I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry.
One of the question was “Show that a harmonic homology whose centre and axis are pole and polar with respect to a point conic $\mathscr C$ keeps $\mathscr C$ invariant.” A harmonic homology just means that the cross ratio is $-1$.
I am unsure on how I can even start the proof. I took on some simple examples. For example. I let the center be $Z(0,0,1)$ and axis be $z(0,0,1)$. I am able to find that the matrix is a symmetric diagonal matrix.
Anyone has any hints on how I can show that it keeps the conic $\mathscr C$ invariant?
I don't have the book, but there is probably a statement and proof of the following: A conic in the projective plane has the following property: If $P$ is a point not on $C$, and if a variable line through $P$ meets $C$ at points $A$ and $B$, then the variable harmonic conjugate of $P$ with respect to $A$ and $B$ traces out a line. The point $P$ is called the pole of that line of harmonic conjugates, and this line is called the polar line of $P$ with respect to the conic. (paraphrased from https://en.wikipedia.org/wiki/Projective_harmonic_conjugate#Projective_conics)
In other words, given a point P not on the conic, draw a line that intersects the conic at $A$ and $B$, and the polar at $P'$. The cross ratio $[AB;PP']=-1$, and $P'$ and $P$ are harmonic conjugates with respect to $A,B$. This defines the polar line of $P$. And it is also true that $A$ and $B$ are harmonic conjugates with respect to $P,P'$.
But this is the definition of the given harmonic homology. It maps every point $A$ of the conic into a point $B$ of the conic. Thus it maps the conic to itself. (I would quibble with the use of the word 'invariant' to describe this, but I assume this is what the author meant.)