Harmonic curves and the beauty of Projective Geometry

Abstract

The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of some classic theorems and a slight reformulation of its axiomatics.

Figures (11)

• Figure 1: a) A harmonic set. b) A harmonic pencil with dashed lines.
• Figure 2: a) Visual proof of the harmonic theorem in 3D. b) Symmetry of harmonic pairs.
• Figure 3: Klein's Triangle Lemma.
• Figure 4: a) Quadrangle $\mathcal{Q}$ with tangent lines to its harmonic curve $\mathcal{C}_\mathcal{Q}$ at its vertices. b) Generic point $Z\in \mathcal{C}_\mathcal{Q}$, where $A,X,B,Y$ is a harmonic set in $A\vee B$.
• Figure 5: Incidence of points in a harmonic curve and lines in its tangent bundle.

Theorems & Definitions (9)

• Theorem 1: Harmonic Theorem
• Lemma 1: Klein's Triangle
• Lemma 2: Duality lemma
• Theorem 2: Polarity
• Lemma 3
• Theorem 3
• Theorem 4: Polarity of ruled surfaces
• Theorem 5
• Theorem 6