The 2D Orientation Interpolation Problem: A Symmetric Space Approach

In this paper, we propose a novel construction of Bézier curves of two-dimensional (2D) orientations using the geometry of real projective plane RP2. Unlike the commonly adopted unit 2-sphere model S2, RP2 is naturally embedded in the 3D special orthogonal group SO(3). It is also a symmetric space that is equipped with a particular class of isometries called geodesic symmetry, which allows us to generate any geodesics using the exponential map of SO(3). We implement the generated geodesics to construct Bézier curves for direction interpolation.