if I understand correctly then, that the projective geometry is the geometry at which the only straight lines are preserved
meaning that points stay points and lines stay lines and conics stay conics (no special points or lines)
and affine geometry where an addition constraint is added which the line at infinity is preserved point at infinity get transformed to another point ant infinity but not ordinary point or in other worlds parallel lines stay parallel but with different slope (we have special line that doesn't change which line at infinity)
meaning that points stay points and lines stay lines and conics stay conics and also trapezoid stay trapezoid
in 2D projective geometry in homogeneous coordinates the transformations 3x3 matrix H has no constrains while in affine geometry H last row must be [0 0 1] to preserve the line at infinity
my questions are:
1- is my understanding correct ?
2- does the projective geometry transformations matrix preserve straightness because it's a matrix multiplication hence it's a linear transformation and linear transformation preserve straightness , or it preserve straightness duo to another reason ?
3- what is the geometry that is more general than projective that doesn't preserve straight lines, and what does the transformations in it look like ?