arXiv subjects

Zijia Li

Publications and source records attributed to Zijia Li.

At least 19 recordsLinked to original sources

arXiv2026

Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$

We study constructive Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$. Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree $2d$ with $d\ge1$ first factors into $d$ quadratic rational curves and then into a product of at most $2d$ planar rotation curves, where each planar rotation curve fixes a two-dimensional plane pointwise. The construction proceeds by extracting left and right isoclinic polynomial parts via Cayley's factorization, factoring the corresponding quaternion polynomials, and pairing linear factors with equal norm polynomials. The resulting algorithms are explicit and are illustrated by examples.

math.AG↗

arXiv2026

Fusion-Poly: A Polyhedral Framework Based on Spatial-Temporal Fusion for 3D Multi-Object Tracking

LiDAR-camera 3D multi-object tracking (MOT) combines rich visual semantics with accurate depth cues to improve trajectory consistency and tracking reliability. In practice, however, LiDAR and cameras operate at different sampling rates. To maintain temporal alignment, existing data pipelines usually synchronize heterogeneous sensor streams and annotate them at a reduced shared frequency, forcing most prior methods to perform spatial fusion only at synchronized timestamps through projection-based or learnable cross-sensor association. As a result, abundant asynchronous observations remain underexploited, despite their potential to support more frequent association and more robust trajectory estimation over short temporal intervals. To address this limitation, we propose Fusion-Poly, a spatial-temporal fusion framework for 3D MOT that integrates asynchronous LiDAR and camera data. Fusion-Poly associates trajectories with multi-modal observations at synchronized timestamps and with single-modal observations at asynchronous timestamps, enabling higher-frequency updates of motion and existence states. The framework contains three key components: a frequency-aware cascade matching module that adapts to synchronized and asynchronous frames according to available detection modalities; a frequency-aware trajectory estimation module that maintains trajectories through high-frequency motion prediction, differential updates, and confidence-calibrated lifecycle management; and a full-state observation alignment module that improves cross-modal consistency at synchronized timestamps by optimizing image-projection errors. On the nuScenes test set, Fusion-Poly achieves 76.5% AMOTA, establishing a new state of the art among tracking-by-detection 3D MOT methods. Extensive ablation studies further validate the effectiveness of each component. Code will be released.

cs.CV↗

arXiv2026

Geometric Stratification for Singular Configurations of the P3P Problem via Local Dual Space

This paper investigates singular configurations of the P3P problem. Using local dual space, a systematic algebraic-computational framework is proposed to give a complete geometric stratification for the P3P singular configurations with respect to the multiplicity $\mu$ of the camera center $O$: for $\mu\ge 2$, $O$ lies on the ``danger cylinder'', for $\mu\ge 3$, $O$ lies on one of three generatrices of the danger cylinder associated with the first Morley triangle or the circumcircle, and for $\mu\ge 4$, $O$ lies on the circumcircle which indeed corresponds to infinite P3P solutions. Furthermore, a geometric stratification for the complementary configuration $O^\prime$ associated with a singular configuration $O$ is studied as well: for $\mu\ge 2$, $O^\prime$ lies on a deltoidal surface associated with the danger cylinder, and for $\mu\ge 3$, $O^\prime$ lies on one of three cuspidal curves of the deltoidal surface.

cs.CV↗

arXiv2025

Quadratic Motion Polynomials With Irregular Factorizations

Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on the invertibility of a certain coefficient occurring in the algorithm. If this coefficient is not invertible, factorizations may or may not exist. In the case of existence we call this an irregular factorization. We characterize quadratic motion polynomials with irregular factorizations in terms of algebraic equations and present examples whose number of unique factorizations range from one to infinitely many. For two special sub-cases we show the unique existence of such polynomials. In case of commuting factors we obtain the conformal Villarceau motion, in case of rigid body motions the circular translation.

math.RA↗

arXiv2024

Motion Polynomials Admitting a Factorization with Linear Factors

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear factors, and we give an algorithm to compute them. We can use those linear factors to construct mechanisms because the factorization corresponds to the decomposition of the rational motion into simple rotations or translations. Bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial. Our criterion for factorizability allows us to improve on earlier algorithms to compute a suitable real or quaternion polynomial co-factor.

