---
title: 'Lissajous Varieties: Algebraic & Trigonometric Insights'
url: https://www.emergentmind.com/topics/lissajous-varieties
type: topic
---

# Lissajous Varieties: Algebraic & Trigonometric Insights

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" [2509.06844], 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 [2509.06844].

## 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
\[
{\mathcal C}_A={\mathcal L}_{A,\mathbf{0}}={\rm cos}(\operatorname{Row}(A)),\qquad {\mathcal S}_A={\mathcal L}_{A,\mathbf{1}}={\rm sin}(\operatorname{Row}(A)).
\]
If \(a_j\) denotes the \(j\)-th column of \(A\), then choosing coordinates \(t=(t_1,\dots,t_d)\in\mathbb{C}^d\) on \(\operatorname{Row}(A)\) yields the parametrization
\[
\phi_{A,b}:\mathbb{C}^d\to\mathbb{C}^n,\qquad t\mapsto \big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),
\]
so that
\[
{\mathcal L}_{A,b}=\operatorname{im}(\phi_{A,b}).
\]

Using Euler’s formula, the same object admits a Laurent-monomial-type parametrization. With
\[
\beta_\ell=e^{-i b_\ell \pi/2}\in \mathbb{C}^*,\qquad v=(v_1,\dots,v_d)\in(\mathbb{C}^*)^d,
\]
one defines
\[
\psi_{A,b}(v)=\left( \frac{\beta_1 v^{a_1}+\beta_1^{-1}v^{-a_1}}{2},\dots, \frac{\beta_n v^{a_n}+\beta_n^{-1}v^{-a_n}}{2} \right),
\]
where
\[
v^{a_j}=\prod_{k=1}^d v_k^{a_{kj}}.
\]
The paper proves
\[
{\mathcal L}_{A,b}=\operatorname{im}(\phi_{A,b})=\operatorname{im}(\psi_{A,b}),
\]
and that \({\mathcal L}_{A,b}\) is a closed irreducible affine variety of dimension \(\operatorname{rank}(A)\) [2509.06844].

The planar classical case is recovered when \(d=1\) and \(n=2\). For example, if \(A=(1\ \ 1)\) and \(b=(0,1)\), then
\[
{\mathcal L}_{A,b}=\{(x,y)\in\mathbb{C}^2:x^2+y^2=1\},
\]
parametrized by
\[
(\cos t,\sin t)=\left(\cos t,\cos\left(t-\frac{\pi}{2}\right)\right)
\]
[2509.06844]. 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 \({\mathcal L}_{A,b}\) with the volume of a centrally symmetric polytope. For \(A\in\mathbb{Z}^{d\times n}\) of full rank \(d\), one sets
\[
P_A=\operatorname{conv}\{\pm a_1,\dots,\pm a_n\}\subset\mathbb{R}^d.
\]
If \(\mathbb{Z}A=\mathbb{Z}^d\), then for generic \(b\in\mathbb{C}^n\),
\[
\deg({\mathcal L}_{A,b})=2^{-\mathrm{CL}_A}\, d!\,\operatorname{vol}(P_A),
\]
where \(d!\operatorname{vol}(P_A)\) is the normalized volume of \(P_A\), and \(\mathrm{CL}_A\) is the number of coloops of \(A\), equivalently the number of zero entries in a generic vector in \(\ker(A)\) [2509.06844].

Without the simplifying assumptions, the degree formula becomes
\[
\deg({\mathcal L}_{A,b})=\frac{d!\,\operatorname{vol}(P_A)}{\deg(\pi_{A,b})\cdot [\mathbb{Z}^d:\mathbb{Z}A]},
\]
where
\[
\pi_{A,b}:{\mathcal Y}_{A,b}\to{\mathcal L}_{A,b}
\]
is the projection from an auxiliary toric-type variety \({\mathcal Y}_{A,b}\). Under the circuit nondegeneracy condition
\[
m^C\cdot b=\sum_{j\in C}m_j^C b_j \notin 2\mathbb{Z} \quad\text{for every circuit }C\in\operatorname{Circ}(A),
\]
the map \(\pi_{A,b}\) has degree
\[
\deg(\pi_{A,b})=2^{\mathrm{CL}_A},
\]
so if \(\mathbb{Z}A=\mathbb{Z}^d\),
\[
\deg({\mathcal L}_{A,b})=\frac{d!\,\operatorname{vol}(P_A)}{2^{\mathrm{CL}_A}}.
\]

The examples recorded in the paper illustrate the geometric meaning of this theorem. For \(A=(1\ 1)\), \(P_A=[-1,1]\), so \(d!\operatorname{vol}(P_A)=2\), matching the degree of the circle. For
\[
A=\begin{pmatrix}
1 & 0 & -1\\
-1 & 1 & 0
\end{pmatrix},
\]
\(P_A\) is a hexagon and
\[
\deg({\mathcal S}_A)=6.
\]
In the same example, \({\mathcal C}_A\) is Cayley’s cubic surface,
\[
1+2xyz-x^2-y^2-z^2=0.
\]
For the cycle graph incidence matrix \(A_n\), the degree of the cycle polynomial \(\Gamma_n'\) is
\[
\deg(\Gamma_n')=\frac{n}{2}\binom{n-1}{\lfloor (n-1)/2\rfloor}
\]
[2509.06844].

