1.Caustics of light rays and Euler’s angle of inclination (arXiv)
Abstract : Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve optically, as the caustic (envelope) of light rays normal to it, and study the Euler’s problem for general caustics. The resulting curves are characterized when the rays are at a constant angle to the curve, generalizing the case of evolutes. Aside from analogs of classical solutions we encounter some new types of curves. We also consider caustics of parallel rays reflected by a curved mirror, where Euler’s problem leads to a novel pantograph equation, and describe its analytic solutions
2. Fused Angles and the Deficiencies of Euler Angles(arXiv)
Abstract : Just like the well-established Euler angles representation, fused angles are a convenient parameterisation for rotations in three-dimensional Euclidean space. They were developed in the context of balancing bodies, most specifically walking bipedal robots, but have since found wider application due to their useful properties. A comparative analysis between fused angles and Euler angles is presented in this paper, delineating the specific differences between the two representations that make fused angles more suitable for representing orientations in balance-related scenarios. Aspects of comparison include the locations of the singularities, the associated parameter sensitivities, the level of mutual independence of the parameters, and the axisymmetry of the parameters.
3. Quantum-limited Euler angle measurements using anticoherent states(arXiv)
Abstract : Many protocols require precise rotation measurement. Here we present a general class of states that surpass the shot noise limit for measuring rotation around arbitrary axes. We then derive a quantum Cramér-Rao bound for simultaneously estimating all three parameters of a rotation (e.g., the Euler angles), and discuss states that achieve Heisenberg-limited sensitivities for all parameters; the bound is saturated by “anticoherent” states [Zimba, Electron. J. Theor. Phys. 3, 143 (2006)] (we are reluctant to use “anticoherent” to describe the states, but the name has become commonplace over the last decade). Anticoherent states have garnered much attention in recent years, and we elucidate a geometrical technique for finding new examples of such states. Finally, we discuss the potential for divergences in multiparameter estimation due to singularities in spherical coordinate systems. Our results are useful for a variety of quantum metrology and quantum communication applications
4. Euler angles based loss function for camera relocalization with Deep learning(arXiv)
Abstract : Deep learning has been applied to camera relocalization, in particular, PoseNet and its extended work are the convolutional neural networks which regress the camera pose from a single image. However there are many problems, one of them is expensive parameter selection. In this paper, we directly explore the three Euler angles as the orientation representation in the camera pose regressor. There is no need to select the parameter, which is not tolerant in the previous works. Experimental results on the 7 Scenes datasets and the King’s College dataset demonstrate that it has competitive performances