How did Girard Desargues use points at infinity in 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.
Girard Desargues was a pioneer in projective geometry who introduced the concept of points at infinity to unify geometric principles that were otherwise treated as separate cases in Euclidean space.
His primary use of this concept included:
- Representing Parallel Intersections: Desargues introduced a single point at infinity to represent the projected intersection of parallel lines. By doing so, he allowed parallel lines to be treated as lines that intersect at a specific, infinitely distant point, rather than as lines that never meet.
- Defining the Horizon: He extended this idea by collecting all such points along the horizon into a single "line at infinity." This provided a consistent framework for perspective drawings and geometric projections.
- Simplifying Geometry: By incorporating these points, Desargues enabled a more unified approach to geometry. This allowed properties of figures to be studied through their projections without needing to account for the special cases where lines are parallel.
While his work was initially overlooked, his introduction of points at infinity became a foundational element for later mathematicians like Jean-Victor Poncelet, who used it to formally develop the modern field of projective geometry.123