math.RA↗

arXiv2024

Rational Curves on Real Classical Groups

This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify quadratic rational curves on $\mathrm{U}_n$, $\mathrm{O}_n(\mathbb{R})$, $\mathrm{O}_{n-1,1}(\mathbb{R})$ and $\mathrm{O}_{n-2,2}(\mathbb{R})$. (ii) We prove a decomposition theorem for rational curves on real classical groups, which can be regarded as a non-commutative generalization of the fundamental theorem of algebra and partial fraction decomposition. (iii) As an application of (i) and (ii), we generalize Kempe's Universality Theorem to rational curves on homogeneous spaces.

math.AG↗

arXiv2024

Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic Sets

Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn's work, we refine the stratification of iterated singular loci to a Whitney (a) stratification, which gives a list of candidates of varieties whose dual is an irreducible component of the algebraic boundary of the dual convex body. We also present an algorithm based on Teissier's criterion to compute Whitney (a) stratifications, which employs conormal spaces and prime decomposition.

math.OC↗

arXiv2024

Quadratic Spinor Polynomials with Infinitely Many Factorizations

Spinor polynomials are polynomials with coefficients in the even sub-algebra of conformal geometric algebra whose norm polynomial is real. They describe rational conformal motions. Factorizations of spinor polynomial corresponds to the decomposition of the rational motion into elementary motions. Generic spinor polynomials allow for a finite number of factorizations. We present two examples of quadratic spinor polynomials that admit infinitely many factorizations. One of them, the circular translation, is well-known. The other one has only been introduced recently but in a different context. We not only compute all factorizations of these conformal motions but also interpret them geometrically.

math.RA↗

arXiv2023

A Geometric Algorithm for the Factorization of Spinor Polynomials

We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions. The factorization algorithm is based on the "kinematics at infinity" of the underlying rational motion. Factorizations exist generically but not generally and are typically not unique. We prove that generic multiples of non-factorizable spinor polynomials admit factorizations and we demonstrate at hand of an example how our ideas can be used to tackle the hitherto unsolved problem of "factorizing" algebraic motions.

math.RA↗

arXiv2022

The integral closure of a primary ideal is not always primary

In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings with any characteristics. We also find that the Jacobian ideal $J$ of the polynomial $f = x^6 + y^6 + x^4 z t + z^3$ given by Briançon and Speder in 1975 is a counterexample to Krull's question. Let $V_1$ be the hypersurface defined by $f = 0$ and $V_2$ be its singular locus. Briançon and Speder proved that Whitney equisingularity does not imply Zariski equisingularity by showing that the pair $(V_1 \setminus V_2,\ V_2)$ satisfies Whitney's conditions around the origin but fails Zariski's equisingular conditions. We discover that the pair $(V_1 \setminus V_2,\ V_2)$ fails Whitney's conditions at the variety of the embedded prime of the integral closure $\bar{J}$, which means that $V_1$ is not Whitney regular along $V_2$. Moreover, we also show that Whitney stratification of this hypersurface is different from the stratification of isosingular sets given by Hauenstein and Wampler, which is related to Thom-Boardman singularity.

math.AC↗

arXiv2022

The Study Variety of Conformal Kinematics

We introduce the Study variety of conformal kinematics and investigate some of its properties. The Study variety is a projective variety of dimension ten and degree twelve in real projective space of dimension 15, and it generalizes the well-known Study quadric model of rigid body kinematics. Despite its high dimension, co-dimension, and degree it is amenable to concrete calculations via conformal geometric algebra (CGA) associated to three-dimensional Euclidean space. Calculations are facilitated by a four quaternion representation which extends the dual quaternion description of rigid body kinematics. In particular, we study straight lines on the Study variety. It turns out that they are related to a class of one-parametric conformal motions introduced by L. Dorst in 2016. Similar to rigid body kinematics, straight lines (that is, Dorst's motions) are important for the decomposition of rational conformal motions into lower degree motions via the factorization of certain polynomials with coefficients in CGA.

