Solution
= Solution
An appropriate <projective transformation> is
$$
\boxed{\tau_\lambda[x:y:z]=[z-x:iy:z].}
$$
It fixes $[0:1:0]$ and has an invertible representing matrix. For $X=z-x$, $Y=iy$, $Z=z$,
$$
X(X-Z)(X-(1-\lambda)Z)
=(z-x)(-x)(\lambda z-x)
=-x(x-z)(x-\lambda z)=Y^2Z.
$$
Hence its restriction is an isomorphism $C_\lambda\to C_{1-\lambda}$.