The 2D Orientation Interpolation Problem: A Symmetric Space Approach

In this paper, we propose a novel construction of Bezier curves of two-dimensional (\(2\)D) orientations using the geometry of real projective plane \(\mathrm{\mathbb {R}P^{2}}\). Unlike the commonly adopted unit 2-sphere model \(S^{2}\), \(\mathrm{\mathbb {R}P^{2}}\) is naturally embedded in the \(3\)D special orthogonal group \(\mathrm{SO(3)}\). 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 \(\mathrm{SO(3)}\). We implement the generated geodesics to construct Bezier curves for direction interpolation.

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