Lissajous Varieties: Algebraic & Trigonometric Insights

• Lissajous varieties are affine algebraic sets created by applying the cosine function coordinatewise to affine linear spaces determined by a matrix, thereby generalizing classical planar Lissajous figures.
• They connect multiple areas such as enumerative geometry, toric parametrizations, and convex optimization, with properties like dimension and degree linked to matrix rank and polytope volume.
• Their study provides practical insights into oscillator-network equilibria, interpolation node design, and bifurcation analysis, reflecting broad applications in algebraic geometry and dynamical systems.

Lissajous varieties are affine algebraic varieties obtained by applying coordinatewise cosine, or sine as a special case, to affine linear spaces determined by a matrix $A$. In the formulation of "Lissajous Varieties" (Mascarin et al., 8 Sep 2025), one starts with $A\in \mathbb{Q}^{d\times n}$ and $b\in \mathbb{C}^n$, defines

$$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$

and then sets

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$

where cosine is taken coordinatewise. This construction generalizes classical planar Lissajous figures to higher-dimensional affine algebraic geometry, and it connects enumerative geometry, toric-type parametrizations, determinantal equations, oscillator-network equilibria, convex optimization, and bifurcation theory (Mascarin et al., 8 Sep 2025).

1. Definition and basic parametrizations

The defining map of a Lissajous variety is

$${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$

Given $A\in \mathbb{Q}^{d\times n}$ and $b\in\mathbb{C}^n$, the affine space

$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$

produces the Lissajous variety

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$

Two distinguished special cases are

$A\in \mathbb{Q}^{d\times n}$0

If $A\in \mathbb{Q}^{d\times n}$1 denotes the $A\in \mathbb{Q}^{d\times n}$2-th column of $A\in \mathbb{Q}^{d\times n}$3, then choosing coordinates $A\in \mathbb{Q}^{d\times n}$4 on $A\in \mathbb{Q}^{d\times n}$5 yields the parametrization

$A\in \mathbb{Q}^{d\times n}$6

so that

$A\in \mathbb{Q}^{d\times n}$7

Using Euler’s formula, the same object admits a Laurent-monomial-type parametrization. With

$A\in \mathbb{Q}^{d\times n}$8

one defines

$A\in \mathbb{Q}^{d\times n}$9

where

$b\in \mathbb{C}^n$0

The paper proves

$b\in \mathbb{C}^n$1

and that $b\in \mathbb{C}^n$2 is a closed irreducible affine variety of dimension $b\in \mathbb{C}^n$3 (Mascarin et al., 8 Sep 2025).

The planar classical case is recovered when $b\in \mathbb{C}^n$4 and $b\in \mathbb{C}^n$5. For example, if $b\in \mathbb{C}^n$6 and $b\in \mathbb{C}^n$7, then

$b\in \mathbb{C}^n$8

parametrized by

$b\in \mathbb{C}^n$9

(Mascarin et al., 8 Sep 2025). In this sense, Lissajous varieties are algebraic generalizations of the classical trigonometric figures traced by harmonic oscillators.

2. Degree, polytope volume, and algebraic structure

A central theorem in the theory identifies the degree of $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$0 with the volume of a centrally symmetric polytope. For $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$1 of full rank $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$2, one sets

$$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$3

If $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$4, then for generic $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$5,

$$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$6

where $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$7 is the normalized volume of $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$8, and $$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$9 is the number of coloops of ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$0, equivalently the number of zero entries in a generic vector in ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$1 (Mascarin et al., 8 Sep 2025).

Without the simplifying assumptions, the degree formula becomes

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$2

where

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$3

is the projection from an auxiliary toric-type variety ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$4. Under the circuit nondegeneracy condition

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$5

the map ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$6 has degree

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$7

so if ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$8,

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}),$9

The examples recorded in the paper illustrate the geometric meaning of this theorem. For $${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$0, $${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$1, so $${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$2, matching the degree of the circle. For

$${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$3

$${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$4 is a hexagon and

$${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$5

In the same example, $${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$6 is Cayley’s cubic surface,

$${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$7

For the cycle graph incidence matrix $${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$8, the degree of the cycle polynomial $${\rm cos}:\mathbb{C}^n\to\mathbb{C}^n,\qquad (x_1,\dots,x_n)\mapsto (\cos x_1,\dots,\cos x_n).$$9 is

$A\in \mathbb{Q}^{d\times n}$0

(Mascarin et al., 8 Sep 2025).

