Fixed straight line
In affine and projective geometry, a fixed line is a straight line that is mapped onto itself under an ( affine ) or projective mapping. In the case of a fixed line - unlike a fixed point line - not all points of the straight line have to be mapped onto itself; it is sufficient if each point of the fixed line is mapped to a point of this straight line. Therefore every fixed point line is a fixed line, but not vice versa. A fixed line is a one-dimensional fixed space . The presence or absence of fixed spaces (more specifically: fixed lines) is an important feature that is used to classify affinities , affine maps , projectivities, and projective mappings .
Definitions
Affine fixed line
be a mapping in coordinate representation. is a fixed line of f if:
- 1.) s is an eigenvector of A with an eigenvalue not equal to 0
- 2.)
Here t is a starting point of g.
Projective fixed line
A projective fixed point line is generated by two linearly independent eigenvectors with the same eigenvalue not equal to 0 in the space of the coordinate vectors.