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 \(A\).

## 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
\[
J=\langle 2x_1-y_1-y_1^{-1},\dots,2x_n-y_n-y_n^{-1}\rangle \subset K[y^{\pm1}],
\]
where \(K=\overline{\mathbb{C}(x)}\), and the quotient algebra
\[
\mathcal{A}=K[y^{\pm1}]/J,
\]
which has dimension \(2^n\). Multiplication by \(g\in K[y^{\pm1}]\) defines a linear map
\[
M_g:\mathcal{A}\to\mathcal{A}.
\]
For one variable,
\[
M_{y_1}= \begin{pmatrix} 0 & -1\\ 1 & 2x_1 \end{pmatrix},
\]
and in general
\[
M_{y_1}= \begin{pmatrix} 0 & -1\\ 1 & 2x_1 \end{pmatrix} \otimes I_2\otimes\cdots\otimes I_2.
\]
The analogous matrices \(M_{y_2},\dots,M_{y_n}\) commute, so every Laurent polynomial \(g\) yields a matrix \(M_g\in \mathbb{C}[x]^{2^n\times 2^n}\) [2509.06844].

If \(g_1,\dots,g_r\) generate the ideal of
\[
Y_{A,\beta}\cap(\mathbb{C}^*)^n,
\]
then the paper proves the rank-condition theorem
\[
{\mathcal L}_{A,b} = \left\{x\in\mathbb{C}^n: \operatorname{rank}(M_{g_1}\mid\cdots\mid M_{g_r})<2^n \right\}.
\]
Equivalently, \({\mathcal L}_{A,b}\) is cut out set-theoretically by the maximal minors of the block matrix
\[
(M_{g_1}\mid\cdots\mid M_{g_r}).
\]
If \(m_1,\dots,m_r\) generate \(\ker_{\mathbb{Z}}(A)\), one can take
\[
g_j=y^{m_j}-\beta^{m_j},
\]
or after clearing denominators,
\[
g_j=\beta^{w_j}y^{u_j}-\beta^{u_j}y^{w_j},\qquad m_j=u_j-w_j,\; u_j,w_j\in\mathbb{N}^n.
\]

In the hypersurface case \(\operatorname{rank}(A)=n-1\), the toric piece \(Y_{A,\beta}\) is a hypersurface and the Lissajous variety admits a determinantal representation:
\[
{\mathcal L}_{A,b}=\{x\in\mathbb{C}^n:\det(M_g)=0\}.
\]
If the prime ideal of \({\mathcal L}_{A,b}\) is \(\langle f\rangle\), then
\[
\det(M_g)=c\, f^{\deg(\pi_{A,b})} \qquad(c\in\mathbb{C}^*).
\]
For the 3-cycle example and \(b=\mathbf{0}\),
\[
\det(M_g)=16\,(x_1^2-2x_1x_2x_3+x_2^2+x_3^2-1)^2,
\]
so the set-theoretic equation is the square of Cayley’s cubic equation [2509.06844].

The use of Chebyshev polynomials is natural in related elimination problems. For classical Lissajous parametrizations
\[
x=\cos(mt+a),\qquad y=\cos(nt+b),
\]
Chebyshev polynomials satisfy
\[
T_n(\cos\theta)=\cos(n\theta),\qquad T_m(T_n(x))=T_{mn}(x),
\]
and eliminating the parameter yields
\[
F_{m,n,\delta}(x,y)=T_n^2(x)-2T_n(x)T_m(y)\cos\delta+T_m^2(y)-\sin^2\delta,
\qquad \delta=mb-na.
\]
When \(m,n\) are coprime positive integers and \(\sin\delta\neq0\), this polynomial is irreducible over both \(\mathbb{R}\) and \(\mathbb{C}\); when \(m\) and \(n\) are not coprime, it factors into components corresponding to lower-frequency Lissajous curves [2204.00083]. 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" [1711.00557], the affine real Chebyshev variety is
\[
\mathcal{C}^{({m})}_{\kappa} = \left\{ x\in [-1,1]^{\mathsf d} \;\middle|\; (-1)^{\kappa_1}T_{m_1}(x_1)=\cdots = (-1)^{\kappa_{\mathsf d}}T_{m_{\mathsf d}}(x_{\mathsf d}) \right\},
\]
and the paper proves that
\[
\mathcal{C}^{({m})}_{\kappa}
=
\bigcup_{\ell\in \mathfrak L^{(m^\sharp,m^\flat)}_\kappa}
\ell([0,2\pi)).
\]
The interpolation nodes are obtained by sampling these curves at
\[
t_l^{(m)}=\frac{l\pi}{\operatorname{lcm}[m]}.
\]
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 [1711.00557].