This degree–volume correspondence places Lissajous varieties close to toric and polyhedral geometry. A plausible implication is that the trigonometric origin of the parametrization does not obscure the combinatorial control of degree; rather, the relevant combinatorics is encoded by the centrally symmetric polytope built from the columns of $A\in \mathbb{Q}^{d\times n}$1.

3. Defining equations, rank constraints, and Chebyshev elimination

The defining equations of a Lissajous variety can be extracted from rank conditions on multiplication matrices. The basic algebraic setup introduces the ideal

$A\in \mathbb{Q}^{d\times n}$2

where $A\in \mathbb{Q}^{d\times n}$3, and the quotient algebra

$A\in \mathbb{Q}^{d\times n}$4

which has dimension $A\in \mathbb{Q}^{d\times n}$5. Multiplication by $A\in \mathbb{Q}^{d\times n}$6 defines a linear map

$A\in \mathbb{Q}^{d\times n}$7

For one variable,

$A\in \mathbb{Q}^{d\times n}$8

and in general

$A\in \mathbb{Q}^{d\times n}$9

The analogous matrices $b\in\mathbb{C}^n$0 commute, so every Laurent polynomial $b\in\mathbb{C}^n$1 yields a matrix $b\in\mathbb{C}^n$2 (Mascarin et al., 8 Sep 2025).

If $b\in\mathbb{C}^n$3 generate the ideal of

$b\in\mathbb{C}^n$4

then the paper proves the rank-condition theorem

$b\in\mathbb{C}^n$5

Equivalently, $b\in\mathbb{C}^n$6 is cut out set-theoretically by the maximal minors of the block matrix

$b\in\mathbb{C}^n$7

If $b\in\mathbb{C}^n$8 generate $b\in\mathbb{C}^n$9, one can take

$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$0

or after clearing denominators,

$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$1

In the hypersurface case $L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$2, the toric piece $L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$3 is a hypersurface and the Lissajous variety admits a determinantal representation: $L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$4 If the prime ideal of $L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$5 is $L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$6, then

$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$7

For the 3-cycle example and $L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$8,

$L_{A,b}=\operatorname{Row}(A)-\frac{b\pi}{2}$9

so the set-theoretic equation is the square of Cayley’s cubic equation (Mascarin et al., 8 Sep 2025).

The use of Chebyshev polynomials is natural in related elimination problems. For classical Lissajous parametrizations

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$0

Chebyshev polynomials satisfy

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$1

and eliminating the parameter yields

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$2

When ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$3 are coprime positive integers and ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$4, this polynomial is irreducible over both ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$5 and ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$6; when ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$7 and ${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$8 are not coprime, it factors into components corresponding to lower-frequency Lissajous curves (Zhang, 2022). This classical algebraic theory is not identical to the matrix-parametrized theory of Lissajous varieties, but it clarifies the elimination mechanisms that underlie both.

4. Chebyshev varieties, interpolation nodes, and multivariate Lissajous geometry

The literature on Lissajous-Chebyshev nodes provides a closely related geometric framework in which varieties are described simultaneously by Chebyshev equations and by Lissajous parametrizations. In "A unifying theory for multivariate polynomial interpolation on general Lissajous-Chebyshev nodes" (Dencker et al., 2017), the affine real Chebyshev variety is

${\mathcal L}_{A,b}={\rm cos}(L_{A,b}).$9

and the paper proves that

$A\in \mathbb{Q}^{d\times n}$00

The interpolation nodes are obtained by sampling these curves at

$A\in \mathbb{Q}^{d\times n}$01

Singular points are characterized by the drop of the Jacobian rank at special Chebyshev grid points, and the same framework yields discrete orthogonality, unique interpolation, and a Chebyshev-weight quadrature rule (Dencker et al., 2017).

An earlier treatment, "Multivariate polynomial interpolation on Lissajous-Chebyshev nodes" (Dencker et al., 2015), studies multivariate Lissajous curves

$A\in \mathbb{Q}^{d\times n}$02

classifies degenerate and non-degenerate cases, and identifies interpolation nodes with singular sets of Chebyshev varieties such as

$A\in \mathbb{Q}^{d\times n}$03

The resulting node sets support exact discrete orthogonality and unique interpolation in explicitly defined spaces of multivariate Chebyshev polynomials (Dencker et al., 2015).

