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 \cal{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 \cal{L} as another copy of P^{2}: the dual projective plane.
A key characteristic of this duality is, firstly, that a point in \cal{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 \cal{L} is the set of “points” (a, b, c) in \cal{L} -- with each such “point” corresponding to the line in P^{2} having the equation a x + b y + c z = 0 -- satisfying a homogeneous linear equation A a + B b + C c = 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 a x + b y + c z = 0 in P^{2} for which (a, b, c) is a “point” on the line A a + B b + C c = 0 in \cal{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 \cal{L} with equation A a + B b + C c = 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 \cal{L} (the dual of the dual), which is the set of lines in \cal{L}, is essentially the same thing as P^{2}.
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?
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}) . |
Find all fixed points and stabilized lines of the projective transformation given by the matrix
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Find all fixed points and stabilized lines of the projective transformation given by the matrix
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Indicate what pencils occur among these lines.
Explain why every point of P^{2} lies on at least one of the lines in every pencil of lines in P^{2}.
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?