The general case is that given 2 points or 3 lines it is possible to find the line or plane joining them using the determinant of the matrix where the additional row or column is the X,Y,Z parameters of the line or question. Projective Geometry Is a useful mathematical expression of what we use in graphics to project the world into a 2-d plane. It contains the point at infinity, and the observer.
All curves are but generalizations of the conic in projective space.
All Euclidean and affine transformations unified into projective map duality. All theorems about points are about lines as well. In non-oriented projective geometry there are no sides of line. There is no notion of opposite sides because you can go to infinity and loop back. A quaternion is a way to turn R 4 into algebra.