These interpolation papers do not define Lissajous varieties in the matrix-cosine sense of (Mascarin et al., 8 Sep 2025). However, they show that the phrase “Lissajous” in multivariate algebraic geometry already encompassed unions of parametrized cosine images satisfying Chebyshev relations. This suggests a broad geometric continuum: classical plane Lissajous figures, Chebyshev varieties sampled along Lissajous curves, and the affine algebraic Lissajous varieties $A\in \mathbb{Q}^{d\times n}$04 all encode trigonometric parametrizations as algebraic sets.

5. Dynamical systems, optimization, and discriminants

A principal motivation for Lissajous varieties is the Kuramoto model of coupled oscillators,

$A\in \mathbb{Q}^{d\times n}$05

Using

$A\in \mathbb{Q}^{d\times n}$06

the steady-state equations become bilinear: $A\in \mathbb{Q}^{d\times n}$07 For a connected graph $A\in \mathbb{Q}^{d\times n}$08, with reduced incidence matrix $A\in \mathbb{Q}^{d\times n}$09, the steady states correspond to the intersection

$A\in \mathbb{Q}^{d\times n}$10

Equivalently, with $A\in \mathbb{Q}^{d\times n}$11, one solves Laurent equations

$A\in \mathbb{Q}^{d\times n}$12

The degree of $A\in \mathbb{Q}^{d\times n}$13 bounds the number of isolated equilibria; for the triangle graph $A\in \mathbb{Q}^{d\times n}$14, the paper exhibits a case with exactly six real steady states, matching $A\in \mathbb{Q}^{d\times n}$15 (Mascarin et al., 8 Sep 2025).

The positive part of a Lissajous variety is characterized by a strictly convex optimization problem. For the sine case,

$A\in \mathbb{Q}^{d\times n}$16

and

$A\in \mathbb{Q}^{d\times n}$17

if and only if $A\in \mathbb{Q}^{d\times n}$18 is the unique minimizer of

$A\in \mathbb{Q}^{d\times n}$19

subject to

$A\in \mathbb{Q}^{d\times n}$20

For general $A\in \mathbb{Q}^{d\times n}$21, the positive part is

$A\in \mathbb{Q}^{d\times n}$22

and

$A\in \mathbb{Q}^{d\times n}$23

if and only if $A\in \mathbb{Q}^{d\times n}$24 is the unique minimizer of

$A\in \mathbb{Q}^{d\times n}$25

subject to

$A\in \mathbb{Q}^{d\times n}$26

If $A\in \mathbb{Q}^{d\times n}$27 is the minimizer, then the corresponding steady state $A\in \mathbb{Q}^{d\times n}$28 is linearly stable, and the Jacobian has the explicit form

$A\in \mathbb{Q}^{d\times n}$29

which is negative definite at the positive solution (Mascarin et al., 8 Sep 2025).

Bifurcation theory enters through the Lissajous discriminant. The incidence variety is

$A\in \mathbb{Q}^{d\times n}$30

with toric Jacobian

$A\in \mathbb{Q}^{d\times n}$31

The ramification locus is

$A\in \mathbb{Q}^{d\times n}$32

and the Lissajous discriminant is

$A\in \mathbb{Q}^{d\times n}$33

If it is a hypersurface, its defining polynomial is $A\in \mathbb{Q}^{d\times n}$34, and

$A\in \mathbb{Q}^{d\times n}$35

For $A\in \mathbb{Q}^{d\times n}$36 and $A\in \mathbb{Q}^{d\times n}$37,

$A\in \mathbb{Q}^{d\times n}$38

For the triangle graph $A\in \mathbb{Q}^{d\times n}$39, the paper computes

$A\in \mathbb{Q}^{d\times n}$40

The discriminant marks parameter values where critical points collide; in the Kuramoto setting, it is the bifurcation locus where stability can change (Mascarin et al., 8 Sep 2025).

6. Broader Lissajous families in geometry and physics

The modern theory of Lissajous varieties sits within a larger corpus in which Lissajous-type structures appear in algebraic geometry, integrable systems, celestial mechanics, magnetic dynamics, and exploratory number-theoretic visualization.

