Federico Thomas

We study the algebraic structure of the common normals to two coplanar conics. The normals to a single conic form a quartic curve in the dual plane; hence their common normals are given by the intersection of two such quartics, which by Bézout's theorem yields up to 16 projective solutions. Once the Euclidean structure is accounted for, however, four of these solutions always lie at infinity ---the complex circular points. We show that the remaining finite solutions reduce to intersecting one conic with an auxiliary sextic curve. Parametrizing this conic via a pencil of lines then yields an explicit univariate polynomial of degree at most 12, whose real roots correspond exactly to the common normals. This construction provides a direct algorithm for computing all real common normals between two conics without requiring resultant computations.

The kinematics, dynamics, and control of a unicycle moving without slipping on a plane has been extensively studied in the literature of nonholonomic mechanical systems. However, since planar motion can be seen as a limiting case of the motion on a sphere, we focus our analysis on the more general spherical case. This paper introduces a novel approach to path planning for a unicycle rolling on a sphere while satisfying the non-slipping constraint. Our method is based on a simple yet effective idea: first, we model the system as a linear time-varying dynamic system. Then, leveraging the fact that certain such systems can be integrated under specific algebraic conditions, we derive a closed-form expression for the control variables. This formulation includes three free parameters, which can be tuned to generate a path connecting any two configurations of the unicycle. Notably, our approach requires no prior knowledge of nonholonomic system analysis, making it accessible to a broader audience.

Nowadays, the singular value decomposition (SVD) is the standard method of choice for solving the nearest rotation matrix problem. Nevertheless, many other methods are available in the literature for the 3D case. This article reviews the most representative ones, proposes alternative ones, and presents a comparative analysis to elucidate their relative computational costs and error performances. This analysis leads to the conclusion that some algebraic closed‐form methods are as robust as the SVD, but significantly faster and more accurate.

In this paper, we address the problem of restoring the orthogonality of a numerically noisy 4D rotation matrix by finding its nearest (in Frobenius norm) correct rotation matrix. This problem can be straightforwardly solved using the Singular Value Decomposition (SVD). Nevertheless, to avoid numerical methods, we present two new closed‐form methods. One relies on the direct minimization of the mentioned Frobenius norm, and the other on the passage to double quaternion representation. A comparison of these two methods with respect to the SVD reveals that the method based on a double quaternion representation is superior in all aspects.