An earlier treatment, "Multivariate polynomial interpolation on Lissajous-Chebyshev nodes" [1511.04564], studies multivariate Lissajous curves
\[
\mathfrak l_{\alpha,u}(t) = \bigl(u_1\cos(q_1 t-\alpha_1),\ldots,u_{\mathsf d}\cos(q_{\mathsf d} t-\alpha_{\mathsf d})\bigr),
\]
classifies degenerate and non-degenerate cases, and identifies interpolation nodes with singular sets of Chebyshev varieties such as
\[
\mathcal C^{(n)} = \{x\in[-1,1]^{\mathsf d}: T_{n_1}(x_1)=\cdots=T_{n_{\mathsf d}}(x_{\mathsf d})\}.
\]
The resulting node sets support exact discrete orthogonality and unique interpolation in explicitly defined spaces of multivariate Chebyshev polynomials [1511.04564].

These interpolation papers do not define Lissajous varieties in the matrix-cosine sense of [2509.06844]. 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 \({\mathcal L}_{A,b}\) 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,
\[
\dot{\theta}_k=\omega_k+\sum_{j\in V_k}K_{kj}\sin(\theta_j-\theta_k), \qquad k=1,\dots,m.
\]
Using
\[
x_k=\sin\theta_k,\qquad y_k=\cos\theta_k,
\]
the steady-state equations become bilinear:
\[
f_k=\omega_k+\sum_{v_j\in V_k}K_{kj}(x_jy_k-x_ky_j).
\]
For a connected graph \(G\), with reduced incidence matrix \(A(G)\), the steady states correspond to the intersection
\[
x\in{\mathcal S}_A,\qquad Ax=\omega/K.
\]
Equivalently, with \(v_j=e^{i\theta_j}\), one solves Laurent equations
\[
\frac{\omega_j}{K} +\frac{a_{j1}}{2i}(v^{a_1}-v^{-a_1}) +\cdots+ \frac{a_{jn}}{2i}(v^{a_n}-v^{-a_n}) =0,\qquad j=1,\dots,d.
\]
The degree of \({\mathcal S}_A\) bounds the number of isolated equilibria; for the triangle graph \(C_3\), the paper exhibits a case with exactly six real steady states, matching \(\deg({\mathcal S}_A)=6\) [2509.06844].

The positive part of a Lissajous variety is characterized by a strictly convex optimization problem. For the sine case,
\[
P=\operatorname{Row}(A)\cap\left[-\frac{\pi}{2},\frac{\pi}{2}\right]^n, \qquad {\mathcal S}_A^+=\sin(\operatorname{int} P),
\]
and
\[
x^*\in {\mathcal S}_A^+\cap\{Ax=\omega/K\}
\]
if and only if \(x^*\) is the unique minimizer of
\[
\min\ \sum_{j=1}^n\Big(x_j\arcsin(x_j)+\sqrt{1-x_j^2}\Big)
\]
subject to
\[
Ax=\omega/K,\qquad x\in(-1,1)^n.
\]
For general \({\mathcal L}_{A,b}\), the positive part is
\[
{\mathcal L}_{A,b}^+=\cos\big(L_{A,b}\cap(0,\pi)^n\big),
\]
and
\[
x^*\in{\mathcal L}_{A,b}^+\cap\{Ax=\omega\}
\]
if and only if \(x^*\) is the unique minimizer of
\[
\min\ \frac{-\pi}{2}b^Tx-\sum_{j=1}^n\Big(x_j\arccos(x_j)-\sqrt{1-x_j^2}\Big)
\]
subject to
\[
Ax=\omega,\qquad x\in(-1,1)^n.
\]
If \(x^*\in{\mathcal L}_{A,b}^+\) is the minimizer, then the corresponding steady state \(\theta^*\) is linearly stable, and the Jacobian has the explicit form
\[
J_{\Phi_{A,b}(\theta)} = -A\,\operatorname{diag}\Big(\sin(a_1\cdot\theta-b_1\tfrac{\pi}{2}),\dots,\sin(a_n\cdot\theta-b_n\tfrac{\pi}{2})\Big)\,A^t,
\]
which is negative definite at the positive solution [2509.06844].