On the algebraic side, the elimination of parameters from

$A\in \mathbb{Q}^{d\times n}$41

produces Chebyshev-Lissajous polynomials. The irreducibility criterion established in (Zhang, 2022) states that

$A\in \mathbb{Q}^{d\times n}$42

is irreducible over $A\in \mathbb{Q}^{d\times n}$43 and $A\in \mathbb{Q}^{d\times n}$44 precisely in the coprime case when $A\in \mathbb{Q}^{d\times n}$45, while the non-coprime case decomposes into finitely many irreducible Lissajous curves. This makes the arithmetic of $A\in \mathbb{Q}^{d\times n}$46 a direct determinant of algebraic decomposition.

In classical and quantum mechanics on the sphere, a Smorodinsky–Winternitz-type Hamiltonian can produce trajectories that are Lissajous-like when two angular motions have rationally related frequencies. In "Superintegrable Lissajous systems on the sphere" (Calzada et al., 2014), rational $A\in \mathbb{Q}^{d\times n}$47 forces the orbit to close inside a bounded angular domain, and the symmetry structure explains both the classical closed trajectories and the quantum degeneracies.

In the circular restricted three-body problem, a unified center-manifold treatment identifies Lissajous, halo, and quasihalo orbits as branches of one coupled construction. The leading-order Lissajous motion is

$A\in \mathbb{Q}^{d\times n}$48

and the paper introduces a coupling coefficient $A\in \mathbb{Q}^{d\times n}$49 with bifurcation equation $A\in \mathbb{Q}^{d\times n}$50. When $A\in \mathbb{Q}^{d\times n}$51, the solution describes Lissajous orbits; when $A\in \mathbb{Q}^{d\times n}$52, it describes quasihalo orbits, and halo orbits arise as the special case $A\in \mathbb{Q}^{d\times n}$53 (Lin et al., 2024).

In magnetic dynamics, "Emergence of Lissajous trajectories in skyrmion oscillator" (Mukherjee et al., 9 Apr 2026) studies a $A\in \mathbb{Q}^{d\times n}$54 Co/Pt multilayer nanostructure with a $A\in \mathbb{Q}^{d\times n}$55-thick Co free layer, simulated in Mumax$A\in \mathbb{Q}^{d\times n}$56 via the LLG equation with Zhang–Li spin-transfer torque. For a drive

$A\in \mathbb{Q}^{d\times n}$57

the skyrmion center follows

$A\in \mathbb{Q}^{d\times n}$58

which is exactly the structure of classical Lissajous curves. The paper reports that temperature deforms these ideal figures through a temperature-dependent Hall angle and stochastic thermal force (Mukherjee et al., 9 Apr 2026).

An exploratory number-theoretic variant replaces single frequencies by prime-frequency finite Fourier sums,

$A\in \mathbb{Q}^{d\times n}$59

where $A\in \mathbb{Q}^{d\times n}$60 are the first $A\in \mathbb{Q}^{d\times n}$61 primes. Motivated by the Ulam spiral, the authors describe these objects as a toy model of prime Lissajous curves, note slight left-right symmetry breaking, and compare sums over the first $A\in \mathbb{Q}^{d\times n}$62, $A\in \mathbb{Q}^{d\times n}$63, and $A\in \mathbb{Q}^{d\times n}$64 primes (Barna et al., 2020).

A further nonlinear extension appears in separable polynomial potentials

$A\in \mathbb{Q}^{d\times n}$65

where closed Lissajous-type trajectories arise only under energy-dependent nonlinear resonance conditions. For $A\in \mathbb{Q}^{d\times n}$66, the quartic case remains algebraic through Jacobi elliptic multiplication formulas; for $A\in \mathbb{Q}^{d\times n}$67, the orbit is naturally expressed through hyperelliptic phase constraints rather than a universal polynomial orbit equation (Escobar-Ruiz et al., 23 Jun 2026).

Taken together, these works indicate that “Lissajous” now denotes more than a classical planar figure. It names a recurrent structural theme: trigonometric or oscillatory parametrizations whose closure, algebraicity, decomposition, or stability encode arithmetic, combinatorial, or dynamical information. In the specific sense formalized in (Mascarin et al., 8 Sep 2025), Lissajous varieties provide the affine algebraic version of that theme.