Projective Proof and Generalization of Chasles's Theorem Along Non-Torsal Lines

Abstract

We prove Chasles's theorem for non-developable ruled surfaces, originally published in 1839, and generalize it to higher-dimensional projective extensions of the real space for the rst polars along non-torsal lines. We avoid the use of dierential geometry and re-prove the theorem strictly projectively, using only incidence properties for surfaces of higher degree.

Keywords

• Chasles's theorem
• ruled surface
• projective methods
• polar surface