Bifurcation theory enters through the Lissajous discriminant. The incidence variety is
\[
W_{A,b}=\{(v,\omega)\in(\mathbb{C}^*)^d\times\mathbb{C}^d:\ A\psi_{A,b}(v)=\omega\},
\]
with toric Jacobian
\[
J_{A,b}(v)
=
\left(v_j\frac{\partial}{\partial v_j}(A\psi_{A,b}(v))_k\right)_{1\le j,k\le d}
=
\frac12\,A\,\operatorname{diag}\!\Big(\beta_1v^{a_1}-\beta_1^{-1}v^{-a_1},\dots,\beta_nv^{a_n}-\beta_n^{-1}v^{-a_n}\Big)\,A^t.
\]
The ramification locus is
\[
R_{A,b}=\{(v,\omega)\in W_{A,b}:\det J_{A,b}(v)=0\},
\]
and the Lissajous discriminant is
\[
\nabla_{A,b}=\overline{\operatorname{pr}_\omega(R_{A,b})}\subseteq\mathbb{C}^d.
\]
If it is a hypersurface, its defining polynomial is \(\Delta_{A,b}\), and
\[
\deg(\nabla_{A,b})\le (\deg \operatorname{pr}_\omega)^{-1}\cdot d\cdot d!\,\operatorname{vol}(P_A).
\]
For \(A=(1\ 2)\) and \(b=\mathbf{1}\),
\[
\Delta_{A,1}=256\omega^4-2367\omega^2+3375.
\]
For the triangle graph \(C_3\), the paper computes
\[
\deg(\nabla_{A,\mathbf{0}})=6,\qquad \deg(\nabla_{A,\mathbf{1}})=12.
\]
The discriminant marks parameter values where critical points collide; in the Kuramoto setting, it is the bifurcation locus where stability can change [2509.06844].

## 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
\[
x=\cos(mt+a),\qquad y=\cos(nt+b)
\]
produces Chebyshev-Lissajous polynomials. The irreducibility criterion established in [2204.00083] states that
\[
T_n^2(x)-2T_n(x)T_m(y)\cos\delta+T_m^2(y)-\sin^2\delta
\]
is irreducible over \(\mathbb{R}\) and \(\mathbb{C}\) precisely in the coprime case when \(\sin\delta\neq0\), while the non-coprime case decomposes into finitely many irreducible Lissajous curves. This makes the arithmetic of \(\gcd(m,n)\) 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" [1404.7064], rational \(k=m/n\) 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
\[
x(t)=\alpha \cos(\omega_0 t+\varphi_1),\qquad
y(t)=\kappa \alpha \sin(\omega_0 t+\varphi_1),\qquad
z(t)=\beta \cos(v_0 t+\varphi_2),
\]
and the paper introduces a coupling coefficient \(\eta\) with bifurcation equation \(\Delta=0\). When \(\eta=0\), the solution describes Lissajous orbits; when \(\eta\neq0\), it describes quasihalo orbits, and halo orbits arise as the special case \(\beta=0\) [2401.14697].

In magnetic dynamics, "Emergence of Lissajous trajectories in skyrmion oscillator" [2604.08073] studies a \(200 \times 200\ \mathrm{nm}^2\) Co/Pt multilayer nanostructure with a \(1\ \mathrm{nm}\)-thick Co free layer, simulated in Mumax\(^3\) via the LLG equation with Zhang–Li spin-transfer torque. For a drive
\[
\bm{j} = \big(A_1\sin(\omega_1 t),\, A_2\sin(\omega_2 t+\phi),\,0\big),
\]
the skyrmion center follows
\[
x(t)=-\frac{v_{0,1}^{(s)}}{\omega_1}\cos(\omega_1 t), \qquad
y(t)=-\frac{v_{0,2}^{(s)}}{\omega_2}\cos(\omega_2 t+\phi),
\]
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 [2604.08073].

An exploratory number-theoretic variant replaces single frequencies by prime-frequency finite Fourier sums,
\[
x(t)=\sum_{i=1}^N \frac{\sin(a_i t)}{a_i},\qquad
y(t)=\sum_{i=1}^N \frac{\cos(a_i t)}{a_i},
\]
where \(a_i\) are the first \(N\) 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 \(100\), \(1000\), and \(5000\) primes [2009.01687].

A further nonlinear extension appears in separable polynomial potentials
\[
V(x,y)=\tfrac{1}{2}\big(x^{2N}+A\,y^{2N}\big),
\]
where closed Lissajous-type trajectories arise only under energy-dependent nonlinear resonance conditions. For \(N=2\), the quartic case remains algebraic through Jacobi elliptic multiplication formulas; for \(N\ge3\), the orbit is naturally expressed through hyperelliptic phase constraints rather than a universal polynomial orbit equation [2606.25145].

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 [2509.06844], Lissajous varieties provide the affine algebraic version of that theme.

Source: https://www.emergentmind.com/topics/lissajous-varieties