[Paper Review] Geometric algebra and singularities of ruled and developable surfaces

Junki Tanaka, Toru Ohmoto|arXiv (Cornell University)|Aug 31, 2018

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper applies geometric algebra—specifically dual quaternions—to classify local diffeomorphic and topological types of singularities on ruled and developable surfaces in ℝ³. By modeling ruled surfaces as curves of unit dual vectors and analyzing the jet expansion of the parametrization map, the authors show that the topological type of a singular developable surface is completely determined by the vanishing order of the dual torsion 𝜏̌, generalizing Mond’s classical result for tangent developables.

ABSTRACT

Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invariants. In particular, using a theorem of G. Ishikawa, we show that local topological type of singular (non-cylindrical) developable surfaces is completely determined by vanishing order of the dual torsion, that generalizes an old result of D. Mond for tangent developables of non-singular space curves. Our approach would be useful for analysis on singularities arising in differential line geometry related with several applications such as robotics, vision theory and architectural geometry.

Motivation & Objective

• To classify local diffeomorphic and topological types of singularities on ruled and developable surfaces in ℝ³ using geometric algebra.
• To characterize singularities of ruled surfaces via the 𝒜-equivalence and rigid equivalence of map-germs (ℝ²,0)→(ℝ³,0).
• To generalize Mond’s result on tangent developables by showing that the topological type of a developable surface is determined solely by the vanishing order of the dual torsion 𝜏̌.
• To establish a framework where geometric algebra provides a natural language for studying singularities in classical Klein geometries.
• To derive canonical Taylor expansions of ruled surface parametrizations using dual quaternion calculus and Frenet formulas in 𝔻³.

Proposed method

• Represent any ruled surface in ℝ³ as a curve of unit dual vectors in the even Clifford algebra 𝒞ℓ⁺(0,3,1), i.e., the algebra of dual quaternions.
• Use the Frenet formula in the dual space 𝔻³ to define dual curvature 𝜅̌ and dual torsion 𝜏̌, with 𝜏̌ = 𝜏₀ + ε𝜏₁.
• Derive a canonical Taylor expansion of the surface parametrization F(s,t) up to order 3 using the dual Bouquet formula.
• Apply Izumiya-Saji’s criteria and Mond’s classification tools to detect 𝒜-types of map-germs, particularly Sw, cA₄, and cA₅.
• Analyze the singular point set S(F) via the condition (f₂)ᵧ = (f₃)ᵧ = 0 and use vector fields to compute invariants like ηλ(0), ηηλ(0), etc.
• Relate the type (m,n₁,n₂) of the striction curve σ(s) to the vanishing orders of τ₁ and τ₀, using recursive Frenet-type differentiation in the dual setting.

Experimental results

Research questions

• RQ1 How can geometric algebra be used to classify local diffeomorphic types of singular ruled surfaces in ℝ³?
• RQ2 What is the topological classification of singular developable surfaces, and how does it relate to the dual torsion 𝜏̌?
• RQ3 Can the 𝒜-classification of map-germs (ℝ²,0)→(ℝ³,0) be applied to parametrized ruled surfaces to detect singularity types?
• RQ4 How does the vanishing order of the dual torsion 𝜏̌ determine the topological type of a developable surface?
• RQ5 What is the relationship between the jet expansion of the surface parametrization and the singularity type (e.g., Sw, cA₄, cA₅) in the classification?

Key findings

• The local diffeomorphic type of a singular ruled surface is completely determined by the jet of the parametrization map, with types such as Sw, cA₄, and cA₅ detectable via invariants derived from τ₁ and τ₀.
• The topological type of a singular developable surface is completely determined by the vanishing order of the dual torsion 𝜏̌, generalizing Mond’s result for tangent developables.
• If the striction curve σ(s) has type (m, n₁, n₂), then the vanishing order of τ₁ at s=0 is m−1 and the vanishing order of τ₀ is r−1, with n₁ = m+1 and n₂ = m+1+r for some r ≥ 1.
• The jet expansion of F(s,t) up to order 3 is explicitly derived using dual quaternion calculus, showing dependence on κ₁′(0), τ₀(0), τ₁(0), and τ₁′(0).
• The singular point set S(F) is defined by λ = 0, where λ is a function involving s and y, and its derivatives along the kernel vector field η = ∂/∂y yield invariants that detect singularity types.
• Codimension-3 and codimension-4 singularities (T₁ and T₂) arise when τ₀ = 0, while higher-order types (Sw, cA₄, cA₅) occur when τ₀ ≠ 0 and τ₁′(0) ≠ 0.