How does projective geometry unify circles ellipses parabolas and hyperbolas?
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.
Projective geometry unifies circles, ellipses, parabolas, and hyperbolas by treating them all as different projective images of a single circle.1
In this framework, conic sections are viewed as plane sections of a right circular cone. By considering the cone's apex as a projective "eye" and a cutting plane as a "picture plane," each conic section is revealed to be a projection of a circle. The specific type of conic produced depends on the orientation of the cutting plane relative to the cone and the line at infinity in the projective plane:1
- Ellipse: Formed when the line at infinity intersects the reality plane outside the base circle.1
- Parabola: Formed when the line at infinity is tangent to the base circle.1
- Hyperbola: Formed when the line at infinity intersects the base circle.1
By ignoring properties like distance and angle—which are not preserved under projection—and focusing on invariant properties like collinearity, mathematicians such as Jean-Victor Poncelet were able to develop a unified theory where every conic section is considered equivalent to a circle.12