Transformation Geometry — Math 331

March 31, 2004

The Dual Projective Plane

A triple (a, b, c) ≠ (0, 0, 0) of real numbers can represent either a point in P^{2} or the coefficient vector in the equation for a line in P^{2}. In both cases the geometric object, i.e., the point or the line, is unchanged if the triple is multiplied by a non-zero scalar.

Thus, the set L of all lines in P^{2} shares with P^{2} the property that its “points” are represented by homogeneous non-zero triples of real numbers. Another way of viewing this is to regard L as another copy of P^{2}: the dual projective plane.

A key characteristic of this duality is, firstly, that a point in L, i.e., a line in P^{2}, is determined by the set of points in P^{2} that belong to it and, secondly, that a point in P^{2} is determined by the set of lines in P^{2} to which it belongs.

Moreover, a line in L is the set of “points” (a, b, c) in L — with each such “point” corresponding to the line in P^{2} having the equation ax+by+cz = 0 — satisfying a homogeneous linear equation Aa + Bb + Cc = 0 with coefficient vector (A, B, C) ≠ (0, 0, 0). But when (A, B, C) is viewed as the triple of homogeneous coordinates for a point of P^{2}, one sees that this point of P^{2} lies on every line ax + by + cz = 0 in P^{2} for which (a, b, c) is a “point” on the line Aa + Bb + Cc = 0 in L, and, moreover, since a point in P^{2} is determined by the set of lines in P^{2} on which it lies, the lines in P^{2} corresponding to “points” of the line in L with equation Aa + Bb + Cc = 0 are precisely the lines in P^{2} containing (A, B, C) regarded as a point of P^{2}.

Definition. A pencil of lines in P^{2} is the set of all lines in P^{2} containing a given point of P^{2}.

The preceding discussion makes it clear that a pencil of lines in P^{2} is essentially the same thing as a “line” in the dual projective plane. Another way to state this is to say that the projective plane dual to L (the dual of the dual), which is the set of lines in L, is essentially the same thing as P^{2}.

Exercises due Friday, April 2

1. Show that the set of lines in P^{2} stabilized under the translation of R^{2} by the vector (3, 4), treated as a projective transformation when x+y +z = 0 is taken as the line at infinity, is a pencil of lines. What point of P^{2} is the point where the lines of the pencil are coincident?

2. If (a_{1}, a_{2}, a_{3}) and (b_{1}, b_{2}, b_{3}) are triples of homogeneous coordinates for two different points of P^{2}, what is the significance for those points of the “point” of the dual projective plane with homogeneous coordinate vector

(a_{2}b_{3} − a_{3}b_{2}, a_{3}b_{1} − a_{1}b_{3}, a_{1}b_{2} − a_{2}b_{1}) .

3. Find all fixed points and stabilized lines of the projective transformation given by the matrix

( \begin{matrix}0 & −1 & 0 \\ 1 & 0 & 0 \\ 0 & 2 & 1 \end{matrix} ) .

4. Find all fixed points and stabilized lines of the projective transformation given by the matrix

( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 2 & 1 \end{matrix} ) .

Indicate what pencils occur among these lines.

5. Explain why every point of P^{2} lies on at least one of the lines in every pencil of lines in P^{2}.

6. Given a pencil of lines in P^{2} how many points of P^{2} lie on more than one of the lines in the pencil?