Recall that if f is an affine transformation and L is a line in R^{2}, then the image f(L) of L under f is also a line in R^{2}. Thus, f . L = f(L) is an action of the affine group on the set of lines.
Every line in R^{2} has an equation of the form
where a_{1}, a_{2}, and k are constants with (a_{1}, a_{2}) <> (0, 0). This equation may be written with matrix notation in the form
| a_{1} | a_{2} | k |
| x_{1} |
| x_{2} |
| 1 |
Two such equations given by triples (a_{1}, a_{2}, k) and (a_{1}', a_{2}', k') correspond to the same line if and only if one triple is a (non-zero) scalar times the other.
Recall from the discussion of April 12 that if f(x) = M x + b is an affine transformation (of R^{2}), then there is a group homomorphism I from the affine group to the group GL_{3}(R) of invertible 3 \times 3 matrices that carries f to the blocked matrix
| M | b |
| 0 | 1 |
If i is the affine transformation from R^{2} to R^{3} that is defined by x = (x_{1}, x_{2}) -> i(x) =(x_{1}, x_{2}, 1), then this embedding i has the ``equivariance'' property
for each x in R^{2} and each affine transformation f of the plane.
If u = (a_{1} a_{2} k ) is the row vector of coefficients in the equation of a line L, then the equation of the line f(L) has the coefficient vector u I(f)^{-1}. That is, if the equation of L is a^{*} x + k = 0, where a and x are columns and a^{*} is the transpose of a, then the equation of f(L) is
A line L is an invariant line of the affine transformation f if and only if f(L) = L. This is the case if and only if the corresponding equation coefficient triples are parallel, i.e., if and only if there a non-zero scalar t such that
Describe the group of all isometries that leave invariant the x-axis of R^{2}, i.e., the isotropy group of the x-axis in the action of the isometry group on the set of all lines.
Explain why there is an isometry of the plane carrying a given line to any other given line.
Show that if every line is an invariant line of an affine transformation of the plane, then that transformation must be the identity.
Because lines in the plane are represented by triples of scalar coefficients subject to the identification of parallel triples, one might expect that the set of lines in the plane is 2-dimensional.
Show that if a line does not contain the origin, then it has a unique equation of the form a_{1} x_{1} + a_{2} x_{2} = 1.
Explain why the set of lines not containing the origin may be regarded as a plane with a single point removed.
Explain why the set of lines that do contain the origin may be regarded as a line with a single point INFTY added.
For thought. How can one characterize the entire set of lines in the plane as a geometric object?