How does projection preserve collinearity of points?
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
In projective geometry, projection is a mapping process where points from one figure are projected onto another surface using a focal point or origin to construct sight lines.12
The preservation of collinearity is a fundamental invariant property of these mappings. Because projective transformations map lines onto lines, any set of points that share a common line (collinear points) will maintain that relationship in their projected image. Essentially, if three points are collinear in the original figure, their corresponding projections will also lie on a single line in the new plane. This property is essential to the discipline, as it allows mathematicians to understand generic geometric properties while ignoring measurements like distance and angle, which are not preserved under projection.1