math.RA↗

arXiv2022

Classification of higher Mobility closed-loop Linkages

We provide a complete classification of paradoxical closed-loop $n$-linkages, where $n\geq6$, of mobility $n-4$ or higher, containing revolute, prismatic or helical joints. We also explicitly write down strong necessary conditions for $nR$-linkages of mobility $n-5$. Our main new tool is a geometric relation between a linkage $L$ and another linkage $L'$ resulting from adding equations to the configuration space of $L$. We then lift known classification results for $L'$ to $L$ using this relation.

cs.CG↗

arXiv2019

Mechanism Singularities Revisited from an Algebraic Viewpoint

It has become obvious that certain singular phenomena cannot be explained by a mere investigation of the configuration space, defined as the solution set of the loop closure equations. For example, it was observed that a particular 6R linkage, constructed by a combination of two Goldberg 5R linkages, exhibits kinematic singularities at a smooth point in its configuration space. Such problems are addressed in this paper. To this end, an algebraic framework is used in which the constraints are formulated as polynomial equations using Study parameters. The algebraic object of study is the ideal generated by the constraint equations (the constraint ideal). Using basic tools from commutative algebra and algebraic geometry (primary decomposition, Hilbert's Nullstellensatz), the special phenomenon is related to the fact that the constraint ideal is not a radical ideal. With a primary decomposition of the constraint ideal, the associated prime ideal of one primary ideal contains strictly into the associated prime ideal of another primary ideal which also gives the smooth configuration curve. This analysis is extended to shaky and kinematotropic linkages, for which examples are presented.

cs.RO↗

arXiv2018

Factorization Results for Left Polynomials in Some Associative Real Algebras: State of the Art, Applications, and Open Questions

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples.

math.RA↗

arXiv2018

A Survey on the Theory of Bonds

Many researchers tried to understand/explain the geometric reasons for paradoxical mobility of a mechanical linkage, i.e. the situation when a linkage allows more motions than expected from counting parameters and constraints. Bond theory is a method that aims at understanding paradoxical mobility from an algebraic point of view. Here we give a self-contained introduction of this theory and discuss its results on closed linkages with revolute or prismatic joints.

eess.SY↗

arXiv2018

The Geometry of Quadratic Quaternion Polynomials in Euclidean and Non-Euclidean Planes

We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split quaternion polynomials of degree two are related to the coupler motion of "four-bar linkages" with equal opposite sides in universal hyperbolic geometry. A factorization corresponds to a leg of the four-bar linkage and during the motion the legs intersect in points of a conic whose focal points are the fixed revolute joints. The number of factorizations is related by the number of real focal points which can, indeed, be six in universal hyperbolic geometry.

math.MG↗

arXiv2018

Degree-optimal moving frames for rational curves

A $\mathit{\text{moving frame}}$ at a rational curve is a basis of vectors moving along the curve. When the rational curve is given parametrically by a row vector $\mathbf{a}$ of univariate polynomials, a moving frame with important algebraic properties can be defined by the columns of an invertible polynomial matrix $P$, such that $\mathbf{a} P=[\gcd(\mathbf{a}),0\ldots,0]$. A $\mathit{\text{degree-optimal moving frame}}$ has column-wise minimal degree, where the degree of a column is defined to be the maximum of the degrees of its components. Algebraic moving frames are closely related to the univariate versions of the celebrated Quillen-Suslin problem, effective Nullstellensatz problem, and syzygy module problem. However, this paper appears to be the first devoted to finding an efficient algorithm for constructing a degree-optimal moving frame, a property desirable in various applications. We compare our algorithm with other possible approaches, based on already available algorithms, and show that it is more efficient. We also establish several new theoretical results concerning the degrees of an optimal moving frame and its components. In addition, we show that any deterministic algorithm for computing a degree-optimal algebraic moving frame can be augmented so that it assigns a degree-optimal moving frame in a $GL_n(\mathbb{K})$-equivariant manner. This crucial property of classical geometric moving frames, in combination with the algebraic properties, can be exploited in various problems.

math.AG↗

arXiv2017

Kempe's Universality Theorem for Rational Space Curves

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 9/2 d - 6c + 1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicity construction of the linkage.

cs.CG↗