1.4.5—Invariant points and lines

An invariant point maps to itself. An invariant line maps every point on the line to a point on the same line, even if individual points move along it.

For a matrix transformation, solve T(v)=v for invariant vectors or use the eigenvalue 1 condition. For an invariant line, check that its direction is mapped to a parallel direction and that the line’s location is preserved.

A reflection leaves every point on its mirror line invariant; a rotation about the origin leaves the origin fixed but usually has no other invariant points.

A line being drawn in the same place is not enough: points must map within it, and “invariant” does not mean every point is fixed.