eye is (for the last two, part of the circle must be level with or behind the painter).

Where else infinity becomes a point

The same idea appears in many places. In the complex numbers, adding a single point \(\infty\) turns the plane into a sphere, the Riemann sphere, on which \(\tfrac10 = \infty\) is allowed (while \(\tfrac00\) is still not). There are even finite projective planes. The smallest, the Fano plane, has 7 points and 7 lines, with 3 points on each line, and still every two lines meet in exactly one point.

Worked examples

1. Where do parallel lines meet? Find the meeting point of \(y = 2x\) and \(y = 2x + 1\).

In homogeneous form: \(2X - Y = 0\) and \(2X - Y + Z = 0\). Subtract the first from the second: \(Z = 0\). Then \(Y = 2X\), so the point is \([1 : 2 : 0]\): the point at infinity of slope 2, as expected.

2. Ordinary lines still meet where they did. Find where \(x + y = 3\) and \(x - y = 1\) meet, in homogeneous coordinates.

The lines are \(X + Y = 3Z\) and \(X - Y = Z\). Adding: \(2X = 4Z\), so \(X = 2Z\). Then \(Y = X - Z = Z\). The point is \([2Z : Z : Z] = [2 : 1 : 1]\), the ordinary point \((2, 1)\). Check: \(2 + 1 = 3\) and \(2 - 1 = 1\). ✓

3. Converting. \([6 : 4 : 2]\): divide by 2 to get \([3 : 2 : 1]\), the point \((3, 2)\). \([3 : -1 : 0]\): the third number is 0, so it is a point at infinity, in the direction of slope \(-\tfrac13\).

4. A hyperbola's points at infinity. The hyperbola \(xy = 1\) becomes \(\tfrac XZ \cdot \tfrac YZ = 1\), that is \(XY = Z^2\). Its points at infinity have \(Z = 0\), so \(XY = 0\): either \(X = 0\), giving \([0 : 1 : 0]\) (the vertical direction), or \(Y = 0\), giving \([1 : 0 : 0]\) (the horizontal direction). These are exactly the directions of its two asymptotes, the \(y\)-axis and the \(x\)-axis.

5. A parabola's point at infinity. \(y = x^2\) becomes \(\tfrac YZ = \tfrac{X^2}{Z^2}\), that is \(YZ = X^2\). At \(Z = 0\): \(X^2 = 0\), so \(X = 0\), and the only point is \([0 : 1 : 0]\). One point: the parabola touches the line at infinity where both arms are heading, straight up.

6. A circle. \(x^2 + y^2 = 1\) becomes \(X^2 + Y^2 = Z^2\). At \(Z = 0\): \(X^2 + Y^2 = 0\), which forces \(X = Y = 0\), not allowed. No points at infinity: a circle (like any ellipse) stays in the finite part of the plane.

Try it

  1. Which point at infinity lies on every line of slope \(-3\)?
  2. Where do the vertical lines \(x = 1\) and \(x = 4\) meet?
  3. Convert \([10 : -5 : 5]\) to an ordinary point. Is \([2 : -1 : 0]\) an ordinary point?
  4. Find where \(y = 3x + 1\) and \(y = -x + 5\) meet, first in ordinary coordinates, then as a homogeneous triple.
  5. How many points at infinity does the hyperbola \(x^2 - y^2 = 1\) have? Find them.
  6. How many points at infinity does the parabola \(y = x^2 + 1\) have?
  7. Why does a painter put the vanishing points of all the ground's straight lines on the horizon line?
  8. Challenge: find the line through the two points at infinity \([1 : 0 : 0]\) and \([0 : 1 : 0]\).
Answers
  1. \([1 : -3 : 0]\) (1 across, \(-3\) up).
  2. In homogeneous form, \(X = Z\) and \(X = 4Z\). Subtracting: \(3Z = 0\), so \(Z = 0\), then \(X = 0\). The point is \([0 : 1 : 0]\), the vertical point at infinity.
  3. \([10 : -5 : 5] = [2 : -1 : 1]\), the point \((2, -1)\). \([2 : -1 : 0]\) has third coordinate 0, so it is not ordinary: it is the point at infinity of slope \(-\tfrac12\).
  4. \(3x + 1 = -x + 5\) gives \(4x = 4\), \(x = 1\), \(y = 4\). The point \((1, 4)\) is \([1 : 4 : 1]\).
  5. \(X^2 - Y^2 = Z^2\). At \(Z = 0\): \(X^2 = Y^2\), so \(Y = X\) or \(Y = -X\). Two points, \([1 : 1 : 0]\) and \([1 : -1 : 0]\): the directions of its asymptotes \(y = x\) and \(y = -x\).
  6. \(YZ = X^2 + Z^2\). At \(Z = 0\): \(X^2 = 0\), so one point, \([0 : 1 : 0]\).
  7. The horizon is the picture of the line at infinity of the ground: every direction along the ground vanishes at its own point on the horizon, and every point of the horizon is the vanishing point of some direction along the ground.
  8. A line \(aX + bY + cZ = 0\) through \([1 : 0 : 0]\) needs \(a = 0\); through \([0 : 1 : 0]\) needs \(b = 0\). So the line is \(cZ = 0\), that is \(Z = 0\): the line at infinity itself, as it should be.

Common mistake

"The point at infinity is a very far ordinary point." It is not on the ordinary plane at all; it is a new point added for a whole direction. Its coordinates have \(Z = 0\), which no ordinary point has.

Thinking a line has two points at infinity, one at each end. In the projective plane both ends of a line head for the same point at infinity (\([1 : 2 : 0]\) and \([-1 : -2 : 0]\) are the same point). That is what closes each line into a loop.

Where it leads