How are shadows related to projective geometry?
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, shadows serve as a foundational concept for understanding how geometric figures relate to their images when projected onto a surface. Mathematician Jean-Victor Poncelet famously investigated the properties that remain unchangedâor invariantâbetween an object and its shadow.12
Key relationships include:
- Preserved Properties: Certain geometric characteristics survive the projection process. For instance, a straight line always casts a straight shadow, and collinearity is preserved; if three points lie on a single line, their projected images will also be collinear. Additionally, a tangent to a curve casts a shadow that remains tangent to the shadow of that curve.21
- Lost Properties: Distances and angles are generally not preserved, meaning the size and shape of an object often appear distorted in its shadow.321
- Projective Invariants: While lengths and angles change, the cross ratio of four collinear points remains constant, providing a critical mathematical link between an object and its projection.1
By focusing on these invariant properties rather than measurements, projective geometry allows for a more generalized study of geometric objects, effectively unifying concepts like conic sections through their shared projective relationships.21