ar X iv : m at h / 04 05 18 9 v 1 [ m at h . D G ] 1 1 M ay 2 00 4 ON THE SPACE OF ORIENTED AFFINE LINES IN R 3

We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean R 3 and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in R 3 , as we illustrate with a number of explicit examples. The correspondence between oriented affine lines in R 3 and the tangent bundle to the 2-sphere has a long history and has been used in various contexts. In particular, it has been used in the construction of minimal surfaces [2], solutions to the wave equation [3] and the monopole equation [1]. The Euclidean group of rotations and translations acts upon the space of oriented lines L and in this paper we freeze out this group action by introducing a particular set of coordinates on L. Our aim is to provide a local coordinate representation for the correspondence, thereby making it accessible to further applications. One application is the construction of canonical coordinates on surfaces S in R 3 which come from the description of the normal lines of S as local sections of the tangent bundle of the 2-sphere. We illustrate this explicitly by considering the ellipsoid and the symmetric torus. Definition 1. Let L be the set of oriented (affine) lines in Euclidean R 3. Definition 2. Let Φ : T S 2 → L be the map that identifies L with the tangent bundle to the unit 2-sphere in Euclidean R 3 , by parallel translation. This bijection gives L the structure of a differentiable 4-manifold. Let (ξ, η) be holomorphic coordinates on T S 2 , where ξ is obtained by stereo-graphic projection from the south pole onto the plane through the equator, and we identify (ξ, η) with the vector η ∂ ∂ξ + η ∂ ∂ξ ∈ T ξ S 2. Theorem 1. The map Φ takes (ξ, η)∈ T S 2 to the oriented line given by z = 2(η − ηξ 2) + 2ξ(1 + ξξ)r (1 + ξξ) 2 (0.1) t = −2(ηξ + ηξ) + (1 − ξ 2 ξ 2)r (1 + ξξ) 2 , (0.2)