How does projection preserve concurrency of lines?
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, the preservation of concurrency is a fundamental property of perspective mappings. Because projective transformations map lines onto lines, the structural relationships between these lines remain consistent during the projection process.1
Specifically, if three or more lines meet at a common point in the original figure, their corresponding images in the projected plane will also meet at a single, common point. This occurs because collinearity and the intersection of lines are invariant properties under projective mappings. While other geometric attributesāsuch as distances and anglesāare not preserved, the incidence relationship (the fact that lines intersect at a specific point) is maintained, ensuring that the concurrency of the lines is preserved in the resulting image.1